Learn Extracted exam questions AP Physics C: Electricity and Magnetism 2023 Free Response · Set 2
2023 Free Response · Set 2
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[Figure: A vertical setup. At the top, Sphere A (charge $+q$) hangs from a fixed support on a thin rod, positioned directly above Sphere B. A vertical distance $H$ separates the centers of Sphere A and Sphere B. Sphere B (mass $M$, charge $+Q$) rests on an insulating platform of negligible mass, which sits on top of a vertical ideal spring with spring constant $k_s$. The spring rests on a fixed base. Note: Figure not drawn to scale.]
Students perform an experiment to determine the value of vacuum permittivity $\varepsilon_0$. Sphere A is nonconducting with charge $+q$ and is attached to an insulating rod. Sphere B is nonconducting with charge $+Q$, and has mass $M$. Sphere B rests on an insulating platform of negligible mass that is attached to a vertical ideal spring with spring constant $k_s$. Sphere B and the spring are initially at rest.
Sphere A is then brought near Sphere B without touching. When the centers of the spheres are separated by a vertical distance $H$, the spring has been compressed a distance $y$, as shown in the figure. The students measure $y$ for different values of $H$.
On the following dot that represents Sphere B in the figure on the previous page, draw and label the forces (not components) that are exerted on Sphere B. Each force must be represented by a distinct arrow starting on, and pointing away from, the dot.
[Diagram: a single dot with a short dashed horizontal line and a short dashed vertical line crossing through it, indicating axes for drawing force vectors. No forces are pre-drawn; the student draws them.]
Derive the relationship between $y$ and $H$ to show that
The students plot collected data of $y$ as a function of $\dfrac{1}{H^2}$, as shown in the graph.
[Graph: y-axis labeled "y (m)" ranging from 0 to 0.18 in increments of 0.02 (gridlines at 0, 0.02, 0.04, 0.06, 0.08, 0.10, 0.12, 0.14, 0.16, 0.18); x-axis labeled "$\dfrac{1}{H^2}\left(\dfrac{1}{\text{m}^2}\right)$" ranging from 0 to 30 in increments of 5. Data points (approximate): (0, 0.10), (7, 0.125), (10, 0.12), (13, 0.135), (17, 0.14), (20, 0.135), (23, 0.16), (27, 0.165), (29, 0.165).]
Draw the best-fit line for the data.
Using the best-fit line, calculate an experimental value for the vacuum permittivity $\varepsilon_0$ when $Q = q = 2.00 \times 10^{-6}\text{ C}$ and $k_s = 25\text{ N/m}$.
Using the best-fit line, calculate an experimental value for the mass of Sphere B.
The students modify the experiment by replacing nonconducting Sphere B with conducting Sphere C that has the same charge $+Q$ and mass $M$. Sphere A is brought near Sphere C without touching, compressing the spring. Sphere C comes to rest.
In the original experiment, when the centers of spheres A and B are a vertical distance $H_1$ apart, the spring is compressed a distance $y_1$. In the modified experiment, when the centers of spheres A and C are a vertical distance $H_1$ apart, the spring is compressed a distance $y_2$.
Is $y_2$ greater than, less than, or equal to $y_1$?
____ $y_2 > y_1$ ____ $y_2 < y_1$ ____ $y_2 = y_1$
Justify your answer.
[Figure: Sphere A (charge $+q$) hangs above Sphere C. Sphere C rests on the platform above the spring, with a wire connected from Sphere C down to the ground/base.]
Sphere C is then grounded with a wire. On the following figure, draw an arrow indicating the direction that the platform will move immediately after being grounded. If the platform remains stationary, write "does not move."
[Figure, Perspective View (at time $t=0$): A sloped section of two parallel conducting rails at height $H$, with a resistor $R$ at the top of the slope. The rails transition to a horizontal section. A region of magnetic field $B_1$ (dots, pointing out of the page, +y-direction) covers part of the horizontal rails near the bottom of the slope. Farther along, a separate region of magnetic field $B_2$ (arrows pointing into the page, +z-direction, shown as downward-slanting field lines with arrowheads) covers another part of the rails. Axes shown: $+x$, $+y$, $+z$ in a right-handed orientation.]
[Figure, Top View (at time $t_B$): Two horizontal parallel rails separated by distance $d$, viewed from above. A resistor $R$ is at the left end (position $x_0$). The bar is shown as a vertical rectangle. Positions marked along the x-axis: $x_0$, $x_1$, $x_B$, $x_2$, $x_3$, $x_4$. The region between $x_1$ and $x_2$ is shaded with dots labeled $B_1$ (field out of the page). The region between $x_3$ and $x_4$ has downward arrows labeled $B_2$ (field into the page). The bar is shown at position $x_B$, between $x_1$ and $x_2$.]
Two parallel conducting rails are separated by distance $d = 0.30\text{ m}$. A resistor of resistance $R = 0.20\ \Omega$ connects the rails. A conducting bar is placed on a sloped section of the rails at height $H$ above the horizontal section of the rails. Frictional forces and the resistances of the bar and rails are negligible.
- At time $t = 0$, the bar is released from rest from position $x_0$ and slides down the sloped section of the rails, as shown in the Perspective View.
- At time $t_1$, the bar reaches position $x_1$ and smoothly transitions to the horizontal section of the rails and enters a uniform magnetic field of magnitude $B_1 = 0.40\text{ T}$ that is directed in the +y-direction.
- At time $t_2$, the bar reaches position $x_2$ and enters a region with no magnetic field.
- At time $t_3$, the bar reaches position $x_3$ and enters a uniform magnetic field of magnitude $B_2 = 0.60\text{ T}$ that is directed in the +z-direction.
- At time $t_4$, the bar reaches position $x_4$ and enters a region with no magnetic field.
The bar is at position $x_B$ (shown in Top View) at time $t_B$ such that $t_1 < t_B < t_2$.
[Diagram: a vertical rectangle representing the bar, viewed from the Top View perspective, with no arrows drawn.]
On the following diagram of the bar, as observed from the Top View, draw an arrow indicating the direction of the net force $F_{\text{net}}$ exerted on the bar at time $t_B$. If the net force is zero, write $F_{\text{net}} = 0$.
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 $F_{\text{net}}$ 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: y-axis labeled "$v$", x-axis labeled "$t$" with tick marks at $t_1$, $t_2$, $t_3$, $t_4$ (dashed vertical gridlines at each); axes otherwise unlabeled with numeric values, origin at $O$.]
The original scenario is repeated but with a new bar that has the same mass but with a nonnegligible resistance $R = 0.20\ \Omega$. The new bar is released from rest and smoothly transitions to the horizontal section of the rails and enters the first uniform magnetic field.
Determine the total resistance of the closed circuit.
In the original scenario, the magnitude of the acceleration of the bar immediately after the bar enters the first uniform magnetic field is $a_{\text{original}}$. In the new scenario, the magnitude of the acceleration of the bar immediately after the bar enters the first uniform magnetic field is $a_{\text{new}}$. Is $a_{\text{new}}$ greater than, less than, or equal to $a_{\text{original}}$? Justify your answer.
Describe a modification to $H$, $B_1$, or $d$ that will result in a larger induced current in the new bar immediately after the bar enters the first magnetic field. Justify your answer.
[Figure: A circuit diagram. A battery of emf $\mathcal{E}$ is on the left side of the circuit. A switch at the top has two positions, A and B (A on the left, B on the right). From position A, a branch leads down through Resistor 1 (resistance $R$) to a node; from position B, a branch leads down through Capacitor 2 (capacitance $C$) to the same node level on the right side. Capacitor 1 (capacitance $C$) is connected across the middle, below Resistor 1. Resistor 2 (resistance $R$) is at the bottom of the circuit, connecting back to the battery. The battery, Resistor 2, and the parallel combination of [Resistor 1 with Capacitor 1 below it] and [the switch branch to Capacitor 2] form the overall loop.]
The circuit shown consists of an ideal battery of emf $\mathcal{E}$, resistors 1 and 2 each with resistance $R$, capacitors 1 and 2 each with capacitance $C$, and a switch. The switch is initially open and both capacitors are uncharged.
At time $t = 0$, the switch is closed to Position 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 $\mathcal{E}, R, C, Q, t$, and fundamental constants, as appropriate.
On the axes shown, sketch graphs of the current $I_R$ in Resistor 1 and the energy $U_C$ stored in Capacitor 1 as functions of time $t$ from time $t = 0$ until steady-state conditions are nearly reached.
[Graph 1: y-axis labeled "$I_R$", x-axis labeled "$t$", origin at $O$, no numeric scale shown.]
[Graph 2: y-axis labeled "$U_C$", x-axis labeled "$t$", origin at $O$, no numeric scale shown.]
A long time after the switch is closed to Position A, the total 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 $t_1$, is the direction of the current in Resistor 1 directed up, directed down, 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 dissipated by Resistor 1 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, a dielectric material with dielectric constant $\kappa = 2$ is inserted between the plates of Capacitor 2.
Determine the charge on the positive plate of Capacitor 2 a long time after the dielectric has been inserted. Express your answer in terms of $Q_0$ and fundamental constants, as appropriate.
[Figure: The same circuit as before (battery $\mathcal{E}$, switch positions A/B, Resistor 1, Capacitor 1, Resistor 2, Capacitor 2), now with a dielectric slab labeled "Dielectric" shown inserted between the plates of Capacitor 2. An additional wire of negligible resistance is shown connected between two corners of the circuit (from the node between the switch and Resistor 1/Capacitor 1, curving around to the node below Capacitor 2/Resistor 2), as described below.]
With the switch still closed to Position B, a wire of negligible resistance is connected between two corners of the circuit, as shown.
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 in Resistor 2 immediately after the wire is connected to the circuit.
Determine the current in Resistor 2 a long time after the wire is connected to the circuit.