Learn Extracted exam questions AP Physics C: Electricity and Magnetism 2025 Free Response
2025 Free Response
Source PDF on the left, extracted YAML on the right. Compare numbering, marks, options and text.
An isolated, air-filled, charged capacitor consists of two conducting, coaxial, cylindrical shells that each have length $L$. The inner shell has radius $R_1$ and the outer shell has radius $R_2$, as shown in Figure 1, where $R_1 < R_2 \ll L$. The surface charge densities (amounts of charge per unit area) of the inner and outer shells are $+\sigma_1$ and $-\sigma_2$, respectively. The absolute values of the total charges on the shells are equal.
[Figure 1: Two diagrams. "Side View" shows a coaxial cylindrical capacitor of length $L$, with the inner shell labeled $+\sigma_1$ and the outer shell labeled $-\sigma_2$. "Cross-Sectional View" shows two concentric circles, the inner one with radius $R_1$ and the outer one with radius $R_2$. Note: Figures not drawn to scale.]
Using Gauss's law, derive an expression for the magnitude $E$ of the electric field as a function of the radial distance $r$ from the center of the capacitor for the region $R_1 < r < R_2$. Express your answer in terms of $R_1$, $\sigma_1$, $r$, and physical constants, as appropriate.
Derive an expression for the absolute value $|\Delta V|$ of the potential difference between the outer and inner shells in terms of $R_1$, $R_2$, $\sigma_1$, and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.
On the axes shown in Figure 2, sketch a graph of $E$ as a function of $r$ from $r = 0$ to a position that is outside the outer shell.
[Figure 2: A blank set of axes with vertical axis labeled $E$ and horizontal axis labeled $r$, origin $O$. Two vertical dashed reference lines are marked on the $r$-axis at positions labeled $R_1$ and $R_2$.]
A material of dielectric constant $\kappa$ is inserted into the isolated, charged capacitor such that the material fills the region $R_1 < r < R_2$, as shown in Figure 3.
[Figure 3: "Cross-Sectional View" showing two concentric circles, the inner one with radius $R_1$ and the outer one with radius $R_2$; the annular region between them is shaded and labeled with $\kappa$, indicating the dielectric material fills that region.]
Derive an expression for the capacitance $C$ of the capacitor with the material inserted in terms of $L$, $R_1$, $R_2$, $\kappa$, and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.
A rotating, circular, conducting loop of area $A$ and resistance $R$ is in an external uniform magnetic field of magnitude $B$ that is directed in the $-z$-direction. At time $t = 0$, the magnetic field is perpendicular to the plane of the loop, as shown in Figure 1. The loop is rotating with constant angular speed $\omega$ and period $T$ about the dashed line that is along the diameter of the loop. The value of the magnetic flux through the loop as a function of time $t$ is $\Phi = BA\cos(\omega t)$.
[Figure 1: A circular loop shown face-on with uniform magnetic field $B$ (into the page, shown by an array of $\times$ symbols) filling the region around and through the loop. A dashed vertical line along a diameter of the loop is the rotation axis, with angular velocity $\omega$ indicated by a curved arrow at the top. A small 3D axis to the right shows $+y$ up, $+x$ to the right, and $+z$ out of the page (circle with a dot).]
The absolute value of the induced emf in the loop is $|\varepsilon|$. The partially completed bar chart in Figure 2 shows a bar that represents $|\varepsilon|$ at $t = \dfrac{3}{4}T$. In Figure 2, draw bars to represent $|\varepsilon|$ at times $t = 0$, $\dfrac{1}{4}T$, and $\dfrac{1}{2}T$ relative to $|\varepsilon|$ shown at $\dfrac{3}{4}T$. If $|\varepsilon| = 0$, write a "0" in that column.
[Figure 2: A bar chart with vertical axis $|\varepsilon|$ and horizontal axis showing four columns labeled $t=0$, $\frac{1}{4}T$, $\frac{1}{2}T$, $\frac{3}{4}T$. Only the $\frac{3}{4}T$ column already has a bar drawn, reaching to a height roughly midway up the chart; the other three columns are blank for the student to fill in.]
Derive an expression for the maximum induced current in the loop in terms of $A$, $R$, $B$, $\omega$, and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.
On the axes shown in Figure 3, sketch a graph of the instantaneous power $P$ dissipated by the loop as a function of $t$ during the time interval $0 \le t \le T$.
[Figure 3: A blank set of axes with vertical axis labeled $P$ and horizontal axis labeled $t$, origin $O$. Two vertical dashed reference lines are marked on the $t$-axis at positions labeled $\frac{1}{2}T$ and $T$.]
Indicate whether the sketch you drew in part C is or is not consistent with the bars that you drew in part A. Briefly justify your answer by referencing the functional dependence between $P$ and $|\varepsilon|$.
In Experiment 1, students are asked to use a graph to determine the resistivity $\rho_1$ of a circuit element that is connected to a variable power supply, as shown in Figure 1. The circuit element is cylindrical and has uniform resistivity. The students have access to a voltmeter, an ammeter, and a ruler.
[Figure 1: A simple circuit diagram — a rectangular loop containing a variable power supply (battery symbol with an arrow through it, indicating adjustable voltage) at the top, connected by wires down both sides to a cylindrical circuit element (shown as a shaded rectangular bar) at the bottom, labeled "Circuit Element".]
Describe a procedure for collecting data that would allow the students to use a graph to determine $\rho_1$, including any steps necessary to reduce experimental uncertainty.
Describe how the collected data could be graphed and how that graph would be analyzed to determine $\rho_1$.
In Experiment 2, the students are asked to use a graph to determine the resistivity $\rho_2$ of solid, cylindrical resistors made of the same material but of different lengths $L$. The cross-sectional area of each resistor is $5.0 \times 10^{-6}\ \text{m}^2$. The students directly measure the resistance $R$ between the ends of each resistor. Table 1 provides $L$ and $R$ for each resistor.
Table 1
| $L$ (m) | $R$ ($\Omega$) |
|---|---|
| 0.010 | 0.90 |
| 0.020 | 1.6 |
| 0.030 | 2.5 |
| 0.040 | 3.2 |
| 0.050 | 4.0 |
Indicate two quantities, either measured quantities from Table 1 or additional calculated quantities, that could be graphed to produce a straight line that could be used to determine $\rho_2$.
Vertical axis: ______ Horizontal axis: ______
On the grid provided, create a graph of the quantities indicated in part C (i).
- Use Table 2 to record the measured or calculated quantities that you will plot.
- Clearly label the axes, including units as appropriate.
- Plot the points you recorded in Table 2.
[Figure: A blank square grid (graph paper) for plotting the data.]
Draw a best-fit line for the data graphed in part C (ii).
Using the best-fit line that you drew in part C (iii), calculate an experimental value for $\rho_2$.
Long, parallel wires S and T are a distance $2d$ apart. Both wires carry equal currents $I$, but the currents are in opposite directions. Both wires are parallel to the $x$-axis. At the instant shown in Figure 1, Sphere 1 is a distance $d$ above Wire S, Sphere 2 is a distance $d$ below Wire S, and both spheres are moving with speed $v$ in the $+x$-direction. Each sphere has positive charge $+Q$. Gravitational effects are negligible.
[Figure 1: Sphere 1 is shown a distance $d$ above Wire S, moving with velocity $v$ in the $+x$-direction and carrying charge $+Q$. Wire S runs horizontally carrying current $I$ to the right. A distance $d$ below Wire S is Sphere 2, also moving with velocity $v$ in the $+x$-direction and carrying charge $+Q$. A further distance $d$ below Sphere 2 is Wire T, carrying current $I$ to the left. A small 3D axis to the right shows $+y$ up, $+x$ to the right, and $+z$ out of the page (circle with a dot).]
$F_1$ is the magnitude of the magnetic force exerted on Sphere 1 due to the currents in wires S and T. $F_2$ is the magnitude of the magnetic force exerted on Sphere 2 due to the currents in wires S and T.
Indicate whether $F_2$ is greater than, less than, or equal to $F_1$ by writing one of the following.
- $F_2 > F_1$
- $F_2 < F_1$
- $F_2 = F_1$
Justify your answer.
Derive an expression for the magnitude $B_{\text{tot}}$ of the magnetic field at the location of Sphere 2 due to the currents in wires S and T in terms of $d$, $I$, and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.
Later, Wire T carries current $3I$ in the $+x$-direction. At the instant shown in Figure 2, Sphere 2 is a distance $d$ below Wire S and is moving with speed $v$ in the $+x$-direction. $F_{\text{new}}$ is the new magnitude of the magnetic force exerted on Sphere 2 due to the currents in wires S and T.
[Figure 2: Wire S runs horizontally carrying current $I$ to the right. A distance $d$ below Wire S is Sphere 2, moving with velocity $v$ in the $+x$-direction and carrying charge $+Q$. A further distance $d$ below Sphere 2 is Wire T, now carrying current $3I$ to the right (same direction as Wire S). A small 3D axis to the right shows $+y$ up, $+x$ to the right, and $+z$ out of the page (circle with a dot).]
Indicate whether $F_{\text{new}}$ is greater than, less than, or equal to $F_2$ by writing one of the following.
- $F_{\text{new}} > F_2$
- $F_{\text{new}} < F_2$
- $F_{\text{new}} = F_2$
Briefly justify your answer by referencing your derivation in part B.