Learn Extracted exam questions AP Physics C: Electricity and Magnetism 2024 Free Response · Set 1
2024 Free Response · Set 1
Source PDF on the left, extracted YAML on the right. Compare numbering, marks, options and text.
A nonconducting rod of uniform positive linear charge density is near a sphere with charge $-2.0\ \text{nC}$. 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 $0.40\ \text{m}$ apart.
[Figure 1: An $x$-$y$ coordinate system (origin at lower left, $+y$ up, $+x$ to the right). On the left side, a horizontal rod lies along the $x$-axis, surrounded by concentric circular equipotential lines labeled (from innermost/rod outward) $50.0\text{ V}$, $40.0\text{ V}$, $30.0\text{ V}$, $20.0\text{ V}$, $10.0\text{ V}$. Position A is marked near the rod, and Position B is marked further out among the rod's equipotential circles. On the right side, a small sphere sits on the $x$-axis, surrounded by concentric circular equipotential lines labeled (from outermost inward toward the sphere) $0.0\text{ V}$, $-10.0\text{ V}$, $-20.0\text{ V}$, $-30.0\text{ V}$. Position C is at the sphere; Position D is below and between the rod and sphere; Position E is above and between the rod and sphere. A horizontal double-headed arrow near the sphere is labeled $0.40\text{ m}$, spanning from the sphere to a point along $+x$. Tick marks along the $x$-axis and $y$-axis are spaced $0.40\text{ m}$ apart.]
Calculate the absolute value of the electric flux through the Gaussian surface whose cross section is the $-20.0\text{ V}$ 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$\rightarrow$E$\rightarrow$D$\rightarrow$A. The test charge is momentarily held at rest at each position.
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. Complete the following bar graph in 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: A bar-chart grid with a vertical axis labeled "Work" (unmarked scale, gridlines shown) and three labeled columns along the horizontal axis: $W_{CE}$, $W_{ED}$, $W_{DA}$. The bar for $W_{CE}$ is already drawn at a fixed reference height (shaded); the bars for $W_{ED}$ and $W_{DA}$ are to be drawn by the student.]
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.
Indicate the direction (not components) of the net electric force exerted on the test charge immediately after the test charge is released from rest.
$\underline{\hspace{1cm}} +x$ $\underline{\hspace{1cm}} +y$ $\underline{\hspace{1cm}}$ Directly away from the sphere
$\underline{\hspace{1cm}} -x$ $\underline{\hspace{1cm}} -y$ $\underline{\hspace{1cm}}$ Directly toward the sphere
Without using equations, justify your answer using physics principles.
[Figure 3: A horizontal rod labeled "Rod" lies along the $x$-axis from $x=0$ to $x=4L$, with linear charge density $+\lambda$ indicated by a bracket labeled $+\lambda$ above the rod near $x=0$. The $x$-axis is marked with tick labels $0, 2L, 4L, 6L, 8L, 10L, 12L$.]
The sphere and the test charge are removed. The rod remains. The rod has length $4L$ and uniform positive linear charge density $+\lambda$. 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 $x_P > 4L$.
The electric potential $V_P$ at $x_P$ is $V_P = k\lambda \ln\!\left[\dfrac{x_P}{x_P - 4L}\right]$.
Using integral calculus, derive the expression for $V_P$ provided.
[Figure 4: A blank graph with vertical axis $E_x$ (unlabeled scale) and horizontal axis $x$ marked with tick labels $2L, 4L, 6L, 8L, 10L, 12L$; a shaded region spans from $x=4L$ to $x=12L$ indicating the region to be graphed.]
On Figure 4, sketch a graph of the $x$-component of the electric field from the rod as a function of $x$ in the region $4L < x < 12L$.
[Figure 1: A circuit diagram. A battery of emf $\mathcal{E}$ is on the left. From the battery's top terminal, a wire goes right to a switch, then to a node. From that node, two identical resistors, each of resistance $R$, are connected in parallel between the node and a second node further right (drawn as two parallel horizontal branches, each containing a resistor $R$ with a zigzag symbol, one above the other). From the second node, a wire continues right and down through an inductor of inductance $L$ (coil symbol), and back to the battery's bottom terminal, completing the loop.]
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 $\mathcal{E}$, 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 $R$.
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: a circle containing the letter "V", with two connection leads.]
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.
[Figure 2: A blank graph with a vertical axis (unlabeled) and a horizontal axis (unlabeled, shown as a dashed line), both with arrowheads, meeting at the origin.]
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.
Describe how the information from the graph in part (b)(i) would be used to determine the experimental value for $R$.
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 $R$, $\mathcal{E}$, $L$, $t$, and physical constants, as appropriate.
After reaching steady state, the absolute value of the potential difference across the inductor is $|\Delta V_1|$. 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 $|\Delta V_2|$.
Indicate whether $|\Delta V_2|$ is greater than, less than, or equal to $|\Delta V_1|$.
$\underline{\hspace{1cm}} |\Delta V_2| > |\Delta V_1|$ $\underline{\hspace{1cm}} |\Delta V_2| < |\Delta V_1|$ $\underline{\hspace{1cm}} |\Delta V_2| = |\Delta V_1|$
Justify your answer.
[Figure 1: An $x$-$z$ coordinate system indicator (with $+z$ up, $+x$ to the right) is shown at top right. A rigid rectangular loop of width $L$ and height $2L$ lies in the plane of the page, with corners forming a rectangle; the loop's leading (right) vertical edge is labeled with an arrow pointing right labeled $v$, indicating the loop moves in the $+x$-direction with constant speed $v$. Inside the loop near its left edge, a resistor $R$ (zigzag symbol) is shown in the left vertical side, and point S is marked at the midpoint of the loop's right (leading) edge. To the right of the loop, the page is divided into two vertical field regions along the $x$-axis: "Region 1" spans from $x=L$ to $x=2L$, shown with dots (indicating field out of the page, $+z$-direction); "Region 2" spans from $x=2.5L$ to $x=3.5L$, shown with alternating dot/cross symbols (two external uniform magnetic fields of equal magnitude $B$ directed in opposite $z$-directions). The horizontal axis below is marked with tick labels $0, L, 2L, 3L, 4L$.]
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 first 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 $x = 2L$. The loop later enters Region 2 with two external, uniform magnetic fields, each of magnitude $B$, but directed in opposite $z$-directions. Region 2 has boundaries $x = 2.5L$ and $x = 3.5L$. Point S is the midpoint of the loop's leading edge $B$ that is directed at the horizontal boundary in Region 2 that separates the two magnetic fields.
[Figure: A blank graph with vertical axis $\Phi$ (unlabeled scale) and horizontal axis $x$ marked with gridlines and tick labels $0, L, 2L, 3L, 4L, 4.5L$ (position of Point S from $x=0$ to $x=4.5L$).]
On the following axes, sketch a graph of the magnetic flux $\Phi$ 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 $+\Phi$.
Consider the instant when Point S reaches $x = 1.5L$.
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.
$\underline{\hspace{1cm}}$ Clockwise $\underline{\hspace{1cm}}$ Counterclockwise $\underline{\hspace{1cm}}$ Zero
Briefly justify your answer.
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.
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 $E_{\text{original}}$.
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 $E_{\text{new}}$.
[Figure 2: Same setup as Figure 1, but Region 1 (dots, $+z$ field out of page) now spans from $x=L$ to $x=1.5L$, and Region 2 (alternating dot/cross field) spans from $x=1.5L$ to $x=2.5L$. The loop, resistor $R$, and point S are shown in the same configuration as before, moving in $+x$ with speed $v$. Axis tick labels: $0, L, 2L, 3L, 4L$.]
Indicate whether $E_{\text{new}}$ is greater than, less than, or equal to $E_{\text{original}}$.
$\underline{\hspace{1cm}} E_{\text{new}} > E_{\text{original}}$ $\underline{\hspace{1cm}} E_{\text{new}} < E_{\text{original}}$ $\underline{\hspace{1cm}} E_{\text{new}} = E_{\text{original}}$
Briefly justify your answer.
[Figure 3: A right-triangular loop with base $L$ and height $2L$ is shown, with a resistor $R$ on its vertical left side and point S at the lower-right corner (lower leading corner) of the loop. To the right, the region $L < x < 3.5L$ contains a uniform magnetic field of magnitude $B$ shown with cross symbols (directed in the $-z$-direction). The loop moves in the $+x$-direction with speed $v$. Axis tick labels: $0, L, 2L, 3L, 4L$.]
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.
[Figure: A blank graph with vertical axis $I_S$ (unlabeled scale) and horizontal axis $x$ marked with gridlines and tick labels $0, L, 2L, 3L, 4L$.]
On the following axes, sketch a graph of the induced current $I_S$ in the triangular loop as Point S moves from $x = L$ to $x = 3L$.