Learn Extracted exam questions AP Physics C: Electricity and Magnetism 2023 Free Response · Set 1
2023 Free Response · Set 1
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Students perform an experiment to determine the value of vacuum permittivity $\varepsilon_0$. Sphere 1 is nonconducting with charge $+q$ and is attached to an insulating rod. Sphere 2 is nonconducting with charge $+Q$ and has mass $M$. Sphere 2 is hung from a string of negligible mass and length $L$. Sphere 1 is brought near, without touching, Sphere 2, as shown. Equilibrium is established when the centers of the two spheres have the same vertical position, are a horizontal distance $d$ apart, and the string is at an angle $\theta$ from the vertical. The experiment is repeated.
[Diagram: A rigid horizontal support at the top with a string of length $L$ hanging at angle $\theta$ from the vertical, ending at Sphere 2 (nonconducting, mass $M$, charge $+Q$). Sphere 1 (nonconducting, charge $+q$) is mounted on an insulating rod held near Sphere 2 at the same height, separated by horizontal distance $d$.]
On the following dot that represents Sphere 2 at the position shown in the previous figure, draw and label the forces (not components) that act on Sphere 2. Each force must be represented by a distinct arrow starting on, and pointing away from, the dot.
[Diagram: A single dot with a dashed horizontal line and a dashed vertical line through it, representing Sphere 2 at its equilibrium position, for the student to draw force vectors on.]
Derive the relationship between the distance $d$ and the angle $\theta$ to show that
These values are collected in one trial: $Q = q = 6.0 \times 10^{-8}\,\text{C}$, $\theta = 12°$, and $d = 0.057\,\text{m}$. Calculate the expected force of tension exerted on Sphere 2 by the string.
The students vary $d$ and measure $\theta$ after equilibrium is reached. The students use the collected data to plot the following graph of $d^2$ vs. $\dfrac{1}{\tan\theta}$.
[Graph: y-axis $d^2$ (m$^2$) from 0.000 to 0.010 in increments of 0.002; x-axis $\dfrac{1}{\tan\theta}$ from 0 to 12 in increments of 2. Data points (approximate grid positions): (1.5, 0.0015), (2.3, 0.0022), (4.3, 0.0038), (6.2, 0.0050), (8.0, 0.0068), (11.3, 0.0087).]
Draw the best-fit line for the data.
Using the best-fit line, calculate an experimental value for the vacuum permittivity $\varepsilon_0$ when $M = 0.0050\,\text{kg}$ and $Q = q = 6.0 \times 10^{-8}\,\text{C}$.
The students modify the experiment by replacing Sphere 1 with a conducting Sphere 3 that has the same size and charge $+q$. The experiment is repeated.
The circle in the following figure represents Sphere 3 when spheres 2 and 3 are at equilibrium. On the circle, draw a single "+" sign to represent the location of highest concentration of the excess positive charges.
[Diagram: A circle labeled "Sphere 3" mounted on a thick insulating rod (shown as a solid black wedge), positioned as Sphere 1 was in the original figure, for the student to mark the "+" location.]
Briefly explain your reasoning for the sketch drawn in part (e)(i).
In the original experiment, when the centers of the two spheres are a horizontal distance $d_1$ apart, the string makes an angle $\theta_1$ from the vertical. In the modified experiment, when the centers of the two spheres are a horizontal distance $d_1$ apart, the string makes an angle $\theta_2$ from the vertical.
Is $\theta_2$ greater than, less than, or equal to $\theta_1$?
____ $\theta_2 > \theta_1$ ____ $\theta_2 < \theta_1$ ____ $\theta_2 = \theta_1$
Briefly justify your answer.
Two horizontal, parallel, conducting rails are separated by distance $L = 0.40\,\text{m}$. A resistor of resistance $R = 0.30\,\Omega$ connects the rails. A horizontal ideal spring is located between the rails. The right end of the spring is free to move and the left end is fixed in place. A conducting bar of mass $m = 0.23\,\text{kg}$ is placed on the rails and is in contact with the spring, which is initially compressed. Frictional forces and the resistance of the bar and rails are negligible.
- At time $t = 0$, the bar is released from rest and is pushed to the right by the spring.
- At time $t_1$, the bar loses contact with the spring and slides to the right.
- At time $t_2$, the bar enters and travels through a uniform magnetic field of magnitude $B = 0.50\,\text{T}$ that is directed into the page, as shown.
- At time $t_3$, the bar enters a region where the magnitude of the uniform magnetic field is still $B = 0.50\,\text{T}$ but is directed out of the page.
- At time $t_4$, the bar enters a region with no magnetic field.
Consider time $t_B$ such that $t_2 < t_B < t_3$.
[Diagram: Top view at time $t_B$. Two horizontal parallel rails separated by distance $L$, connected on the left by a resistor $R$ and a coiled spring, with the Conducting Bar shown between two field regions. The left field region (between the spring/bar and the bar's current position) is shaded with $\times$ symbols labeled $B$, indicating a uniform magnetic field into the page. The right region is marked with $\cdot$ (dot) symbols labeled $B$, indicating a uniform magnetic field out of the page.]
On the following diagram of the bar, draw an arrow indicating the direction of the net force $F_{net}$ exerted on the bar at time $t_B$. If the net force is zero, write $F_{net} = 0$.
[Diagram: A vertical rectangle representing the conducting bar, for the student to draw a force arrow on.]
At time $t_B$, the speed of the bar is $v = 2.5\,\text{m/s}$.
Calculate the magnitude of the current in the bar at time $t_B$.
Calculate the magnitude of the net force $F_{net}$ exerted on the bar at time $t_B$.
On the following axes, sketch a graph of the speed $v$ of the bar as a function of time $t$ between $t = 0$ and $t_4$.
[Graph: Blank axes with vertical axis $v$ and horizontal axis $t$, origin labeled $O$. Four vertical dashed reference lines are marked on the horizontal axis at $t_1$, $t_2$, $t_3$, and $t_4$, for the student to sketch the speed-time curve.]
The scenario is repeated but an additional resistor of resistance $R = 0.30\,\Omega$ is connected, as shown.
[Diagram: Top view at time $t_B$, same rail/bar/field setup as before, but now with a second resistor $R$ connected in the circuit next to the first resistor $R$ and the spring, both on the left end of the rails.]
Determine the total resistance $R_{total}$ of the closed circuit for the new scenario.
In the original scenario, the magnitude of the acceleration of the bar immediately after the bar enters the first uniform magnetic field is $a_{original}$. In the new scenario, the magnitude of the acceleration of the bar immediately after the bar enters the first uniform magnetic field is $a_{new}$.
Is $a_{new}$ greater than, less than, or equal to $a_{original}$? Justify your answer.
Describe a modification to $m$, $B$, or $L$ that will result in a smaller induced potential difference across the original resistor immediately after the bar enters the first uniform magnetic field. Justify your answer.
The circuit shown consists of a battery of emf $\mathcal{E}$, resistors 1 and 2 each with resistance $R$, capacitors 1 and 2 with capacitances $C$ and $2C$, respectively, and a switch. The switch is initially open and both capacitors are uncharged.
At time $t = 0$, the switch is closed to Position A.
[Diagram: A circuit with a battery of emf $\mathcal{E}$ on the left. A switch at the top can connect to Position A or Position B. Resistor 1 and Resistor 2 (each resistance $R$) are in parallel branches below the switch positions. Capacitor 1 (capacitance $C$) is connected at the bottom of the Resistor 1 branch. Capacitor 2 (capacitance $2C$) is connected on the right side of the circuit, in the branch reached via Position B and Resistor 2.]
Write, but do NOT solve, a differential equation that can be used to determine the charge $Q$ on the positive plate of Capacitor 1 as a function of time $t$ after the switch is closed to Position A. Express your answer in terms of $\mathcal{E}$, $R$, $C$, $Q$, $t$, and fundamental constants, as appropriate.
On the axes shown, sketch graphs of the surface charge density $\sigma$ on the positive plate of Capacitor 1 and the total power $P$ dissipated by the resistors as functions of time $t$ from time $t = 0$ until steady-state conditions are nearly reached.
[Graph 1: Blank axes, vertical axis $\sigma$, horizontal axis $t$, origin $O$, for the student to sketch surface charge density vs. time.]
[Graph 2: Blank axes, vertical axis $P$, horizontal axis $t$, origin $O$, for the student to sketch total power dissipated vs. time.]
A long time after the switch is closed to Position A, the charge on the positive plate of Capacitor 1 is $Q_0$ and Capacitor 2 is uncharged.
At time $t_1$, the switch is closed to Position B.
Immediately after time $t_1$, is the direction of the current in the switch directed toward the left, directed toward the right, or is there no current? Briefly justify your answer.
Determine an expression for the total charge on the positive plate of Capacitor 2 a long time after $t_1$. Express your answer in terms of $Q_0$ and fundamental constants, as appropriate.
Derive an expression for the total energy $E_R$ dissipated by resistors 1 and 2 from immediately after time $t_1$ until new steady-state conditions have been reached. Express your answer in terms of $C$, $Q_0$, and fundamental constants, as appropriate.
With the switch still closed to Position B, the parallel plates of Capacitor 2 are moved so that the separation distance increases by a factor of 2.
Determine the ratio $\dfrac{U_2}{U_1}$ of the energy $U_2$ stored in Capacitor 2 to the energy $U_1$ stored in Capacitor 1 a long time after the plates of Capacitor 2 have been moved. Briefly justify your answer.
With the capacitors still charged as in part (d), the switch is now closed to Position A.
Express your answers to part (e)(i) and part (e)(ii) in terms of $R$, $C$, $Q_0$, and fundamental constants, as appropriate.
Derive an expression for the current $I_0$ from the battery immediately after the switch is closed to Position A.
With the capacitors still charged as in part (d), the switch is now closed to Position A.
Express your answers to part (e)(i) and part (e)(ii) in terms of $R$, $C$, $Q_0$, and fundamental constants, as appropriate.
Determine the current $I_\infty$ from the battery a long time after the switch is closed to Position A.