Learn Extracted exam questions AP Physics C: Electricity and Magnetism 2024 Free Response · Set 2
2024 Free Response · Set 2
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A nonconducting rod of uniform negative linear charge density is near a sphere with charge $+1.0\,\text{nC}$. The rod and sphere are held at rest on the $y$-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 on the $y$-axis are $0.40\text{ m}$ apart.
[Figure 1: A set of equipotential lines around a sphere and a rod on the $y$-axis, in the $xy$-plane with $+x$ to the right and $+y$ upward. The rod is a thick vertical bar below the origin, labeled "Rod". Above the origin sits a small circle labeled "Sphere". Equipotential lines are curved, roughly circular/oval loops surrounding the sphere and rod, labeled from innermost to outermost with potential values: $-10.0\text{ V}$ (innermost, near sphere, marked with point D), $0.0\text{ V}$ (marked with point E, near the sphere), $-20.0\text{ V}$, $-30.0\text{ V}$, $-40.0\text{ V}$ (marked with point C, to the left of the rod). Point P is marked near the sphere at the top. Points B and A lie further out along the outermost equipotential loops on the lower left, with A below B. A horizontal double-headed arrow near the $x$-axis on the right indicates a spacing of $0.40\text{ m}$ between adjacent tick marks.]
Calculate the absolute value of the electric flux through the Gaussian surface whose cross section is the $0.0\text{ V}$ equipotential line.
A positive test charge (not shown) is placed and held at rest at Position C. An external force is applied to move the test charge to different positions in the order of C$\rightarrow$E$\rightarrow$D$\rightarrow$A. The test charge is held momentarily at rest at each position.
The bar shown in Figure 2 represents the absolute value of the work $W_{\text{CE}}$ done by the external force on the test charge to move the test charge from Position C to Position E.
Draw a bar to represent the relative absolute value of the work $W_{\text{ED}}$, 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 $W_{\text{DA}}$ 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 $W_{\text{CE}}$. If $W_{\text{ED}} = 0$ and/or $W_{\text{DA}} = 0$, write a "0" in the corresponding columns, as appropriate.
[Figure 2: A bar chart with vertical axis labeled "Work" (unlabeled scale, dashed horizontal gridlines) and three columns on the horizontal axis labeled $W_{\text{CE}}$, $W_{\text{ED}}$, $W_{\text{DA}}$. A single shaded bar of medium height is already drawn above the $W_{\text{CE}}$ column; the $W_{\text{ED}}$ and $W_{\text{DA}}$ columns are empty for the student to complete.]
Calculate the approximate magnitude of the $x$-component of the electric field at Position B.
The positive test charge is placed at Position C. 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.
_____ $+x$ _____ $+y$ _____ Directly away from the sphere
_____ $-x$ _____ $-y$ _____ Directly toward the sphere
Without using equations, justify your answer using physics principles.
[Figure 3: A vertical $y$-axis with tick marks labeled from bottom to top: $-\lambda$ at $0$, then $2L$, $4L$, $6L$, $8L$, $10L$, $12L$. A thick vertical bar labeled "Rod" spans from $0$ to $2L$ on the axis, adjacent to the label $-\lambda$.]
The sphere and the test charge are removed. The rod has length $2L$ and uniform negative linear charge density $-\lambda$. The rod is held at rest on the $y$-axis in the orientation shown in Figure 3. Position P (not shown) is located on the $y$-axis a distance $y_P$ from the origin, where $y_P > 2L$.
The electric potential $V_{\text{P}}$ at $y_P$ is $V_{\text{P}} = -k\lambda \ln\!\left(\dfrac{y_P}{y_P - 2L}\right)$.
Using integral calculus, derive the expression for $V_{\text{P}}$ provided.
On Figure 4, sketch a graph of the $y$-component $E_y$ of the electric field resulting from the rod as a function of $y$ in the region $2L < y < 12L$.
[Figure 4: A blank graph with vertical axis labeled $E_y$ (no numeric scale shown) and horizontal axis labeled $y$ with tick marks at $2L$, $4L$, $6L$, $8L$, $10L$, $12L$. Axes are otherwise empty for the student to sketch a curve.]
[Figure 1: A circuit diagram. A battery of emf $\mathcal{E}$ is on the left side of the loop. Along the top wire, moving right from the battery, is a switch, then resistor $R_A$, then resistor $R_B$ (both drawn as zigzag resistor symbols). The right side of the loop contains an inductor of inductance $L$ (drawn as a coil symbol), connecting back down to the battery, completing the rectangular circuit loop.]
Students are asked to determine the resistance $R$ of identical resistors $R_A$ and $R_B$. The resistors are connected in series with each other, 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 increases 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 increases with time.
[Voltmeter Symbol: a circle containing the letter "V", with a wire lead extending from each side, labeled "Voltmeter Symbol".]
Describe a procedure for collecting data that would allow the students to graphically determine the experimental value for $R$ using a measured quantity that increases with time. Provide enough detail so that another student could replicate the experiment.
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: A blank graph with a vertical axis (upward arrow, unlabeled) and a horizontal axis (rightward arrow, unlabeled, drawn dashed). No scale or curve shown; axes are empty for the student to label and sketch.]
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 $R_A$ 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 $R_A$ is $|\Delta V_2|$.
Indicate whether $|\Delta V_2|$ is greater than, less than, or equal to $|\Delta V_1|$.
_____ $|\Delta V_2| > |\Delta V_1|$ _____ $|\Delta V_2| < |\Delta V_1|$ _____ $|\Delta V_2| = |\Delta V_1|$
Justify your answer.
[Figure 1: A diagram on the left shows a square loop of side length $D$ approaching a rectangular field region from the left, with $+x$ to the right and $+y$ upward indicated by a small axis. The loop has a resistor $R$ on its left side and width $D$, moving right with velocity $v$ in the $+x$ direction. To the right is a number line marked $0, D, 2D, 3D, 4D$. Above the number line, two adjacent rectangular field regions are shown: "Region 1" spans from $x=D$ to $x=2D$, containing an array of small circles with dots (field out of the page, magnitude $B$); "Region 2" spans from $x=2D$ to $x=3D$, containing an array of "×" symbols (field into the page, magnitude $2B$). Point S is the leading (right) edge of the loop.]
A wire is connected to a resistor of resistance $R$ to form a rigid square loop of side length $D$. 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 enters Region 1 of external uniform magnetic field of magnitude $B$ that is directed in the $+z$-direction. Region 1 has boundaries $x=D$ and $x=2D$. The loop later enters Region 2 of external uniform magnetic field of magnitude $2B$ that is directed in the $-z$-direction. Region 2 has boundaries $x=2D$ and $x=3.5D$. Point S is the midpoint of the leading edge of the loop.
On the following axes, sketch a graph of the magnetic flux $\Phi$ through the square loop as a function of the position $x$ of Point S from $x=0$ to $x=4.5D$. The $+z$-direction indicated in Figure 1 corresponds to $+\Phi$.
[Blank graph: vertical axis labeled $\Phi$, horizontal axis labeled $x$ with tick marks at $D$, $2D$, $3D$, $4D$, gridlines shown, origin labeled $O$. No curve plotted; axes are empty for the student to sketch.]
Consider the instant when Point S reaches $x=1.5D$.
Indicate whether the current $I_0$ that is induced in the square loop when Point S reaches $x=1.5D$ is clockwise, counterclockwise, or zero.
_____ Clockwise _____ Counterclockwise _____ Zero
Briefly justify your answer.
Derive an expression for $I_0$ when Point S reaches $x=1.5D$. If $I_0=0$, indicate how the derived expression shows that $I_0=0$. Express your answer in terms of $R$, $D$, $v$, $B$, and physical constants, as appropriate.
Derive an expression for the power $P$ dissipated by the resistor when Point S reaches $x=1.5D$. Express your answer in terms of $R$, $D$, $v$, $B$, and physical constants, as appropriate.
The total energy dissipated by the resistor in the square loop as Point S moves from $x=0$ to $x=4.5D$ is $E_{\text{original}}$.
The vertical boundary between regions 1 and 2 is now shifted, so the boundary is at $x=2.5D$. After the boundary is shifted, the square 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 $x=4.5D$ is $E_{\text{new}}$.
[Figure 2: Same layout as Figure 1, but the boundary between Region 1 (dots, field out of page, magnitude $B$) and Region 2 (×'s, field into page, magnitude $2B$) is now at $x=2.5D$; Region 1 spans $x=D$ to $x=2.5D$, Region 2 spans $x=2.5D$ to $x=3.5D$ (approximately). Number line marked $0, D, 2D, 3D, 4D$. Square loop with resistor $R$ and side length $D$ shown at the left, moving in $+x$ direction with speed $v$.]
Indicate whether $E_{\text{new}}$ is greater than, less than, or equal to $E_{\text{original}}$.
_____ $E_{\text{new}} > E_{\text{original}}$ _____ $E_{\text{new}} < E_{\text{original}}$ _____ $E_{\text{new}} = E_{\text{original}}$
Briefly justify your answer.
The original magnetic fields are modified so that the region $D < x < 3.5D$ contains an external uniform magnetic field $B$ 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 $D$ and height $D$, as shown in Figure 3. 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 upper-left corner of the loop.
[Figure 3: A right triangular loop with a resistor $R$ on its vertical left side of height $D$, base length $D$ along the bottom, and hypotenuse sloping from the top-left down to the bottom-right, with Point S marked at the top-left corner. The triangle moves in the $+x$ direction with speed $v$. To the right, a single rectangular field region spans from $x=D$ to $x=3.5D$, filled with dots (field out of the page, magnitude $B$). Number line marked $0, D, 2D, 3D, 4D$, with small axis indicator showing $+x$ and $+z$ directions.]
On the following axes, sketch a graph of the induced current $I_2$ in the loop as Point S moves from $x=D$ to $x=3D$.
[Blank graph: vertical axis labeled $I_2$, horizontal axis with tick marks at $D$, $2D$, $3D$, $4D$; shaded vertical bands appear at the far left (near $0$–$D$) and in the middle region (around $2D$–$3D$) as reference/shading, with the rest of the axes empty for the student to sketch a curve.]