2023 AP Physics C: Electricity and Magnetism Free-Response Questions

Directions: Answer all three questions. The suggested time is about 15 minutes for answering each of the questions, which are worth 15 points each. The parts within a question may not have equal weight. Show all your work in this booklet in the spaces provided after each part. 1. Students perform an experiment to determine the value of vacuum permittivity ε₀. 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 θ from the vertical. [Figure: Sphere 1, labeled +q, is on an insulating rod to the left of Sphere 2. Sphere 2 is labeled M, +Q, and is suspended by a string of length L. The centers are separated horizontally by d, and the string makes angle θ from the vertical.] (a) 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. (b) Derive the relationship between the distance d and the angle θ to show that d = √(Qq/(4πε₀Mg tanθ)). (c) These values are collected in one trial: Q = q = 6.0 × 10⁻⁸ C, θ = 12°, and d = 0.057 m. Calculate the expected force of tension exerted on Sphere 2 by the string. (d) The students vary d and measure θ after equilibrium is reached. The students use the collected data to plot the following graph of d² vs. 1/tanθ. [Graph: d² (m²) versus 1/tanθ, with plotted data points.] i. Draw the best-fit line for the data. ii. Using the best-fit line, calculate an experimental value for the vacuum permittivity ε₀ when M = 0.0050 kg and Q = q = 6.0 × 10⁻⁸ C. (e) 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. i. 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. [Figure: A circle labeled Sphere 3 attached to an insulating rod, with space on the circle for a single plus sign.] ii. Briefly explain your reasoning for the sketch drawn in part (e)(i). iii. In the original experiment, when the centers of the two spheres are a horizontal distance d₁ apart, the string makes an angle θ₁ from the vertical. In the modified experiment, when the centers of the two spheres are a horizontal distance d₁ apart, the string makes an angle θ₂ from the vertical. Is θ₂ greater than, less than, or equal to θ₁? _____ θ₂ > θ₁ _____ θ₂ < θ₁ _____ θ₂ = θ₁ Briefly justify your answer.

2. Two horizontal, parallel, conducting rails are separated by distance L = 0.40 m. A resistor of resistance R = 0.30 Ω 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 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. Figure: Top view showing the rails separated by L, the resistor R, the compressed spring, and the conducting bar. The bar moves through regions with magnetic field B directed into the page (×) and B directed out of the page (•). • At time t = 0, the bar is released from rest and is pushed to the right by the spring. • At time t₁, the bar loses contact with the spring and slides to the right. • At time t₂, the bar enters and travels through a uniform magnetic field of magnitude B = 0.50 T that is directed into the page, as shown. • At time t₃, the bar enters a region where the magnitude of the uniform magnetic field is still B = 0.50 T but is directed out of the page. • At time t₄, the bar enters a region with no magnetic field. Consider time t_B such that t₂ < t_B < t₃. (a) 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. (b) At time t_B, the speed of the bar is v = 2.5 m/s. i. Calculate the magnitude of the current in the bar at time t_B. ii. Calculate the magnitude of the net force F_net exerted on the bar at time t_B. (c) On the following axes, sketch a graph of the speed v of the bar as a function of time t between t = 0 and t₄. The axes show v versus t, with marked times t₁, t₂, t₃, and t₄. (d) The scenario is repeated but an additional resistor of resistance R = 0.30 Ω is connected, as shown. i. Determine the total resistance R_total of the closed circuit for the new scenario. ii. 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. (e) 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.

3. The circuit shown consists of a battery of emf 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. [Circuit diagram showing the battery, switch positions A and B, Resistor 1, Resistor 2, Capacitor 1, and Capacitor 2.] At time t = 0, the switch is closed to Position A. (a) 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 ℰ, R, C, Q, t, and fundamental constants, as appropriate. (b) On the axes shown, sketch graphs of the surface charge density σ 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. [Axes shown for σ versus t and P versus t.] A long time after the switch is closed to Position A, the charge on the positive plate of Capacitor 1 is Q₀ and Capacitor 2 is uncharged. (c) At time t₁, the switch is closed to Position B. i. Immediately after time t₁, 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. ii. Determine an expression for the total charge on the positive plate of Capacitor 2 a long time after t₁. Express your answer in terms of Q₀ and fundamental constants, as appropriate. iii. Derive an expression for the total energy E_R dissipated by resistors 1 and 2 from immediately after time t₁ until new steady-state conditions have been reached. Express your answer in terms of C, Q₀, 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. (d) Determine the ratio U₂/U₁ of the energy U₂ stored in Capacitor 2 to the energy U₁ 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. (e) Express your answers to part (e)(i) and part (e)(ii) in terms of R, C, Q₀, and fundamental constants, as appropriate. i. Derive an expression for the current I₀ from the battery immediately after the switch is closed to Position A. ii. Determine the current I∞ from the battery a long time after the switch is closed to Position A.