Learn Extracted exam questions AP Physics C: Electricity and Magnetism 2026 Free Response
2026 Free Response
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Two very long, cylindrical, current-carrying Wires S and T are oriented parallel to the $y$-axis, as shown in Figure 1. The centers of the wires are in the $xy$-plane. The wires are described as follows.
- Wire S has radius $2a$ and is centered at $x = 0$. Wire S has nonuniform current density and carries total current $I_0$ in the $+y$-direction. The magnitude of the current density in Wire S as a function of the radial distance $r$ from the center of the wire is described by $J = Cr$, where $C$ is a positive constant.
- Wire T has radius $a$ and is centered at $x = 10a$. Wire T has uniform current density and carries total current $I_0$ in the $-y$-direction.
[Figure 1: Cross-section diagram in the $xy$-plane. Wire S is a shaded cylindrical bar centered at $x=0$ spanning from $-2a$ to $2a$, with a $y$-axis arrow through its center pointing up (current $I_0$ in $+y$-direction) and a label $J = Cr$ pointing to the wire. Point P is marked inside Wire S, above the center axis, at horizontal position $x = a$. Wire T is a shaded cylindrical bar centered at $x = 10a$ spanning from $9a$ to $11a$, with a downward arrow through its center (current $I_0$ in $-y$-direction). The $x$-axis is marked with gridlines at $-2a, 0, 2a, 4a, 6a, 8a, 10a$.]
Point P is located inside of Wire S at horizontal position $x = a$. Derive an expression for the magnitude of the magnetic field at Point P due only to the current in Wire S in terms of $C$, $a$, 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 the magnitude $B$ of the magnetic field due to the currents in both wires as a function of $x$ along the $x$-axis in the region $2a \le x \le 9a$. Sketches made in the shaded region will not be graded.
[Figure 2: A blank graph grid with vertical axis $B$ and horizontal axis $x$, gridlines marked at $O, 2a, 3a, 4a, 5a, 6a, 7a, 8a, 9a$. The region from $O$ to $2a$ is shaded (not to be used); the region from $2a$ to $9a$ is the graded sketch area.]
At the instant shown in Figure 3, a small sphere with charge $Q$ is at horizontal position $x = 5a$ and moving in the $xy$-plane with speed $v$ in the $+x$-direction. At this instant, the net magnetic force exerted on the sphere due to the currents in Wires S and T has magnitude $F$.
[Figure 3: Same wire configuration as Figure 1 (Wire S centered at $x=0$ with current $I_0$ in $+y$-direction, Wire T centered at $x=10a$ with current $I_0$ in $-y$-direction). A small circle labeled $Q$ is shown at $x = 5a$, midway between the wires, with an arrow labeled $v$ pointing in the $+x$-direction.]
Derive an expression for $F$ in terms of $a$, $Q$, $I_0$, $v$, and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.
Thin, nonconducting, charged Rods S and T are held fixed in the $xy$-plane, as shown in Figure 1. Rod S is oriented along the $x$-axis in the region $-3L \le x \le -L$. Rod T is oriented along the $y$-axis in the region $L \le y \le 3L$. Each rod has uniform, positive linear charge density $+\lambda$. Point P is located at $(-2L, 2L)$.
[Figure 1: In the $xy$-plane, Rod S is a thick horizontal segment on the $x$-axis from $x=-3L$ to $x=-L$, labeled "Rod S" with charge density $+\lambda$. Rod T is a thick vertical segment on the $y$-axis from $y=L$ to $y=3L$, labeled "Rod T" with charge density $+\lambda$. Point P is marked with a dot at $(-2L, 2L)$. Dashed gridlines mark $x = -3L, -2L, -L, O, L$ and $y = L, 2L, 3L$.]
Complete the following tasks in Figure 2.
- Draw an arrow starting on the dot that indicates the direction of the net electric field $\vec{E}_P$ at Point P due to Rods S and T.
- Draw an arrow starting on the dot that indicates the direction of the net electric field $\vec{E}_O$ at the origin $O$ due to Rods S and T.
- Draw an arrow starting on the dot that indicates the direction of the net acceleration $\vec{a}_O$ of a small, negatively charged sphere (not shown) immediately after the sphere is released from rest at the origin $O$.
[Figure 2: Three separate blank coordinate-axis boxes, each showing a dashed horizontal and vertical axis crossing at a labeled dot: the first labeled "Direction of $\vec{E}_P$" with the dot labeled P; the second labeled "Direction of $\vec{E}_O$" with the dot labeled O; the third labeled "Direction of $\vec{a}_O$" with the dot labeled O. Students draw an arrow from each dot in the appropriate direction.]
Derive an expression for the magnitude $E_O$ of the net electric field at the origin $O$ due to Rods S and T in terms of $L$, $\lambda$, and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.
Rod T is removed. Rod S is then moved such that the right end of Rod S is at the origin $O$. Rod S is then moved with speed $v$ in the $-x$-direction, as shown in Figure 3.
[Figure 3: Rod S is a thick horizontal segment lying along the negative $x$-axis with its right end at the origin $O$, labeled with charge density $+\lambda$ and moving with speed $v$ in the $-x$-direction (arrow pointing left below the rod). The $x$-axis extends in both directions from $O$ with a dashed line.]
On the axes shown in Figure 4, sketch a graph of the electric potential $V_O$ at the origin $O$ due to Rod S as a function of the distance $r$ between the origin and the right end of Rod S.
[Figure 4: A blank graph with vertical axis $V_O$ and horizontal axis $r$, both starting from the origin; axes shown with no scale marked, for the student to sketch a curve on.]
A new thin, nonconducting, charged Rod W has the same length as Rod S. Rod W has uniform, negative linear charge density $-\lambda$ and is oriented along the $y$-axis. Rods S and W are initially held fixed with one end at the origin. Starting at the same time, Rod S is moved with speed $v$ in the $-x$-direction and Rod W is moved with speed $v$ in the $+y$-direction, as shown in Figure 5.
[Figure 5: Rod S is a thick horizontal segment along the negative $x$-axis with its right end at the origin $O$, labeled $+\lambda$, moving with speed $v$ in the $-x$-direction. Rod W is a thick vertical segment along the positive $y$-axis with its bottom end at the origin $O$, labeled $-\lambda$, moving with speed $v$ in the $+y$-direction.]
Indicate how a sketch of $V_O$ at the origin $O$ due to both Rods S and W as a function of $r$ would be different from the sketch made in part C. Briefly justify your answer.
The following information applies to parts A and B.
In Experiment 1, students are asked to collect data that can be graphed to determine the inductance $L_1$ of an inductor. The students have access to multiple charged capacitors, each of a different known capacitance, and a voltmeter that measures potential difference as a function of time. In each trial, the inductor is connected to a charged capacitor and an initially open switch, as shown in Figure 1.
[Figure 1: A circuit diagram showing a charged capacitor (parallel-plate symbol) on the left connected via a switch (initially open, shown as a break with two contact dots) to an inductor $L_1$ (coil symbol) on the right, forming a closed loop.]
The following information applies to parts C and D.
In Experiment 2, the students are asked to use a graph to determine the inductance $L_2$ of a new inductor. The inductor is connected to a 12 V battery, a $10\ \Omega$ resistor, and an initially open switch, as shown in Figure 2.
[Figure 2: A circuit diagram showing a 12 V battery on the left connected via a switch (initially open) at the top to a loop containing an inductor $L_2$ (coil symbol) on the right side and a $10\ \Omega$ resistor along the bottom, all forming a single closed loop.]
After the switch is closed, the current $I$ in the circuit is measured at different times and the corresponding rate of change of the current, $\dfrac{dI}{dt}$, is determined. The collected data are shown in Table 1.
Table 1
| $I$ (A) | $\dfrac{dI}{dt}$ (A/s) |
|---|---|
| 0.40 | 280 |
| 0.50 | 230 |
| 0.75 | 150 |
| 0.90 | 100 |
| 1.00 | 68 |
Indicate quantities that could be measured by the students that would allow them to determine $L_1$ using a linear graph.
Briefly describe a method to reduce experimental uncertainty.
Indicate what quantities the students could graph on the horizontal and vertical axes to create a linear graph that can be used to determine $L_1$.
Briefly describe the relationship between $L_1$ and a feature of the graph from part B(i). Your answer may include an equation that relates $L_1$ and the chosen feature of the graph.
Label the axes of the grid provided with measured or calculated quantities. Include units, as appropriate. The graphed quantities should yield a linear graph that can be used to determine $L_2$.
On the grid provided, create a graph of the quantities indicated in part C(i).
- Clearly label each axis with a numerical scale.
- Plot the corresponding data points on the grid.
- Table 2 is provided in your booklet for scratch work and will not be scored.
[A blank labeled grid is provided for the student to plot the graph, with blank axis-label lines above/below reading "Quantity (units, if appropriate)" for both the horizontal and vertical axes.]
Draw a best-fit line for the data plotted in part C(ii).
Using the best-fit line that you drew in part C(iii), calculate an experimental value for $L_2$.
In Scenario 1, a square conducting loop with side length $s$ has total resistance $R$. The loop is held fixed in the $xy$-plane in an external, uniform, time-varying magnetic field that is initially directed in the $+z$-direction, as shown. The $z$-component of the magnetic field changes as a function of time $t$ as described by $B_z = B_0 \cos\!\left(2\pi \dfrac{t}{t_1}\right)$, where $B_0$ and $t_1$ are positive constants.
[Figure: A grid of dots (representing a uniform magnetic field $B_z$ pointing out of the page, in the $+z$-direction) fills the region. A square conducting loop of side length $s$ and total resistance $R$ is drawn within the field region, with $R$ labeled pointing to the loop and $s$ labeling the loop's side length via a vertical double-arrow. A small 3D axis indicator to the right shows $+y$ pointing up, $+x$ pointing right, and $+z$ pointing out of the page (circled dot).]
Indicate whether the induced current $I$ in the loop is clockwise, counterclockwise, or zero during the time interval $0 < t < \dfrac{t_1}{4}$ by writing one of the following.
- $I$ is clockwise.
- $I$ is counterclockwise.
- $I$ is zero.
Justify your answer without only manipulating equations.
Derive an expression for $I$ as a function of $t$ in terms of $s$, $R$, $B_0$, $t_1$, $t$, and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.
In Scenario 2, a new square conducting loop with side length $2s$ and total resistance $2R$ is placed in the same external, uniform, time-varying magnetic field as in Scenario 1.
Indicate whether the maximum induced current $I_2$ in Scenario 2 is greater than, less than, or equal to the maximum induced current $I_1$ in Scenario 1 by writing one of the following.
- $I_2 > I_1$
- $I_2 < I_1$
- $I_2 = I_1$
Briefly justify your answer by referencing the expression you derived in part B.