trajectory_id string | problem_id string | domain string | turn int64 | accumulated_directives list | prompt string | target_output string | source_candidate_ids list | dataset_tier string | instruction string | input string | output string |
|---|---|---|---|---|---|---|---|---|---|---|---|
E4_solver:physgym_0010:turn0_sol1 | physgym_0010 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
The region in space where $x > 0$ and $y > 0$ is a vacuum, while... | The infinite conductor and the point at infinity are equipotential, $U{=}0$, with the boundary condition that the tangential electric field $E_{\tau}{=}0$. By the uniqueness theorem of electrostatics, the induced charges on the conductor can be equivalently replaced by image charges to determine the contribution to the... | [
"physgym_0010_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
The region in space where $x > 0$ and $y > 0$ is a vacuum, while... | The infinite conductor and the point at infinity are equipotential, $U{=}0$, with the boundary condition that the tangential electric field $E_{\tau}{=}0$. By the uniqueness theorem of electrostatics, the induced charges on the conductor can be equivalently replaced by image charges to determine the contribution to the... | |
E4_solver:physgym_0043:turn0_sol1 | physgym_0043 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Two dielectric materials with base areas both equal to $\pmb{A}$... | Let the magnitudes of the electric field intensity in the two media be $E_{1}, E_{2}$ respectively, and according to the relationship between the electric potentials, we have
$$
E_{1}h_{1} + E_{2}h_{2} = U_{0}
$$
The magnitudes of the current density in the two media are
$$
\begin{aligned}
j_1 &= \sigma_1 E_1, \\
j_... | [
"physgym_0043_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Two dielectric materials with base areas both equal to $\pmb{A}$... | Let the magnitudes of the electric field intensity in the two media be $E_{1}, E_{2}$ respectively, and according to the relationship between the electric potentials, we have
$$
E_{1}h_{1} + E_{2}h_{2} = U_{0}
$$
The magnitudes of the current density in the two media are
$$
\begin{aligned}
j_1 &= \sigma_1 E_1, \\
j_... | |
E4_solver:physgym_0250:turn0_sol1 | physgym_0250 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
There is now an electrolyte with thickness $L$ in the $z$ direct... | Given the potential difference $u$, it can be seen:
$$
\varphi_{+}=u/2,\varphi_{-}=-u/2,\lambda=\frac{2\pi(\varepsilon_{1}+\varepsilon_{2})\varphi_{+}}{2\xi_{+}}=\frac{\pi(\varepsilon_{1}+\varepsilon_{2})u}{\operatorname{arccosh}(\mathrm{D/R})}
$$
Select a surface encapsulating the cylindrical surface and examine Gau... | [
"physgym_0250_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
There is now an electrolyte with thickness $L$ in the $z$ direct... | Given the potential difference $u$, it can be seen:
$$
\varphi_{+}=u/2,\varphi_{-}=-u/2,\lambda=\frac{2\pi(\varepsilon_{1}+\varepsilon_{2})\varphi_{+}}{2\xi_{+}}=\frac{\pi(\varepsilon_{1}+\varepsilon_{2})u}{\operatorname{arccosh}(\mathrm{D/R})}
$$
Select a surface encapsulating the cylindrical surface and examine Gau... | |
E4_solver:physgym_0636:turn0_sol1 | physgym_0636 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Establish a Cartesian coordinate system Oxyz, with a hypothetica... | Consider the case of a single circular loop, establishing a spherical coordinate system $(r, \alpha)$ with the loop axis as the polar axis. Then: $$ V = {\frac{1}{4\pi\varepsilon_{0}}}\int_{0}^{2\pi}{\frac{Q d\theta}{2\pi}}{\frac{1}{\sqrt{r^{2}+R^{2}-2R r \sin\alpha \cos\Theta}}}={\frac{Q}{4\pi\varepsilon_{0}}}\left({\... | [
"physgym_0636_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Establish a Cartesian coordinate system Oxyz, with a hypothetica... | Consider the case of a single circular loop, establishing a spherical coordinate system $(r, \alpha)$ with the loop axis as the polar axis. Then: $$ V = {\frac{1}{4\pi\varepsilon_{0}}}\int_{0}^{2\pi}{\frac{Q d\theta}{2\pi}}{\frac{1}{\sqrt{r^{2}+R^{2}-2R r \sin\alpha \cos\Theta}}}={\frac{Q}{4\pi\varepsilon_{0}}}\left({... | |
E4_solver:physgym_0653:turn0_sol1 | physgym_0653 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A massless insulating string of length $a$ is fixed at one end, ... | If the ring is a superconducting ring, when it moves in a magnetic field, due to the properties of the superconductor, the magnetic flux in the ring remains unchanged, i.e.: $$ L I + B S = B \cdot \pi r^2 $$ Thus, the current in the ring is: $$ \begin{array}{l}{I = \displaystyle\frac{B}{L}(\pi r^{2} - S)}\ = \displayst... | [
"physgym_0653_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A massless insulating string of length $a$ is fixed at one end, ... | If the ring is a superconducting ring, when it moves in a magnetic field, due to the properties of the superconductor, the magnetic flux in the ring remains unchanged, i.e.: $$ L I + B S = B \cdot \pi r^2 $$ Thus, the current in the ring is: $$ \begin{array}{l}{I = \displaystyle\frac{B}{L}(\pi r^{2} - S)}\ = \displa... | |
E4_solver:physgym_0039:turn0_sol1 | physgym_0039 | MECHANICS | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A rectangular wooden block of height $H$ and density $\rho_{1}$ ... | Taking the top at equilibrium as the origin, establish the coordinate system as shown.
Let the position of the top of the block be $y$. The net force on the block is $-\rho_{1}H S g + \rho_{2}(H-h_{0}-y)S g = -\rho_{2}S g y$. The net force on the block is a linear restoring force, and its motion is simple harmonic mot... | [
"physgym_0039_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A rectangular wooden block of height $H$ and density $\rho_{1}$ ... | Taking the top at equilibrium as the origin, establish the coordinate system as shown.
Let the position of the top of the block be $y$. The net force on the block is $-\rho_{1}H S g + \rho_{2}(H-h_{0}-y)S g = -\rho_{2}S g y$. The net force on the block is a linear restoring force, and its motion is simple harmonic mot... | |
E4_solver:physgym_0041:turn0_sol1 | physgym_0041 | MECHANICS | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
The rocket ascends vertically from the Earth's surface with a co... | At the moment of takeoff, $N_1=(1+k_1)mg_0$, where $m$ is the mass of the tester.
When ascending to a height $h$, the gravitational acceleration changes to $g=g_0(\dfrac{R}{R+h})^2$
The pressure at this time is $N_2=mg_0(\dfrac{R}{R+h})^2+k_1mg_0$
Since $N_2=k_2N_1$, solving the equations together gives
$h=R(\dfrac{1}{... | [
"physgym_0041_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
The rocket ascends vertically from the Earth's surface with a co... | At the moment of takeoff, $N_1=(1+k_1)mg_0$, where $m$ is the mass of the tester.
When ascending to a height $h$, the gravitational acceleration changes to $g=g_0(\dfrac{R}{R+h})^2$
The pressure at this time is $N_2=mg_0(\dfrac{R}{R+h})^2+k_1mg_0$
Since $N_2=k_2N_1$, solving the equations together gives
$h=R(\dfrac{1}{... | |
E4_solver:physgym_0367:turn0_sol1 | physgym_0367 | MECHANICS | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
There is a smooth elliptical track, and the equation of the trac... | It is evident that at this point, the small object is in a stable equilibrium position at the two vertices of the major axis of the elliptical orbit. When the small object experiences a slight deviation along the orbit from the stable equilibrium position, as shown in the figure below, the distance from the small objec... | [
"physgym_0367_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
There is a smooth elliptical track, and the equation of the trac... | It is evident that at this point, the small object is in a stable equilibrium position at the two vertices of the major axis of the elliptical orbit. When the small object experiences a slight deviation along the orbit from the stable equilibrium position, as shown in the figure below, the distance from the small objec... | |
E4_solver:physgym_0401:turn0_sol1 | physgym_0401 | MECHANICS | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A certain planet has a smooth and elastic surface, perfect for p... | The two components of the initial throw speed are:
$$
v_{r}=k v_{0}\sin\theta\quad,\quad v_{\theta}=k v_{0}\cos\theta
$$
Based on the symmetry of the starting and landing points with respect to the major axis of the elliptical orbit, and combining the definition of the restitution coefficient under the conditions of ... | [
"physgym_0401_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A certain planet has a smooth and elastic surface, perfect for p... | The two components of the initial throw speed are:
$$
v_{r}=k v_{0}\sin\theta\quad,\quad v_{\theta}=k v_{0}\cos\theta
$$
Based on the symmetry of the starting and landing points with respect to the major axis of the elliptical orbit, and combining the definition of the restitution coefficient under the conditions of ... | |
E4_solver:physgym_0654:turn0_sol1 | physgym_0654 | MECHANICS | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
The forty-second batch of Trisolarian immigrants arrived on Eart... | There are two reference frames; let's set up the agreements first. The reference frame of the Earth observer is called the $\Sigma$ frame, and the reference frame of the spaceship is called the $\Sigma$ frame. We agree that: $$\beta_{1}=\frac{v_{1}}{c},\gamma_{1}=\frac{1}{\sqrt{1-\beta_{1}^{2}}},\beta_{2}=\frac{v_{2}}{... | [
"physgym_0654_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
The forty-second batch of Trisolarian immigrants arrived on Eart... | There are two reference frames; let's set up the agreements first. The reference frame of the Earth observer is called the $\Sigma$ frame, and the reference frame of the spaceship is called the $\Sigma$ frame. We agree that: $$\beta_{1}=\frac{v_{1}}{c},\gamma_{1}=\frac{1}{\sqrt{1-\beta_{1}^{2}}},\beta_{2}=\frac{v_{2}}... | |
E4_solver:physgym_0674:turn0_sol1 | physgym_0674 | MECHANICS | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A coin is placed at rest on the edge of a smooth tabletop, with ... | The moment of inertia of the coin about the axis is \[J=\frac14mr^2\] According to the impulse and impulse-momentum theorem, about the center of mass we have \[I+I_N=m v_0, Ir-I_Nr=J\omega_0\] There is also the velocity relation \[v_0=\omega_0 r\] Therefore, we have \[v_0=\frac{8I}{5m}, \omega_0=\frac{8I}{5mr}\] After ... | [
"physgym_0674_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A coin is placed at rest on the edge of a smooth tabletop, with ... | The moment of inertia of the coin about the axis is \[J=\frac14mr^2\] According to the impulse and impulse-momentum theorem, about the center of mass we have \[I+I_N=m v_0, Ir-I_Nr=J\omega_0\] There is also the velocity relation \[v_0=\omega_0 r\] Therefore, we have \[v_0=\frac{8I}{5m}, \omega_0=\frac{8I}{5mr}\] After... | |
E4_solver:physgym_0766:turn0_sol1 | physgym_0766 | MECHANICS | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A small ring $A$ with mass $m$ is placed on a smooth horizontal ... | 【Solution】Let the angle between the rope and the rod be $\theta$, the velocity of the small ring A be $\boldsymbol{v}_{A}$, and the velocity of the small ball B relative to the small ring be $v'$. Then the velocity of the small ball B relative to the bottom surface $\boldsymbol{v}_{B}$ is given by the relative motion f... | [
"physgym_0766_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A small ring $A$ with mass $m$ is placed on a smooth horizontal ... | 【Solution】Let the angle between the rope and the rod be $\theta$, the velocity of the small ring A be $\boldsymbol{v}_{A}$, and the velocity of the small ball B relative to the small ring be $v'$. Then the velocity of the small ball B relative to the bottom surface $\boldsymbol{v}_{B}$ is given by the relative motion f... | |
E4_solver:physgym_0284:turn0_sol1 | physgym_0284 | OPTICS | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Fiber-optic communication has greatly advanced our information t... | Considering that the optical signal should propagate without loss, the light transmission in the optical fiber should involve total internal reflection at the interface. Assume the incident electric field is represented by ${\widetilde{E}}_{i}$, and the reflected electric field by $\widetilde{E}_{r}$; we have $\widetil... | [
"physgym_0284_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Fiber-optic communication has greatly advanced our information t... | Considering that the optical signal should propagate without loss, the light transmission in the optical fiber should involve total internal reflection at the interface. Assume the incident electric field is represented by ${\widetilde{E}}_{i}$, and the reflected electric field by $\widetilde{E}_{r}$; we have $\widetil... | |
E4_solver:lit_em_0001:turn0_sol1 | lit_em_0001 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A hydrogen atom consists of an electron and a proton. For our
pr... | Electricity is an inverse square force, so if
r is smaller by a factor of 206.77, the force will
be greater by a factor of 206 .772 = 42754.
ANSWER: 42754 | [
"lit_em_0001_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A hydrogen atom consists of an electron and a proton. For our
pr... | Electricity is an inverse square force, so if
r is smaller by a factor of 206.77, the force will
be greater by a factor of 206 .772 = 42754.
ANSWER: 42754 | |
E4_solver:lit_em_0003:turn0_sol1 | lit_em_0003 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
In Fullerton, California, the earth’s magnetic field is in the
d... | By/Bx = tan 59 deg = 1.7
ANSWER: 1.7 | [
"lit_em_0003_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
In Fullerton, California, the earth’s magnetic field is in the
d... | By/Bx = tan 59 deg = 1.7
ANSWER: 1.7 | |
E4_solver:lit_em_0004:turn0_sol1 | lit_em_0004 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
The gravitational field g of a typical asteroid is so weak that ... | The total force acting on the particle
is mg +qE, so by Newton’s second law its acceleration is a =g + (q/m)E. This makes sense
because when q = 0, we get a =g. Applying the
constant-acceleration equation v2f = v20 + 2a∆x
gives h =−v2/2(g +qE/m). In the case where
gravity is dominant, the negative value ofg gives
h> 0,... | [
"lit_em_0004_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
The gravitational field g of a typical asteroid is so weak that ... | The total force acting on the particle
is mg +qE, so by Newton’s second law its acceleration is a =g + (q/m)E. This makes sense
because when q = 0, we get a =g. Applying the
constant-acceleration equation v2f = v20 + 2a∆x
gives h =−v2/2(g +qE/m). In the case where
gravity is dominant, the negative value ofg gives
h> 0,... | |
E4_solver:lit_em_0005:turn0_sol1 | lit_em_0005 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Consider the following four potentials, which exist in some
regi... | In each case, the electric field is minus the
derivative of the potential.
Potentials 1 and 2 give the same field,E =−a.
These are uniform fields. The constant a is the
strength of the field, and b is just an arbitrary
constant with no physical significance.
Potentials 3 and 4 both give E = a/x2. This
looks like the fi... | [
"lit_em_0005_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Consider the following four potentials, which exist in some
regi... | In each case, the electric field is minus the
derivative of the potential.
Potentials 1 and 2 give the same field,E =−a.
These are uniform fields. The constant a is the
strength of the field, and b is just an arbitrary
constant with no physical significance.
Potentials 3 and 4 both give E = a/x2. This
looks like the fi... | |
E4_solver:lit_em_0006:turn0_sol1 | lit_em_0006 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A particle with a charge of 1.0 C and a mass of 1.0 kg is
observ... | The electric force vector is
FE =qE
= (1 N)ˆy .
The magnitude of the magnetic force is
|FB| =|qv× B|
=q|v× B|
=q|v||B| sinθvB
= 2 N ,
and its direction is determined by the right hand
rule to be along the z axis, so
FB = (2 N)ˆz .
The total force on the particle is
F = (1 N)ˆy + (2 N)ˆz .
ANSWER: (1 N)ˆy + (2 N)ˆz | [
"lit_em_0006_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A particle with a charge of 1.0 C and a mass of 1.0 kg is
observ... | The electric force vector is
FE =qE
= (1 N)ˆy .
The magnitude of the magnetic force is
|FB| =|qv× B|
=q|v× B|
=q|v||B| sinθvB
= 2 N ,
and its direction is determined by the right hand
rule to be along the z axis, so
FB = (2 N)ˆz .
The total force on the particle is
F = (1 N)ˆy + (2 N)ˆz .
ANSWER: (1 N)ˆy + (2 N)ˆz | |
E4_solver:lit_em_0008:turn0_sol1 | lit_em_0008 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Lightning discharges a cloud during an electrical storm. Suppose... | q =
I dt =
bt dt = 12bt2 +qo
Hereqo is a constant of integration, representing
the cloud’s charge at time zero.
ANSWER: the cloud’s charge at time zero. | [
"lit_em_0008_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Lightning discharges a cloud during an electrical storm. Suppose... | q =
I dt =
bt dt = 12bt2 +qo
Hereqo is a constant of integration, representing
the cloud’s charge at time zero.
ANSWER: the cloud’s charge at time zero. | |
E4_solver:lit_em_0010:turn0_sol1 | lit_em_0010 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Magnetic dipole 1 has its dipole moment oriented along the
z axi... | The field of dipole 1, experienced by
dipole 2, is along the z axis with component
Bz = −(km1/c2)r−3. This gives dipole 2 an
energy U =−Bzm2 = (k/c2)m1m2r−3. They
are stable in the orientation that minimizes U,
which would be the antiparallel one in which the
signs of m1 and m2 are opposite. If they were
parallel rathe... | [
"lit_em_0010_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Magnetic dipole 1 has its dipole moment oriented along the
z axi... | The field of dipole 1, experienced by
dipole 2, is along the z axis with component
Bz = −(km1/c2)r−3. This gives dipole 2 an
energy U =−Bzm2 = (k/c2)m1m2r−3. They
are stable in the orientation that minimizes U,
which would be the antiparallel one in which the
signs of m1 and m2 are opposite. If they were
parallel rathe... | |
E4_solver:lit_em_0011:turn0_sol1 | lit_em_0011 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Suppose that an FM radio tuner for the US commercial broadcast b... | Solving ω≈ 1/
LC for C, we get C =
1/Lω2 = 1/4π2L2f2. Looking at the nearest
handy FM radio dial, the range of frequencies
is about 88-108 MHz, so the capacitor needs to
cover the range from about 2.2 to 3.3 pF.
ANSWER: cover the range from about 2.2 to 3.3 pF. | [
"lit_em_0011_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Suppose that an FM radio tuner for the US commercial broadcast b... | Solving ω≈ 1/
LC for C, we get C =
1/Lω2 = 1/4π2L2f2. Looking at the nearest
handy FM radio dial, the range of frequencies
is about 88-108 MHz, so the capacitor needs to
cover the range from about 2.2 to 3.3 pF.
ANSWER: cover the range from about 2.2 to 3.3 pF. | |
E4_solver:lit_em_0012:turn0_sol1 | lit_em_0012 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Each of the following circuit elements or combination of element... | The impedance of an inductor is iωL,
so the argument is 90◦.
ANSWER: so the argument is 90◦. | [
"lit_em_0012_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Each of the following circuit elements or combination of element... | The impedance of an inductor is iωL,
so the argument is 90◦.
ANSWER: so the argument is 90◦. | |
E4_solver:lit_em_0013:turn0_sol1 | lit_em_0013 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Suppose you are holding your hands in front of you, 10 cm
apart.... | Your body is mostly water, and
water is mosly oxygen by weight, so for the
purposes of this estimate, let’s pretend that
your body just consists of oxygen atoms. The
mass of an oxygen atom is about 16 times
the mass of a proton, which is 1 .7× 10−27
kg. If your hand has a mass of about 0.2
kg, then the number of oxygen... | [
"lit_em_0013_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Suppose you are holding your hands in front of you, 10 cm
apart.... | Your body is mostly water, and
water is mosly oxygen by weight, so for the
purposes of this estimate, let’s pretend that
your body just consists of oxygen atoms. The
mass of an oxygen atom is about 16 times
the mass of a proton, which is 1 .7× 10−27
kg. If your hand has a mass of about 0.2
kg, then the number of oxygen... | |
E4_solver:lit_em_0014:turn0_sol1 | lit_em_0014 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
In our by-now-familiar neuron, the voltage difference between th... | E = V/d = 70 mV/6.0 nm = 1 .2× 107
V/m
ANSWER: 1 .2× 107 V/m | [
"lit_em_0014_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
In our by-now-familiar neuron, the voltage difference between th... | E = V/d = 70 mV/6.0 nm = 1 .2× 107
V/m
ANSWER: 1 .2× 107 V/m | |
E4_solver:lit_em_0016:turn0_sol1 | lit_em_0016 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Show that the equation VL = L ∆I/∆t has the right units. | By definition, one henry is 1 J /A2.
The quantity L dI/ dt therefore has units of
( J
A2
)( C/s
s
)
=
( J
(C/s)2
)( C/s
s
)
= JC
= V
ANSWER: V | [
"lit_em_0016_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Show that the equation VL = L ∆I/∆t has the right units. | By definition, one henry is 1 J /A2.
The quantity L dI/ dt therefore has units of
( J
A2
)( C/s
s
)
=
( J
(C/s)2
)( C/s
s
)
= JC
= V
ANSWER: V | |
E4_solver:lit_em_0020:turn0_sol1 | lit_em_0020 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
The voltage difference between the ends of a AAA battery is
1.5 ... | When a function has a constant derivative, we don’t really need calculus and can find
∆.../ ∆... , which is the same as d.../... Ignoring signs, the field is
E = (ignored−)∆φ∆x
= 1.5 V4.2× 10−2 m
= 36 V/m.
ANSWER: 36 V/m | [
"lit_em_0020_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
The voltage difference between the ends of a AAA battery is
1.5 ... | When a function has a constant derivative, we don’t really need calculus and can find
∆.../ ∆... , which is the same as d.../... Ignoring signs, the field is
E = (ignored−)∆φ∆x
= 1.5 V4.2× 10−2 m
= 36 V/m.
ANSWER: 36 V/m | |
E4_solver:lit_em_0021:turn0_sol1 | lit_em_0021 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Suppose a charged particle is moving through a region of
space i... | The electric force is either in the same direction as the electric field or in the opposite direction, depending on whether the particles are
positively or negatively charged. Since the magnetic force is perpendicular to both v and B
and the question states that the electric field is
perpendicular to both v and B, the ... | [
"lit_em_0021_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Suppose a charged particle is moving through a region of
space i... | The electric force is either in the same direction as the electric field or in the opposite direction, depending on whether the particles are
positively or negatively charged. Since the magnetic force is perpendicular to both v and B
and the question states that the electric field is
perpendicular to both v and B, the ... | |
E4_solver:lit_em_0022:turn0_sol1 | lit_em_0022 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
At a particular point in time and space, an electromagnetic
plan... | Since it’s a plane wave, the direction of
the magnetic field is perpendicular to the electric field and the direction of propagation, so it
must be in the ±y direction. What gives E× B
in the positive z direction is if the magnetic field
is in the +y direction.
The Poynting vector is S = (c2/4πk)E× B.
Because E and B a... | [
"lit_em_0022_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
At a particular point in time and space, an electromagnetic
plan... | Since it’s a plane wave, the direction of
the magnetic field is perpendicular to the electric field and the direction of propagation, so it
must be in the ±y direction. What gives E× B
in the positive z direction is if the magnetic field
is in the +y direction.
The Poynting vector is S = (c2/4πk)E× B.
Because E and B a... | |
E4_solver:lit_em_0023:turn0_sol1 | lit_em_0023 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
In AM (amplitude-modulated) radio, an audio signal f (t) is
mult... | I = dq/ dt =b sinωt +ω(a +bt) cosωt
ANSWER: b sinωt +ω(a +bt) cosωt | [
"lit_em_0023_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
In AM (amplitude-modulated) radio, an audio signal f (t) is
mult... | I = dq/ dt =b sinωt +ω(a +bt) cosωt
ANSWER: b sinωt +ω(a +bt) cosωt | |
E4_solver:lit_em_0025:turn0_sol1 | lit_em_0025 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Deuterium is an isotope of hydrogen in which the nucleus
has one... | The g factor is ( e/m)(|L|/|m|) = 1 .17.
This makes sense. If the nucleus had been made
of two pointlike particles, each with charge e,
then we should have gotten g = 2. Since one
particle is neutral, we would expect to get get
359
half the magnetic moment, cutting the g factor in half, so g≈ 1, which is roughly what w... | [
"lit_em_0025_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Deuterium is an isotope of hydrogen in which the nucleus
has one... | The g factor is ( e/m)(|L|/|m|) = 1 .17.
This makes sense. If the nucleus had been made
of two pointlike particles, each with charge e,
then we should have gotten g = 2. Since one
particle is neutral, we would expect to get get
359
half the magnetic moment, cutting the g factor in half, so g≈ 1, which is roughly what w... | |
E4_solver:lit_em_0026:turn0_sol1 | lit_em_0026 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Find an expression for the energy stored in a capacitor in
terms... | Combining U = q2/2C with q = CV
givesU = (1/2)CV 2.
ANSWER: (1/2)CV 2 | [
"lit_em_0026_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Find an expression for the energy stored in a capacitor in
terms... | Combining U = q2/2C with q = CV
givesU = (1/2)CV 2.
ANSWER: (1/2)CV 2 | |
E4_solver:lit_em_0028:turn0_sol1 | lit_em_0028 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Suppose you are holding your hands in front of you, 10 cm
apart.... | Coulomb’s law gives F =
k(1026e)(1026e)/(0.1 m) 2 = 2 × 1026 N, or
call it 10 26 N.
ANSWER: 2 × 1026 N, or call it 10 26 N | [
"lit_em_0028_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Suppose you are holding your hands in front of you, 10 cm
apart.... | Coulomb’s law gives F =
k(1026e)(1026e)/(0.1 m) 2 = 2 × 1026 N, or
call it 10 26 N.
ANSWER: 2 × 1026 N, or call it 10 26 N | |
E4_solver:lit_em_0030:turn0_sol1 | lit_em_0030 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
If you put four times more current through a solenoid, how
many ... | Four times the current makes four times
the magnetic field, which makes 4 2 = 16 times
the energy.
ANSWER: 16 times the energy | [
"lit_em_0030_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
If you put four times more current through a solenoid, how
many ... | Four times the current makes four times
the magnetic field, which makes 4 2 = 16 times
the energy.
ANSWER: 16 times the energy | |
E4_solver:lit_em_0034:turn0_sol1 | lit_em_0034 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
In an inkjet printer, a series of droplets of ink, each with
mas... | Let thex axis be in the initial direction
of motion andy the direction of the electric field.
Newton’s laws tell us that thex andy motion are
independent, so the x velocity is constant, and
the time taken to pass through the capacitor is
t = w/u. The final velocity in the y diection is
v = at = (q/m)Et = qEw/mu. The de... | [
"lit_em_0034_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
In an inkjet printer, a series of droplets of ink, each with
mas... | Let thex axis be in the initial direction
of motion andy the direction of the electric field.
Newton’s laws tell us that thex andy motion are
independent, so the x velocity is constant, and
the time taken to pass through the capacitor is
t = w/u. The final velocity in the y diection is
v = at = (q/m)Et = qEw/mu. The de... | |
E4_solver:lit_em_0035:turn0_sol1 | lit_em_0035 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A carbon dioxide molecule is structured like O-C-O, with
all thr... | This is the opposite of the previous problem. To find the field given the voltage, we take
the derivative:
E =− ddr kr3
= 3kr4
ANSWER: 3kr4 | [
"lit_em_0035_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A carbon dioxide molecule is structured like O-C-O, with
all thr... | This is the opposite of the previous problem. To find the field given the voltage, we take
the derivative:
E =− ddr kr3
= 3kr4
ANSWER: 3kr4 | |
E4_solver:lit_em_0036:turn0_sol1 | lit_em_0036 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A charged particle is released from rest. We see it start to
mov... | There must be either E or B or both,
since some force caused the particle to accelerate. It can’t be a pure magnetic field, because
F = qv× B is zero for v = 0, so the particle
would not have accelerated from rest. It can’t
be a pure electric field, because a uniform electric field would just have caused the particle t... | [
"lit_em_0036_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A charged particle is released from rest. We see it start to
mov... | There must be either E or B or both,
since some force caused the particle to accelerate. It can’t be a pure magnetic field, because
F = qv× B is zero for v = 0, so the particle
would not have accelerated from rest. It can’t
be a pure electric field, because a uniform electric field would just have caused the particle t... | |
E4_solver:lit_em_0038:turn0_sol1 | lit_em_0038 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
If a typical light bulb draws about 900 mA from a 110 V
househol... | R = ∆V/I = (110 V)/(0.9 A) = 120 Ω
ANSWER: 120 Ω | [
"lit_em_0038_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
If a typical light bulb draws about 900 mA from a 110 V
househol... | R = ∆V/I = (110 V)/(0.9 A) = 120 Ω
ANSWER: 120 Ω | |
E4_solver:lit_em_0040:turn0_sol1 | lit_em_0040 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A Helmholtz coil is defined as a pair of identical circular coil... | The field at any point on the axis can be
found by adding the contributions to the field
from the two loops. The center of the entire apparatus does not, however, coincide with the center of either loop: in each case it is out of that
plane by a distance h/2. By symmetry, both
these fields are along the axis, so we can... | [
"lit_em_0040_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A Helmholtz coil is defined as a pair of identical circular coil... | The field at any point on the axis can be
found by adding the contributions to the field
from the two loops. The center of the entire apparatus does not, however, coincide with the center of either loop: in each case it is out of that
plane by a distance h/2. By symmetry, both
these fields are along the axis, so we can... | |
E4_solver:lit_em_0041:turn0_sol1 | lit_em_0041 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Suppose that a parallel-plate capacitor is constructed so that
t... | The capacitance is proportional to 1 /h.
(a) At fixed voltage, we have U = (1/2)CV 2, so
U∝ 1/h. (b) At fixed charge, U = (1/2C)q2, so
U∝h.
ANSWER: (1/2C)q2, so U∝h | [
"lit_em_0041_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Suppose that a parallel-plate capacitor is constructed so that
t... | The capacitance is proportional to 1 /h.
(a) At fixed voltage, we have U = (1/2)CV 2, so
U∝ 1/h. (b) At fixed charge, U = (1/2C)q2, so
U∝h.
ANSWER: (1/2C)q2, so U∝h | |
E4_solver:lit_em_0044:turn0_sol1 | lit_em_0044 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Show that the magnitude of the electric field produced by a
simp... | Let the dipole consist of a charge q at
a distance 𝓁/2 to the right of the origin, and a
charge−q at 𝓁/2 on the other side. The field at
a point lying at a distance r to the right of the
origin is then
E = kq(r−𝓁/2)2− kq(r +𝓁/2)2
= kqr2
[ 1
(1−𝓁/2r)2− 1(1 +𝓁/2r)2
]
≈ kqr2
[(
1 +𝓁r)−(1− 𝓁r)]
= 2kq𝓁r3
= 2kDr3
AN... | [
"lit_em_0044_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Show that the magnitude of the electric field produced by a
simp... | Let the dipole consist of a charge q at
a distance 𝓁/2 to the right of the origin, and a
charge−q at 𝓁/2 on the other side. The field at
a point lying at a distance r to the right of the
origin is then
E = kq(r−𝓁/2)2− kq(r +𝓁/2)2
= kqr2
[ 1
(1−𝓁/2r)2− 1(1 +𝓁/2r)2
]
≈ kqr2
[(
1 +𝓁r)−(1− 𝓁r)]
= 2kq𝓁r3
= 2kDr3
AN... | |
E4_solver:lit_em_0046:turn0_sol1 | lit_em_0046 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Find the inductance of two identical inductors in parallel. | The definition of inductance is U =
LI2/2. If we think of the pair of inductors as
91
one big inductor, then applying this definition,
I has to be the total current that goes in. This
gets split in half, so the energy stored in a single
inductor is Uone = Lone(I/2)2/2, and the total
energy is U = 2Uone =LoneI2/4. Equat... | [
"lit_em_0046_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Find the inductance of two identical inductors in parallel. | The definition of inductance is U =
LI2/2. If we think of the pair of inductors as
91
one big inductor, then applying this definition,
I has to be the total current that goes in. This
gets split in half, so the energy stored in a single
inductor is Uone = Lone(I/2)2/2, and the total
energy is U = 2Uone =LoneI2/4. Equat... | |
E4_solver:lit_em_0048:turn0_sol1 | lit_em_0048 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Evaluate your result, in radians, for m = 2.0 × 10−10 kg, u =
20... | Converting the given data to SI and plugging
in, the result is θ = 0.98 radians, or a little more
than a 45 degree angle.
ANSWER: 0.98 radians, or a little more than a 45 degree angle | [
"lit_em_0048_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Evaluate your result, in radians, for m = 2.0 × 10−10 kg, u =
20... | Converting the given data to SI and plugging
in, the result is θ = 0.98 radians, or a little more
than a 45 degree angle.
ANSWER: 0.98 radians, or a little more than a 45 degree angle | |
E4_solver:lit_em_0049:turn0_sol1 | lit_em_0049 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A vacuum tube, in its simplest conceptual form, is a parallelpla... | E =−dφ/dx =−(4/3)cx1/3.
ANSWER: −(4/3)cx1/3 | [
"lit_em_0049_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A vacuum tube, in its simplest conceptual form, is a parallelpla... | E =−dφ/dx =−(4/3)cx1/3.
ANSWER: −(4/3)cx1/3 | |
E4_solver:lit_em_0050:turn0_sol1 | lit_em_0050 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A charged particle is in a region of space in which there is a
u... | F =qv×B = 0, so the particle remains
at rest.
ANSWER: 0, so the particle remains at rest | [
"lit_em_0050_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A charged particle is in a region of space in which there is a
u... | F =qv×B = 0, so the particle remains
at rest.
ANSWER: 0, so the particle remains at rest | |
E4_solver:lit_em_0052:turn0_sol1 | lit_em_0052 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A resistor has a voltage difference ∆ V across it, causing a
cur... | P =I∆V =I(IR) =I2R
ANSWER: I2R | [
"lit_em_0052_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A resistor has a voltage difference ∆ V across it, causing a
cur... | P =I∆V =I(IR) =I2R
ANSWER: I2R | |
E4_solver:lit_em_0055:turn0_sol1 | lit_em_0055 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
In example 6, p. 101, we analyzed a spherical capacitor.
Buildin... | Doing this using the relation U =
q2/2C would be fairly complicated, since the energy density is not constant. We would have to
find the energy stored in a spherical shell betweenr andr +dr, and then integrate to findU.
It’s much easier to do this using C =q/V , since
we already know how to relate q andV from the
analy... | [
"lit_em_0055_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
In example 6, p. 101, we analyzed a spherical capacitor.
Buildin... | Doing this using the relation U =
q2/2C would be fairly complicated, since the energy density is not constant. We would have to
find the energy stored in a spherical shell betweenr andr +dr, and then integrate to findU.
It’s much easier to do this using C =q/V , since
we already know how to relate q andV from the
analy... | |
E4_solver:lit_em_0057:turn0_sol1 | lit_em_0057 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Suppose you are holding your hands in front of you, 10 cm
apart.... | A force of 10 26 N is enough to pulverize
a planet, so the cancellation is obviously very
nearly perfect; even a very tiny percentage
difference in charge would lead to a huge
force. If the fractional difference was ϵ, then
the total charge on each hand would be ϵq,
where q = 10 26e as estimated above. The
total force ... | [
"lit_em_0057_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Suppose you are holding your hands in front of you, 10 cm
apart.... | A force of 10 26 N is enough to pulverize
a planet, so the cancellation is obviously very
nearly perfect; even a very tiny percentage
difference in charge would lead to a huge
force. If the fractional difference was ϵ, then
the total charge on each hand would be ϵq,
where q = 10 26e as estimated above. The
total force ... | |
E4_solver:lit_em_0058:turn0_sol1 | lit_em_0058 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Consider the electric field created by a uniform ring of total
c... | Imagine breaking the ring down into
many tiny pieces, like a necklace. These pieces
all contribute to the field at the point we’re talking about, and their contributions add like vectors. The components of these vectors parallel
to the plane of the ring all cancel out, but the
components along the axis don’t. Each cont... | [
"lit_em_0058_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Consider the electric field created by a uniform ring of total
c... | Imagine breaking the ring down into
many tiny pieces, like a necklace. These pieces
all contribute to the field at the point we’re talking about, and their contributions add like vectors. The components of these vectors parallel
to the plane of the ring all cancel out, but the
components along the axis don’t. Each cont... | |
E4_solver:lit_em_0059:turn0_sol1 | lit_em_0059 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Consider two solenoids, one of which is smaller so that it
can b... | The difference in energy is
c2
8πk
[
(B +b)2−B2−b2]xa = c2
4πk·Bbxa .
ANSWER: c2 4πk·Bbxa | [
"lit_em_0059_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Consider two solenoids, one of which is smaller so that it
can b... | The difference in energy is
c2
8πk
[
(B +b)2−B2−b2]xa = c2
4πk·Bbxa .
ANSWER: c2 4πk·Bbxa | |
E4_solver:lit_em_0062:turn0_sol1 | lit_em_0062 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
At time t = 0, a positively charged particle is placed,
at rest,... | F = ma and F = qE, and since the
field is uniform, the force and acceleration will be
constant. For motion with constant acceleration,
we havev =at, and eliminating the unknown F
givesv =qEt/m.
ANSWER: qEt/m | [
"lit_em_0062_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
At time t = 0, a positively charged particle is placed,
at rest,... | F = ma and F = qE, and since the
field is uniform, the force and acceleration will be
constant. For motion with constant acceleration,
we havev =at, and eliminating the unknown F
givesv =qEt/m.
ANSWER: qEt/m | |
E4_solver:lit_em_0063:turn0_sol1 | lit_em_0063 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
In example 8, p. 59, we found that the field of a uniform,
strai... | V =−
E dr
=−2kλ lnr + const.
This is a little bit of a strange-looking expression,
because normally we can’t evaluate a transcendental function on an input that has units. However, we can get away with this in the case of a
log, because ln(a/b) = lna−lnb, so if we wanted
to make the units not look so wrong, we could
wr... | [
"lit_em_0063_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
In example 8, p. 59, we found that the field of a uniform,
strai... | V =−
E dr
=−2kλ lnr + const.
This is a little bit of a strange-looking expression,
because normally we can’t evaluate a transcendental function on an input that has units. However, we can get away with this in the case of a
log, because ln(a/b) = lna−lnb, so if we wanted
to make the units not look so wrong, we could
wr... | |
E4_solver:lit_em_0065:turn0_sol1 | lit_em_0065 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Express the power dissipated by a resistor in terms of R
and ∆V ... | P =I∆V = (∆V/R)∆V = ∆V 2/R
ANSWER: ∆V 2/R | [
"lit_em_0065_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Express the power dissipated by a resistor in terms of R
and ∆V ... | P =I∆V = (∆V/R)∆V = ∆V 2/R
ANSWER: ∆V 2/R | |
E4_solver:lit_em_0068:turn0_sol1 | lit_em_0068 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
In the limit where the gap between a and b is very small, show
t... | In the notation of the parallel-plate capacitor
example, we have A = 4πa2 (or 5πb2, it doesn’t
matter becausea andb are nearly the same) and
b = a +h. Our expression for the capacitance
involves the quantity 1 /a− 1/b, which can be
rewritten as 1/a−1/(a+h) =h/a(a+h)≈h/a2.
We then have C≈ a2/kh = A/4πkh, as for the
para... | [
"lit_em_0068_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
In the limit where the gap between a and b is very small, show
t... | In the notation of the parallel-plate capacitor
example, we have A = 4πa2 (or 5πb2, it doesn’t
matter becausea andb are nearly the same) and
b = a +h. Our expression for the capacitance
involves the quantity 1 /a− 1/b, which can be
rewritten as 1/a−1/(a+h) =h/a(a+h)≈h/a2.
We then have C≈ a2/kh = A/4πkh, as for the
para... | |
E4_solver:lit_em_0070:turn0_sol1 | lit_em_0070 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
The Earth and Moon are bound together by gravity. If, instead, t... | Letq stand for the absolute values of the
charges, which are equal to each other. We can
find the force either by using Newton’s law of
gravity or by using the kinematical equationa =
v2/r for the moon’s acceleration. Choosing the
first approach, we have
kq2
r2 =Gm1m2r2 ,
which we solve for q:
q =
Gm1m2
k
= 5.70× 1013 ... | [
"lit_em_0070_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
The Earth and Moon are bound together by gravity. If, instead, t... | Letq stand for the absolute values of the
charges, which are equal to each other. We can
find the force either by using Newton’s law of
gravity or by using the kinematical equationa =
v2/r for the moon’s acceleration. Choosing the
first approach, we have
kq2
r2 =Gm1m2r2 ,
which we solve for q:
q =
Gm1m2
k
= 5.70× 1013 ... | |
E4_solver:lit_em_0071:turn0_sol1 | lit_em_0071 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Consider the electric field created by a uniform ring of total
c... | For a = 0, the field is zero, which makes
sense, because we expect perfect cancellation at
the center of the ring. For very large values of a,
b becomes negligible compared to a, so the field
is kqa(a2)−3/2 =kq/a2, which is just what we’d
expect for any charge q at a distance a. In other
words, the field doesn’t depend... | [
"lit_em_0071_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Consider the electric field created by a uniform ring of total
c... | For a = 0, the field is zero, which makes
sense, because we expect perfect cancellation at
the center of the ring. For very large values of a,
b becomes negligible compared to a, so the field
is kqa(a2)−3/2 =kq/a2, which is just what we’d
expect for any charge q at a distance a. In other
words, the field doesn’t depend... | |
E4_solver:lit_em_0073:turn0_sol1 | lit_em_0073 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Find the capacitance of two identical capacitors in series. | The capacitors have a total of four
plates, and the two in the middle, which are connected, are isolated, so their total charge will always be zero. Therefore, the two middle plates
must have the same amount of charge but with
opposite signs. Thus the pattern of charges on
the four plates must be +q,−q, +q,−q. If we’re... | [
"lit_em_0073_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Find the capacitance of two identical capacitors in series. | The capacitors have a total of four
plates, and the two in the middle, which are connected, are isolated, so their total charge will always be zero. Therefore, the two middle plates
must have the same amount of charge but with
opposite signs. Thus the pattern of charges on
the four plates must be +q,−q, +q,−q. If we’re... | |
E4_solver:lit_em_0074:turn0_sol1 | lit_em_0074 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Find the angle between the following two vectors:
ˆx + 2ˆy + 3ˆz... | a· b =|a||b| cosθab
θab = cos−1
( a· b
|a||b|
)
= 12.9◦
ANSWER: 12.9◦ | [
"lit_em_0074_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Find the angle between the following two vectors:
ˆx + 2ˆy + 3ˆz... | a· b =|a||b| cosθab
θab = cos−1
( a· b
|a||b|
)
= 12.9◦
ANSWER: 12.9◦ | |
E4_solver:lit_em_0076:turn0_sol1 | lit_em_0076 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
An electric potential is given by φ = ar2, where r is the
distan... | E =−∇φ. In terms of components,
we have
Ex =−2ax,
and similar equations for y and z.
ANSWER: −2ax, and similar equations for y and z | [
"lit_em_0076_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
An electric potential is given by φ = ar2, where r is the
distan... | E =−∇φ. In terms of components,
we have
Ex =−2ax,
and similar equations for y and z.
ANSWER: −2ax, and similar equations for y and z | |
E4_solver:lit_em_0078:turn0_sol1 | lit_em_0078 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Electrical receptacles in your home are mostly 110 V, but circui... | The power is proportional to ∆V 2, so cutting
∆V in half reduces the power by a factor of 4,
to 50 W.
ANSWER: to 50 W. | [
"lit_em_0078_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Electrical receptacles in your home are mostly 110 V, but circui... | The power is proportional to ∆V 2, so cutting
∆V in half reduces the power by a factor of 4,
to 50 W.
ANSWER: to 50 W. | |
E4_solver:lit_em_0082:turn0_sol1 | lit_em_0082 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Find the parallel impedance of a 37 kΩ resistor and a 1.0
nF cap... | Z = (1/ZR + 1/ZC)−1
= [1/R + 1/(−i/ωC)]−1
= (2.70 + 6.28i)−1× 105 Ω
The magnitude of a complex number’s inverse is
the inverse of its magnitude, so
|Z| = 1√
2.702 + 6.282× 105 Ω
= 1.5× 104 Ω
(rounding the final result to two sig figs). The
argument of a complex number’s inverse is minus
the argument of the original num... | [
"lit_em_0082_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Find the parallel impedance of a 37 kΩ resistor and a 1.0
nF cap... | Z = (1/ZR + 1/ZC)−1
= [1/R + 1/(−i/ωC)]−1
= (2.70 + 6.28i)−1× 105 Ω
The magnitude of a complex number’s inverse is
the inverse of its magnitude, so
|Z| = 1√
2.702 + 6.282× 105 Ω
= 1.5× 104 Ω
(rounding the final result to two sig figs). The
argument of a complex number’s inverse is minus
the argument of the original num... | |
E4_solver:lit_em_0083:turn0_sol1 | lit_em_0083 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Each bulb has a resistance of one ohm. How much power is
drawn f... | Like the preceding problem, this one
can’t be solved by breaking the circuit down into
parallel and series parts. Unlike that one, there is
no easy way to solve it by symmetry. We simply
have to use the loop rule and the junction rule
to write down enough equations to solve for all
the unknowns. Let’s label the bulbs A... | [
"lit_em_0083_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Each bulb has a resistance of one ohm. How much power is
drawn f... | Like the preceding problem, this one
can’t be solved by breaking the circuit down into
parallel and series parts. Unlike that one, there is
no easy way to solve it by symmetry. We simply
have to use the loop rule and the junction rule
to write down enough equations to solve for all
the unknowns. Let’s label the bulbs A... | |
E4_solver:lit_em_0087:turn0_sol1 | lit_em_0087 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Find a vector that is perpendicular to both of the following
two... | The easiest way to find such a vector is to
take the cross product.
(A× B)x =AyBz−ByAz =−3
(A× B)y =AzBx−BzAx = 6
(A× B)z =AxBy−BxAy =−3
335
So our vector is −3ˆx + 6ˆy− 3ˆz. We could also
use any other vector lying along the same line,
i.e., any scalar multiple of this one. I checked my
result by computing its dot pro... | [
"lit_em_0087_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Find a vector that is perpendicular to both of the following
two... | The easiest way to find such a vector is to
take the cross product.
(A× B)x =AyBz−ByAz =−3
(A× B)y =AzBx−BzAx = 6
(A× B)z =AxBy−BxAy =−3
335
So our vector is −3ˆx + 6ˆy− 3ˆz. We could also
use any other vector lying along the same line,
i.e., any scalar multiple of this one. I checked my
result by computing its dot pro... | |
E4_solver:lit_em_0088:turn0_sol1 | lit_em_0088 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Suppose that at some instant in time, a wire extending from
x = ... | q =
∫ x=∞
x=0
dq
=
∫ ∞
0
dq
dx dx
=
∫ ∞
0
ae−bx dx
= ab
ANSWER: ab | [
"lit_em_0088_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Suppose that at some instant in time, a wire extending from
x = ... | q =
∫ x=∞
x=0
dq
=
∫ ∞
0
dq
dx dx
=
∫ ∞
0
ae−bx dx
= ab
ANSWER: ab | |
E4_solver:lit_em_0089:turn0_sol1 | lit_em_0089 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
By comparing with the result of example 7, p. 55, show that this... | The result from the example was that for
a sphere of uniformly distributed charge, E =
(4πkρ/3)r, which is the same form as our field,
with a = 2πkρ/3.
ANSWER: 2πkρ/3 | [
"lit_em_0089_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
By comparing with the result of example 7, p. 55, show that this... | The result from the example was that for
a sphere of uniformly distributed charge, E =
(4πkρ/3)r, which is the same form as our field,
with a = 2πkρ/3.
ANSWER: 2πkρ/3 | |
E4_solver:lit_em_0090:turn0_sol1 | lit_em_0090 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
The following data give the results of two experiments in which
... | The second data-point isn’t very surprising, because multiplying charge by −2 and the
velocity by−1 multiplies the magnetic force by
(−2)(−1) = 2. The only thing the second datapoint really tells us is that the field is purely
magnetic. If the F, v, and B vectors had to
be perpendicular, there would only be one possibi... | [
"lit_em_0090_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
The following data give the results of two experiments in which
... | The second data-point isn’t very surprising, because multiplying charge by −2 and the
velocity by−1 multiplies the magnetic force by
(−2)(−1) = 2. The only thing the second datapoint really tells us is that the field is purely
magnetic. If the F, v, and B vectors had to
be perpendicular, there would only be one possibi... | |
E4_solver:lit_em_0091:turn0_sol1 | lit_em_0091 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
You take an LP record out of its sleeve, and it acquires a
stati... | In one revolution, the amount of
charge passing by a certain angular position
is 1 nC. The time required for one revolution
is 1/(33.3 min−1), so I = ∆ q/∆t = 3 .3×
10−8 C/min = 600 pA.
ANSWER: 600 pA | [
"lit_em_0091_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
You take an LP record out of its sleeve, and it acquires a
stati... | In one revolution, the amount of
charge passing by a certain angular position
is 1 nC. The time required for one revolution
is 1/(33.3 min−1), so I = ∆ q/∆t = 3 .3×
10−8 C/min = 600 pA.
ANSWER: 600 pA | |
E4_solver:lit_em_0093:turn0_sol1 | lit_em_0093 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A square current loop with sides of length 2 h carries current
I... | As suggested in the hint, we recycle the
setup from the long, straight wire, which resulted
in
B = kIc2R∫ π0 sinθ dθ.
This can be applied to one side of the square
simply by changing the limits of integration. We
then have
Bsquare = 4· kIc2R∫ 3π/4π/4 sinθ dθ
= 4
2 kIc2R.
The final result for the comparison with the cir... | [
"lit_em_0093_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A square current loop with sides of length 2 h carries current
I... | As suggested in the hint, we recycle the
setup from the long, straight wire, which resulted
in
B = kIc2R∫ π0 sinθ dθ.
This can be applied to one side of the square
simply by changing the limits of integration. We
then have
Bsquare = 4· kIc2R∫ 3π/4π/4 sinθ dθ
= 4
2 kIc2R.
The final result for the comparison with the cir... | |
E4_solver:lit_em_0094:turn0_sol1 | lit_em_0094 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Resistors have a standard color code on them, but capacitors
oft... | We need the trace on the left to show us
where “zero” is on this exponential curve. The
experimenter has adjusted the scope so that this
is on the line above the bottom line. We need
the trace on the right to show us enough time
resolution to get decent precision. In the trace
on the right, the voltage starts at 8.75 d... | [
"lit_em_0094_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Resistors have a standard color code on them, but capacitors
oft... | We need the trace on the left to show us
where “zero” is on this exponential curve. The
experimenter has adjusted the scope so that this
is on the line above the bottom line. We need
the trace on the right to show us enough time
resolution to get decent precision. In the trace
on the right, the voltage starts at 8.75 d... | |
E4_solver:lit_em_0096:turn0_sol1 | lit_em_0096 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Given that the field of a dipole is proportional to D/r3 (proble... | The voltage is the integral of the field
with respect to position. We can use an indefinite integral and not worry about the constant
of integration, since physically it just relates to
the arbitrary choice of a reference point.
V =−
E dr
=−
∫ d
r3 dr
= d2r2
ANSWER: d2r2 | [
"lit_em_0096_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Given that the field of a dipole is proportional to D/r3 (proble... | The voltage is the integral of the field
with respect to position. We can use an indefinite integral and not worry about the constant
of integration, since physically it just relates to
the arbitrary choice of a reference point.
V =−
E dr
=−
∫ d
r3 dr
= d2r2
ANSWER: d2r2 | |
E4_solver:lit_em_0097:turn0_sol1 | lit_em_0097 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Prove that in an electromagnetic plane wave, half the energy
is ... | At a given point in space, the magnitudes of the electric field and magnetic field are
related by E = cB. The energy density of the
magnetic field is c28πkB2, and the energy density of the electric field is 18πkE2 = 18πk (cB)2 =
c2
8πkB2..
ANSWER: 8πkB2.. | [
"lit_em_0097_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Prove that in an electromagnetic plane wave, half the energy
is ... | At a given point in space, the magnitudes of the electric field and magnetic field are
related by E = cB. The energy density of the
magnetic field is c28πkB2, and the energy density of the electric field is 18πkE2 = 18πk (cB)2 =
c2
8πkB2..
ANSWER: 8πkB2.. | |
E4_solver:lit_em_0099:turn0_sol1 | lit_em_0099 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
The earth’s magnetic field measured inside a certain classroom
i... | Applying the Pythagorean theorem
gives 30.0 µT.
ANSWER: gives 30.0 µT. | [
"lit_em_0099_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
The earth’s magnetic field measured inside a certain classroom
i... | Applying the Pythagorean theorem
gives 30.0 µT.
ANSWER: gives 30.0 µT. | |
E4_solver:lit_em_0101:turn0_sol1 | lit_em_0101 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Suppose that the potential had instead been
φ = a(x2 + y2 + z2) ... | It would have had no effect. Mathematically,
the gradient operator ∇ is a kind of derivative,
and derivatives are linear and have the property
that the derivative of a constant is zero. Therefore adding a constant onto an expression never
has any effect on its derivative. Physically, the
potentialφ is a measure of pote... | [
"lit_em_0101_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Suppose that the potential had instead been
φ = a(x2 + y2 + z2) ... | It would have had no effect. Mathematically,
the gradient operator ∇ is a kind of derivative,
and derivatives are linear and have the property
that the derivative of a constant is zero. Therefore adding a constant onto an expression never
has any effect on its derivative. Physically, the
potentialφ is a measure of pote... | |
E4_solver:lit_em_0102:turn0_sol1 | lit_em_0102 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
The following data give the results of two experiments in which
... | It can’t be a purely electric field, because
then the velocity would have no effect on the
345
force. It can’t be purely magnetic, either, because tripling the velocity didn’t simply triple
the force, as it would if the only force was
FB =qv× B.
ANSWER: qv× B | [
"lit_em_0102_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
The following data give the results of two experiments in which
... | It can’t be a purely electric field, because
then the velocity would have no effect on the
345
force. It can’t be purely magnetic, either, because tripling the velocity didn’t simply triple
the force, as it would if the only force was
FB =qv× B.
ANSWER: qv× B | |
E4_solver:lit_em_0103:turn0_sol1 | lit_em_0103 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Although the planetary model of the atom can be made to work
wit... | The idea is the same. The time for one orbit
is ∆t = 2πr/v, so I =e/∆t =ve/2πr = 1 mA.
ANSWER: 1 mA | [
"lit_em_0103_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Although the planetary model of the atom can be made to work
wit... | The idea is the same. The time for one orbit
is ∆t = 2πr/v, so I =e/∆t =ve/2πr = 1 mA.
ANSWER: 1 mA | |
E4_solver:lit_em_0105:turn0_sol1 | lit_em_0105 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A regular polygon with n sides can be inscribed within a circle
... | This is a simple generalization of the
preceding problem involving a square. The
contribution from each side is proportional to
(1/h) cosθ, where θ is the angle between a side
and a line from the center to an adjacent vertex.
Let the perimeter be𝓁, so that cosθ = (𝓁/2n)/R.
Then the field is proportional to 𝓁/hR =𝓁/... | [
"lit_em_0105_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A regular polygon with n sides can be inscribed within a circle
... | This is a simple generalization of the
preceding problem involving a square. The
contribution from each side is proportional to
(1/h) cosθ, where θ is the angle between a side
and a line from the center to an adjacent vertex.
Let the perimeter be𝓁, so that cosθ = (𝓁/2n)/R.
Then the field is proportional to 𝓁/hR =𝓁/... | |
E4_solver:lit_em_0106:turn0_sol1 | lit_em_0106 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Carry out the algebra in example 2, p. 315, to show that
the ind... | We have
B = 4πkc−2IN/𝓁 (5.1)
U = (1/2)LI2 (5.2)
U = (c2/8πk)B2v and (5.3)
v =A𝓁. (5.4)
Combining equations 1, 3, and 4, we have
U = c2
8πk
(4πk
c2 · IN𝓁 )2A𝓁 = 2πkc2 I2N2A/𝓁
Solving equation 2 for L then gives
L = (4πk/c2)N2A/𝓁.
ANSWER: (4πk/c2)N2A/𝓁 | [
"lit_em_0106_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Carry out the algebra in example 2, p. 315, to show that
the ind... | We have
B = 4πkc−2IN/𝓁 (5.1)
U = (1/2)LI2 (5.2)
U = (c2/8πk)B2v and (5.3)
v =A𝓁. (5.4)
Combining equations 1, 3, and 4, we have
U = c2
8πk
(4πk
c2 · IN𝓁 )2A𝓁 = 2πkc2 I2N2A/𝓁
Solving equation 2 for L then gives
L = (4πk/c2)N2A/𝓁.
ANSWER: (4πk/c2)N2A/𝓁 | |
E4_solver:lit_em_0107:turn0_sol1 | lit_em_0107 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A series LRC circuit consists of a 1.000 Ω resistor, a 1.000 F
c... | The resonant frequency is ω ≈ 1/
LC =
1.000 Hz. This is the frequency at which the
magnitude of the impedance is the smallest, so
the largest current will flow for a given voltage.
ANSWER: the largest current will flow for a given voltage. | [
"lit_em_0107_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A series LRC circuit consists of a 1.000 Ω resistor, a 1.000 F
c... | The resonant frequency is ω ≈ 1/
LC =
1.000 Hz. This is the frequency at which the
magnitude of the impedance is the smallest, so
the largest current will flow for a given voltage.
ANSWER: the largest current will flow for a given voltage. | |
E4_solver:lit_em_0108:turn0_sol1 | lit_em_0108 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A proton is in a region in which the electric field is given by
... | The voltage is
V =−
E dx
=−ax− 14bx4 ,
where the constant of integration is irrelevant because the electrical energy is only meaningful up
to an arbitrary additive constant. Conservation
of energy gives
K1 +U1 =K2 +U2
0 = 12mv2 +eV (x2)
v =±
−2eV (x2)m
=±
2e(ax2 + 14bx42)
m .
Note that for certain values of a, b, and x... | [
"lit_em_0108_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A proton is in a region in which the electric field is given by
... | The voltage is
V =−
E dx
=−ax− 14bx4 ,
where the constant of integration is irrelevant because the electrical energy is only meaningful up
to an arbitrary additive constant. Conservation
of energy gives
K1 +U1 =K2 +U2
0 = 12mv2 +eV (x2)
v =±
−2eV (x2)m
=±
2e(ax2 + 14bx42)
m .
Note that for certain values of a, b, and x... | |
E4_solver:lit_em_0114:turn0_sol1 | lit_em_0114 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
We have referred to resistors dissipating heat, i.e., we have
as... | To get negative values ofP , you’d have to
have opposite signs forI and ∆V . But you can’t
control the signs of I and ∆V independently —
switching the wires reverses both. The current
flows in response to the electric force, whose direction is related to the direction of the voltage
drop. Another way of looking at it i... | [
"lit_em_0114_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
We have referred to resistors dissipating heat, i.e., we have
as... | To get negative values ofP , you’d have to
have opposite signs forI and ∆V . But you can’t
control the signs of I and ∆V independently —
switching the wires reverses both. The current
flows in response to the electric force, whose direction is related to the direction of the voltage
drop. Another way of looking at it i... | |
E4_solver:lit_em_0116:turn0_sol1 | lit_em_0116 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A particular solenoid used in my lab classes has roughly N =
300... | Plugging in gives an inductance of 0.9 H.
Given the non-ideal nature of this solenoid, we
expect this to be only a rough estimate. The
actual measured inductance is about 20% lower.
ANSWER: actual measured inductance is about 20% lower. | [
"lit_em_0116_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A particular solenoid used in my lab classes has roughly N =
300... | Plugging in gives an inductance of 0.9 H.
Given the non-ideal nature of this solenoid, we
expect this to be only a rough estimate. The
actual measured inductance is about 20% lower.
ANSWER: actual measured inductance is about 20% lower. | |
E4_solver:lit_em_0117:turn0_sol1 | lit_em_0117 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
At a frequency of 53.1 kHz, a certain series LR circuit has an
i... | In parallel, it will be the inverse
impedances that add. The total inverse
impedance is 1 /Z = (0.4− 0.3i) kΩ−1. The
real part of this can be contributed by the inverse resistance of a resistor with a resistance
of 2.5 kΩ. For the imaginary part, we need a
component whose inverse impedance 1/Z will be
a negative imagin... | [
"lit_em_0117_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
At a frequency of 53.1 kHz, a certain series LR circuit has an
i... | In parallel, it will be the inverse
impedances that add. The total inverse
impedance is 1 /Z = (0.4− 0.3i) kΩ−1. The
real part of this can be contributed by the inverse resistance of a resistor with a resistance
of 2.5 kΩ. For the imaginary part, we need a
component whose inverse impedance 1/Z will be
a negative imagin... | |
E4_solver:lit_em_0118:turn0_sol1 | lit_em_0118 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
To do this problem, you need to understand how to do
volume inte... | Consider a ring extending from r to
r+dr around the wire. Its volume is proportional
to its circumference and to dr, so dV = (... )r dr,
where the ... stands for stuff that we don’t care
about because it won’t affect whether the result is infinite. The magnetic field varies as 1 /r,
and to integrate the energy, we have... | [
"lit_em_0118_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
To do this problem, you need to understand how to do
volume inte... | Consider a ring extending from r to
r+dr around the wire. Its volume is proportional
to its circumference and to dr, so dV = (... )r dr,
where the ... stands for stuff that we don’t care
about because it won’t affect whether the result is infinite. The magnetic field varies as 1 /r,
and to integrate the energy, we have... | |
E4_solver:lit_em_0123:turn0_sol1 | lit_em_0123 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Hybrid and electric cars have been gradually gaining market
shar... | If the power is the same, then P =
I∆V tells us that tripling the voltage has to go
along with cutting down the current to one third
of what it was.
ANSWER: of what it was. | [
"lit_em_0123_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Hybrid and electric cars have been gradually gaining market
shar... | If the power is the same, then P =
I∆V tells us that tripling the voltage has to go
along with cutting down the current to one third
of what it was.
ANSWER: of what it was. | |
E4_solver:lit_em_0125:turn0_sol1 | lit_em_0125 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A capacitance C and an inductance L are connected in
parallel. S... | We have Z−1 = Z−1L +Z−1C , and if
we want no current to flow, then Z should be
infinite, so the right-hand side of this equation
needs to vanish. One term is positive imaginary
and the other negative imaginary, so this cancellation will happen whenever |ZL| =|ZC|. This
results in 1/ωC =ωL, so ω = 1/
LC. This relation l... | [
"lit_em_0125_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A capacitance C and an inductance L are connected in
parallel. S... | We have Z−1 = Z−1L +Z−1C , and if
we want no current to flow, then Z should be
infinite, so the right-hand side of this equation
needs to vanish. One term is positive imaginary
and the other negative imaginary, so this cancellation will happen whenever |ZL| =|ZC|. This
results in 1/ωC =ωL, so ω = 1/
LC. This relation l... | |
E4_solver:lit_em_0127:turn0_sol1 | lit_em_0127 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
The strongest magnetic field at the earth’s surface is 66 µT,
of... | The energy density of the magnetic
field is proportional to the square of the
field, so the ratio of the energy densities is
[(66 µT)/(23 µT)]2 = 8.2.
ANSWER: 8.2 | [
"lit_em_0127_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
The strongest magnetic field at the earth’s surface is 66 µT,
of... | The energy density of the magnetic
field is proportional to the square of the
field, so the ratio of the energy densities is
[(66 µT)/(23 µT)]2 = 8.2.
ANSWER: 8.2 | |
E4_solver:lit_em_0129:turn0_sol1 | lit_em_0129 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Three charges, each of strength Q (Q > 0) form a fixed
equilater... | The basic idea is to find the voltage V
at the center of the triangle (defining the voltage infinitely far away to be zero, as usual). We
then use conservation of energy to find the minimum initial kinetic energy, which corresponds to
some minimum initial speed. A little trig shows
that the distance r from the center t... | [
"lit_em_0129_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Three charges, each of strength Q (Q > 0) form a fixed
equilater... | The basic idea is to find the voltage V
at the center of the triangle (defining the voltage infinitely far away to be zero, as usual). We
then use conservation of energy to find the minimum initial kinetic energy, which corresponds to
some minimum initial speed. A little trig shows
that the distance r from the center t... | |
E4_solver:lit_em_0130:turn0_sol1 | lit_em_0130 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A charged particle of mass m and charge q moves in a circle
due ... | a = Fm
v2
r = qvB sin 90◦m
v
r = qmB
r = mq · vB
ANSWER: mq · vB | [
"lit_em_0130_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A charged particle of mass m and charge q moves in a circle
due ... | a = Fm
v2
r = qvB sin 90◦m
v
r = qmB
r = mq · vB
ANSWER: mq · vB | |
E4_solver:lit_em_0131:turn0_sol1 | lit_em_0131 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Hybrid and electric cars have been gradually gaining market
shar... | By P = I∆V , the only way to get more
power with the same voltage would be to increase the current. This would require fatter
wires, which would be more expensive.
ANSWER: wires, which would be more expensive. | [
"lit_em_0131_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Hybrid and electric cars have been gradually gaining market
shar... | By P = I∆V , the only way to get more
power with the same voltage would be to increase the current. This would require fatter
wires, which would be more expensive.
ANSWER: wires, which would be more expensive. | |
E4_solver:lit_em_0134:turn0_sol1 | lit_em_0134 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
To do this problem, you need to understand how to do
volume inte... | A spherical shell extending from r to r + dr
has a volume proportional to its surface area and
to dr, so dV = (... )r2 dr. The field depends on
1/r2, so the energy density goes like 1 /r4, and
the thing we’re integrating has a 1/r2. Integrating this gives 1 /r, which diverges at zero, but
not at infinity.
ANSWER: not a... | [
"lit_em_0134_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
To do this problem, you need to understand how to do
volume inte... | A spherical shell extending from r to r + dr
has a volume proportional to its surface area and
to dr, so dV = (... )r2 dr. The field depends on
1/r2, so the energy density goes like 1 /r4, and
the thing we’re integrating has a 1/r2. Integrating this gives 1 /r, which diverges at zero, but
not at infinity.
ANSWER: not a... | |
E4_solver:lit_em_0135:turn0_sol1 | lit_em_0135 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
The electric and magnetic fields have different units, so we
can... | Setting (c2/8πk)B2 = (1/8πk)E2 gives
E/B =c.
ANSWER: c | [
"lit_em_0135_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
The electric and magnetic fields have different units, so we
can... | Setting (c2/8πk)B2 = (1/8πk)E2 gives
E/B =c.
ANSWER: c | |
E4_solver:lit_em_0137:turn0_sol1 | lit_em_0137 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A coaxial cable consists of a cylindrical inner conductor
with r... | This is similar to the example of the
spherical capacitor in section 3.6. The electric
field in this geometry was found in section 2.8 to
be proportional to 1/r, by applying Gauss’s law
to a cylindrical Gaussian surface: as the cylinder’s radius grows, its area grows like r to the
first power, so the field must fall of... | [
"lit_em_0137_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A coaxial cable consists of a cylindrical inner conductor
with r... | This is similar to the example of the
spherical capacitor in section 3.6. The electric
field in this geometry was found in section 2.8 to
be proportional to 1/r, by applying Gauss’s law
to a cylindrical Gaussian surface: as the cylinder’s radius grows, its area grows like r to the
first power, so the field must fall of... | |
E4_solver:lit_em_0138:turn0_sol1 | lit_em_0138 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A charged particle of mass m and charge q moves in a circle
due ... | v =ωr
ω = vr
= qBm
ANSWER: qBm | [
"lit_em_0138_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A charged particle of mass m and charge q moves in a circle
due ... | v =ωr
ω = vr
= qBm
ANSWER: qBm | |
E4_solver:lit_em_0139:turn0_sol1 | lit_em_0139 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Wire is sold in a series of standard diameters, called “gauges.”... | Resistance is inversely proportional to the
cross-sectional area. The area depends on the
square of the diameter, so the ratio of resistances
is (1.2)2 = 1.4.
ANSWER: 1.4 | [
"lit_em_0139_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Wire is sold in a series of standard diameters, called “gauges.”... | Resistance is inversely proportional to the
cross-sectional area. The area depends on the
square of the diameter, so the ratio of resistances
is (1.2)2 = 1.4.
ANSWER: 1.4 | |
E4_solver:lit_em_0141:turn0_sol1 | lit_em_0141 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
In an electrical storm, an electric field builds up in the space... | The energy per unit volume is
E
V = 18πk|E|2 .
(E is energy, E is electric field.) At the moment
when the lightning strikes, the field equals Ec.
The volume of the air in which the field exists is
L2h.
E = 18πkE2cL2h
ANSWER: 18πkE2cL2h | [
"lit_em_0141_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
In an electrical storm, an electric field builds up in the space... | The energy per unit volume is
E
V = 18πk|E|2 .
(E is energy, E is electric field.) At the moment
when the lightning strikes, the field equals Ec.
The volume of the air in which the field exists is
L2h.
E = 18πkE2cL2h
ANSWER: 18πkE2cL2h | |
E4_solver:lit_em_0143:turn0_sol1 | lit_em_0143 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Show that your answer makes sense when a = b and when b < a. | When b = a, we have ln( b/a) = 0, and
E =∞. This makes sense because the only way
to have a finite voltage change over an infinitesimal distance is if the field is infinite. When
b < a, the log is negative, which flips the sign
of the field, and that makes sense, because it’s
like we’ve switched the electrodes around.
... | [
"lit_em_0143_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Show that your answer makes sense when a = b and when b < a. | When b = a, we have ln( b/a) = 0, and
E =∞. This makes sense because the only way
to have a finite voltage change over an infinitesimal distance is if the field is infinite. When
b < a, the log is negative, which flips the sign
of the field, and that makes sense, because it’s
like we’ve switched the electrodes around.
... | |
E4_solver:lit_em_0144:turn0_sol1 | lit_em_0144 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A line of charge with density λ is moving at velocity v in
the d... | In time t, the length that passes by
a fixed point is 𝓁 = vt, and the charge in this
portion of the line is q =λ𝓁 =λvt. The current
is I =q/t =λv.
ANSWER: λv | [
"lit_em_0144_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A line of charge with density λ is moving at velocity v in
the d... | In time t, the length that passes by
a fixed point is 𝓁 = vt, and the charge in this
portion of the line is q =λ𝓁 =λvt. The current
is I =q/t =λv.
ANSWER: λv | |
E4_solver:lit_em_0145:turn0_sol1 | lit_em_0145 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
It’s fairly common in electrical circuits for additional, undesi... | Let the voltage of the battery be V .
In the original circuit, we haveP =IV =V 2/R,
353
while in the new one P′ =V 2/(R +r). By forming the ratio of these two quantities, we can eliminate the voltage, and P/P′ = 1 +r/R. The result is r = (P/P′− 1)R.
ANSWER: (P/P′− 1)R | [
"lit_em_0145_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
It’s fairly common in electrical circuits for additional, undesi... | Let the voltage of the battery be V .
In the original circuit, we haveP =IV =V 2/R,
353
while in the new one P′ =V 2/(R +r). By forming the ratio of these two quantities, we can eliminate the voltage, and P/P′ = 1 +r/R. The result is r = (P/P′− 1)R.
ANSWER: (P/P′− 1)R | |
E4_solver:lit_em_0146:turn0_sol1 | lit_em_0146 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
In a series LC circuit driven by a DC voltage ( ω = 0),
compare ... | A DC current can’t flow through a capacitor, so the steady-state behavior of the circuit is that the capacitor is charged up, and there
is no current flowing through the inductor. That
means there’s an electric field in the capacitor,
but no magnetic field in the inductor. All of the
energy is stored in the capacitor, ... | [
"lit_em_0146_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
In a series LC circuit driven by a DC voltage ( ω = 0),
compare ... | A DC current can’t flow through a capacitor, so the steady-state behavior of the circuit is that the capacitor is charged up, and there
is no current flowing through the inductor. That
means there’s an electric field in the capacitor,
but no magnetic field in the inductor. All of the
energy is stored in the capacitor, ... | |
E4_solver:lit_em_0147:turn0_sol1 | lit_em_0147 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Throughout this problem, we will use the standard notation
ρ (Gr... | The energy depends on the square of
the field, so we have
α = (E1 + E2)2
E21
= (E1 + E2)· E1 + E2)E21
= E21 + 2E1· E2 +E22
E21
= 2 + 2 cosθ
θ = cos−1
(α
2− 1
)
.
ANSWER: . | [
"lit_em_0147_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Throughout this problem, we will use the standard notation
ρ (Gr... | The energy depends on the square of
the field, so we have
α = (E1 + E2)2
E21
= (E1 + E2)· E1 + E2)E21
= E21 + 2E1· E2 +E22
E21
= 2 + 2 cosθ
θ = cos−1
(α
2− 1
)
.
ANSWER: . | |
E4_solver:lit_em_0153:turn0_sol1 | lit_em_0153 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Prove the anticommutative property of the vector cross product, ... | The definition of the vector cross product
in terms of the components is (A×B)x =AyBz−
ByAz, and similarly for y and z. Reversing the
roles of A and B interchanges the positive and
negative terms on the right, so the sign flips.
ANSWER: negative terms on the right, so the sign flips. | [
"lit_em_0153_sol1"
] | E4_solver_adversarial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Prove the anticommutative property of the vector cross product, ... | The definition of the vector cross product
in terms of the components is (A×B)x =AyBz−
ByAz, and similarly for y and z. Reversing the
roles of A and B interchanges the positive and
negative terms on the right, so the sign flips.
ANSWER: negative terms on the right, so the sign flips. |
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