2024 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. Figure 1 1. A nonconducting rod of uniform positive linear charge density is near a sphere with charge −2.0 nC. The rod and sphere are held at rest on the x-axis, as shown in Figure 1. Equipotential lines and positions A, B, C, D, and E are labeled. Adjacent tick marks on the x-axis and the y-axis are 0.40 m apart. (a) Calculate the absolute value of the electric flux through the Gaussian surface whose cross section is the −20.0 V equipotential line. A positive test charge (not shown) is placed and held at rest at Position C. An external force is applied to the test charge to move the test charge to different positions in the order of C→E→D→A. The test charge is momentarily held at rest at each position. (b) The bar shown in Figure 2 represents the absolute value of the work W_CE done by the external force on the test charge to move the test charge from Position C to Position E. i. Complete the following tasks on Figure 2. • Draw a bar to represent the relative absolute value of the work W_ED done by the external force on the test charge to move the test charge from Position E to Position D. • Draw a bar to represent the relative absolute value of the work W_DA done by the external force on the test charge to move the test charge from Position D to Position A. • The height of each bar should be proportional to the value of W_CE. If W_ED = 0 and/or W_DA = 0, write a "0" in the corresponding columns, as appropriate. Figure 2 ii. Calculate the approximate magnitude of the x-component of the electric field at Position B. The positive test charge is placed at Position D. The test charge is then released from rest. (c) Indicate the direction (not components) of the net electric force exerted on the test charge immediately after the test charge is released from rest. ___ +x ___ +y ___ Directly away from the sphere ___ −x ___ −y ___ Directly toward the sphere Without using equations, justify your answer using physics principles. Figure 3 The sphere and the test charge are removed. The rod has length 4L and uniform positive linear charge density +λ. The rod is held at rest on the x-axis in the orientation shown in Figure 3. Position P (not shown) is located on the x-axis a distance x_P from the origin, where x_P > 4L. (d) The electric potential V_P at x_P is V_P = kλ ln ( x_P / (x_P − 4L) ). i. Using integral calculus, derive the expression for V_P provided. ii. On Figure 4, sketch a graph of the x-component E_x of the electric field from the rod as a function of x in the region 4L < x < 12L. Figure 4

Begin your response to QUESTION 2 on this page. [Figure 1: Circuit diagram showing a battery ε, switch, two identical resistors R in parallel, and an inductor L in series] Figure 1 2. Students are asked to determine the resistance R of two identical resistors. The resistors are in parallel with each other and are connected in series to a battery of known emf ε, an inductor of known inductance L, and a switch, as shown in Figure 1. The students have access to a voltmeter that can measure potential difference as a function of time. The students are required to measure a quantity that decreases with time to determine R. (a) i. On the circuit diagram shown in Figure 1, **draw** the voltmeter, using the following symbol, with connections that would allow the students to correctly measure a potential difference that decreases with time. [Voltmeter symbol] Voltmeter Symbol ii. **Describe** a procedure for collecting data that would allow the students to graphically determine the experimental value for R using the measured quantity that decreases with time. Provide enough detail so that another student could replicate the experiment. Continue your response to QUESTION 2 on this page. (b) i. On the axes shown in Figure 2, produce a graph that represents the expected trend of the data by completing the following tasks. • **Label** the quantities graphed on the vertical and horizontal axes. • **Sketch** a line or curve that represents the expected trend of the collected data. • **Label** any appropriate intercepts and/or asymptotes in terms of the quantities provided. [Figure 2: Blank vertical and horizontal axes] Figure 2 ii. **Describe** how the information from the graph in part (b)(i) would be used to determine the experimental value for R. Continue your response to QUESTION 2 on this page. (c) Starting with an appropriate application of Kirchhoff's loop rule, **derive**, but do NOT solve, a differential equation that can be used to determine the current I in the inductor at time t after the switch is closed. Express your answer in terms of R, ε, L, t, and physical constants, as appropriate. After reaching steady state, the absolute value of the potential difference across the inductor is |ΔV₁|. The students replace the original inductor with a new inductor that has nonnegligible resistance. The experiment is repeated. After a long time, the absolute value of the potential difference across the new inductor is |ΔV₂|. (d) **Indicate** whether |ΔV₂| is greater than, less than, or equal to |ΔV₁|. ____ |ΔV₂| > |ΔV₁| ____ |ΔV₂| < |ΔV₁| ____ |ΔV₂| = |ΔV₁| **Justify** your answer.

3. A wire is connected to a resistor of resistance R to form a rigid rectangular loop of width L and height 2L. An external force is exerted on the loop so that the loop always moves with constant speed v in the +x-direction, as shown in Figure 1. The loop then enters Region 1 of external uniform magnetic field of magnitude B that is directed in the −z-direction. Region 1 has boundaries x = L and x = 2.5L. The loop later enters Region 2 with two external, uniform magnetic fields, each of magnitude B, that are parallel but are directed in opposite z-directions. Region 2 has boundaries x = 2.5L and x = 3.5L. Point S is the midpoint of the leading edge of the loop and is aligned with the horizontal boundary in Region 2 that separates the two magnetic fields. (a) On the following axes, sketch a graph of the magnetic flux Φ through the rectangular loop as a function of the position x of Point S from x = 0 to x = 4.5L. The +z-direction indicated in Figure 1 corresponds to +Φ. (b) Consider the instant when Point S reaches x = 1.5L. i. Indicate whether the current I_R that is induced in the rectangular loop when Point S reaches x = 1.5L is clockwise, counterclockwise, or zero. ___ Clockwise ___ Counterclockwise ___ Zero Briefly justify your answer. ii. Derive an expression for I_R when Point S reaches x = 1.5L. If I_R = 0, indicate how the derived expression shows that I_R = 0. Express your answer in terms of R, L, v, B, and physical constants, as appropriate. iii. Derive an expression for the power P dissipated by the resistor when Point S reaches x = 1.5L. Express your answer in terms of R, L, v, B, and physical constants, as appropriate. The total energy dissipated by the resistor in the rectangular loop as Point S moves from x = 0 to x = 4.5L is E_original. The vertical boundary between regions 1 and 2 is now shifted to x = 1.5L. After the boundary is shifted, the rectangular loop again moves with speed v in the +x-direction, as shown in Figure 2. The total energy dissipated by the resistor as Point S moves from x = 0 to x = 4.5L is E_new. (c) Indicate whether E_new is greater than, less than, or equal to E_original. _____ E_new > E_original _____ E_new < E_original _____ E_new = E_original Briefly justify your answer. The original magnetic fields are modified so that the region L < x < 3.5L contains an external uniform magnetic field of magnitude B that is directed in the −z-direction. A new wire is connected to a resistor of resistance R to form a rigid triangular loop with base length L and height 2L. An external force is exerted on the loop so that the loop always moves with speed v in the +x-direction, as shown in Figure 3. Point S represents the lower-leading corner of the loop. (d) On the following axes, sketch a graph of the induced current I_T in the triangular loop as Point S moves from x = L to x = 3L. GO ON TO THE NEXT PAGE.