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.