Learn Extracted exam questions AP Physics C: Electricity and Magnetism 2021 Free Response
2021 Free Response
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[Circuit diagram: An ideal 10 V battery (+ terminal on top) connects through a switch $S$ in series with resistor $R_1 = 100\ \Omega$. The circuit then splits into three parallel branches between the top and bottom rails: a branch with resistor $R_2 = 30\ \Omega$ (in series with switch $S$, both shown on the left branch), a branch with resistor $R_3 = 60\ \Omega$, a branch with two capacitors $C_1 = 10\ \mu\text{F}$ and $C_2 = 15\ \mu\text{F}$ stacked in series with each other, and a branch with capacitor $C_3 = 20\ \mu\text{F}$.]
The circuit shown above is composed of an ideal 10 V battery, three resistors and three capacitors with the values shown, and an open switch $S$. The capacitors are initially uncharged. Switch $S$ is now closed.
Calculate the current through $R_1$ immediately after switch $S$ is closed.
Switch $S$ has been closed for a long time, and the circuit has reached a steady state.
Calculate the potential difference across $R_1$.
Calculate the charge stored on the positive plate of capacitor $C_2$.
Is the charge stored on capacitor $C_3$ greater than, less than, or equal to the charge stored on capacitor $C_2$?
_____ Greater than _____ Less than _____ Equal to
Justify your answer.
Switch $S$ is then opened.
Determine the current through $R_1$ immediately after the switch is opened.
Calculate the current through $R_2$ immediately after the switch is opened.
On the axes below, sketch a graph of the potential difference $V$ across capacitor $C_2$ as a function of time $t$ if switch $S$ is opened at time $t = 0$. Label the maximum value.
[Graph axes: vertical axis $V$, horizontal axis $t$; origin labeled $O$; a dashed vertical guideline is shown near the vertical axis, and a dashed horizontal guideline extends across the plot area; no data plotted, blank axes for the student to sketch on.]
Capacitor $C_3$ is replaced by two $10\ \mu\text{F}$ capacitors connected in series, switch $S$ is closed, and the circuit reaches equilibrium. Switch $S$ is then opened at time $t = 0$.
For $t > 0$, would the sketch of a graph of the new voltage across $C_2$ as a function of time be above, below, or the same as the sketch for part (e)?
_____ Above _____ Below _____ The same
Justify your answer.
[Apparatus diagram: A fixed support holds an insulating string from which a Conducting Sphere (upper) hangs, labeled "Insulating String" at top. Directly below it, separated by a distance $d$, is a Conducting Sphere (lower) mounted on an "Insulating Rod," which rests on an "Electronic Balance" (drawn as a rectangular box). The two spheres are identical conducting spheres, vertically aligned with separation $d$ between their centers.]
Students perform an experiment to study the force between two charged objects using the apparatus shown above, which contains two identical conducting spheres. The upper sphere is attached to an insulating string, which can be used to move the sphere downward. The lower sphere sits on an insulating rod, which is on an electronic balance. The electronic balance is zeroed before the lower sphere and insulating rod are in place.
For the first trial, a charge of $Q$ is placed on each sphere and then the upper sphere is slowly moved downward. The students measure the distance $d$ between the centers of the spheres and the magnitude $F$ of the force that appears on the electronic balance. The recorded data are shown on the graph of $F$ as a function of $\dfrac{1}{d^2}$ shown below.
[Graph: vertical axis $F\ (\mu\text{N})$ ranging from 9000 to 13,000 in increments of 1000; horizontal axis $\dfrac{1}{d^2}\left(\dfrac{1}{\text{m}^2}\right)$ ranging from 0 to 100 in increments of 20; plotted data points (approximate): $(5, 9200)$, $(30, 9700)$, $(45, 10,300)$, $(65, 11,400)$, $(90, 12,900)$; no line of best fit drawn yet.]
Draw a line that represents the best fit to the points shown.
Use the graph to calculate the charge $Q$.
[Second, extended graph: vertical axis $F\ (\mu\text{N})$ ranging from 9000 to 14,000 in increments of 1000; horizontal axis $\dfrac{1}{d^2}\left(\dfrac{1}{\text{m}^2}\right)$ ranging from 0 to 160 in increments of 20; a dashed best-fit line/curve is drawn through data points from the origin region up through about $(140, 13,700)$; plotted data points (approximate): $(10, 9100)$, $(30, 9700)$, $(50, 10,000)$, $(65, 11,500)$, $(90, 12,700)$, $(105, 13,000)$, $(120, 13,300)$, $(140, 13,700)$, $(155, 13,800)$.]
On the graph on the previous page, draw a circle around the data point that was taken when the distance between the centers of the spheres was the least.
Determine the distance between the centers of the spheres for the data point indicated above.
What physical quantity does the vertical intercept represent?
Justify your answer.
The experiment is extended by collecting additional data points, which appear on the right side of the graph shown above. The new data points do not follow the linear pattern seen with the first points. The group of students tries to explain this discrepancy.
One student suspects that charge is slowly leaking off the top sphere. Could this explain the discrepancy?
_____ Yes _____ No
Justify your answer.
A second student suspects that the excess charges have rearranged themselves, polarizing the spheres.
On the circles representing the spheres below, use a single "+" sign on each sphere to represent the locations of highest concentration of the excess positive charges.
[Two blank circles are shown, stacked vertically, representing the upper and lower spheres, for the student to mark.]
Explain how this rearrangement could be responsible for the discrepancy.
A third student suggests that the experiment be modified so that the top sphere is given a negative charge that is equal in magnitude to the positive charge given to the bottom sphere.
On the circles representing the spheres below, use a single "+" sign on the bottom sphere to represent the location of highest concentration of the excess positive charges. Use a single "−" sign on the top sphere to represent the location of the highest concentration of the excess negative charges.
[Two blank circles are shown, stacked vertically, representing the upper (top) and lower (bottom) spheres, for the student to mark.]
For a separation distance equal to that of the data point indicated in part (a)(iii), would the magnitude of the force reading with spheres of opposite charges be greater than, less than, or equal to the magnitude of the force reading with spheres of the same charges?
_____ Greater than _____ Less than _____ Equal to
Justify your answer.
[Diagram: A thin conducting ring, shown as an ellipse (viewed at an angle) with "Area of Ring = $A$" and "Resistance of Ring = $R$" labeled above it. Horizontal field lines with arrowheads pass through and around the ring from left to right, representing a uniform magnetic field $\vec{B}(t)$ directed to the right and perpendicular to the plane of the ring, with $\vec{B}(t)$ labeled to the right of the ring.]
A thin, conducting ring of area $A$ and resistance $R$ is aligned in a uniform magnetic field directed to the right and perpendicular to the plane of the ring, as shown. At time $t = 0$, the magnitude of the magnetic field is $B_0$. At $t = 1$ s, the magnitude of the magnetic field begins to decrease according to the equation $B(t) = \dfrac{\beta}{t}$, where $\beta$ has units of T·s.
Derive an equation for the magnitude of the induced current $I$ in the ring as a function of $t$ for $t > 1$. Express your answer in terms of $\beta$, $A$, $R$, $t$, and physical constants, as appropriate.
Assume $A = 0.50\ \text{m}^2$, $R = 2.0\ \Omega$, and $\beta = 0.50\ \text{T·s}$.
Calculate the electrical energy dissipated in the ring from $t = 1$ s to $t = 2$ s.
[Diagram: The same ring, now tilted so its plane makes a $30°$ angle with the horizontal field lines, with the angle labeled "$30°$" at the top between the ring's plane and the field direction. Horizontal field lines with arrowheads pass through and around the tilted ring.]
The ring is then rotated so that the plane of the ring is aligned at a $30°$ angle to the magnetic field, as shown.
The magnitude of the magnetic field is reset to a magnitude of $B_0$ at a new time $t = 0$ and again begins to decrease at $t = 1$ s according to the equation $B(t) = \dfrac{\beta}{t}$, where $\beta$ has units of T·s.
Will the amount of energy dissipated in the ring from $t = 1$ s to $t = 2$ s be greater than, less than, or equal to the energy dissipated in part (b)?
_____ Greater than _____ Less than _____ Equal to
Justify your answer.
[Diagram: The ring (labeled $A = 0.50\ \text{m}^2$, Resistance = $2.00\ \Omega$, $B_0 = 0.50\ \text{T}$) is mounted on an "Axle" (shown as a dashed line through its center) perpendicular to the horizontal magnetic field lines, which pass through and around the ring.]
[Graph: $\mathcal{E}$ (V) on the vertical axis vs. $t$ (s) on the horizontal axis, ranging from 0 to 3; a sinusoidal curve starting at $\mathcal{E} = 0$ at $t = 0$, rising to a positive peak, crossing zero, dipping to a negative trough, and repeating, completing roughly one and a half full cycles by $t = 3$ s.]
The ring is now mounted on an axle that is perpendicular to the magnetic field. The magnitude of the magnetic field is now held at a constant $B_0 = 0.50$ T, as shown. The ring rotates about the axle, and the emf $\mathcal{E}$ induced in the ring as a function of time $t$ is shown on the graph.
Calculate the angular speed $\omega$ of the rotating ring in rad/s.
Calculate the magnitude of the maximum emf $\mathcal{E}_{\text{MAX}}$ induced in the ring.
The ring now begins to rotate at an angular speed $2\omega$.
[Graph: $\mathcal{E}$ (V) on the vertical axis vs. $t$ (s) on the horizontal axis, ranging from 0 to 3; a dashed sinusoidal curve (the original emf from part (d)/(e)) is shown, starting at $\mathcal{E} = 0$ at $t = 0$, completing roughly one and a half cycles by $t = 3$ s; blank space for the student to draw a new solid curve.]
On the graph below, draw a curve to indicate the new induced emf $\mathcal{E}$ in the ring. The dashed curve shows the emf induced under the original conditions.
Justify your sketch, specifically identifying and addressing any similarities or differences between the sketch and the original graph.
[A practice graph with the same axes and the same dashed sinusoidal curve is also provided beneath, labeled "PRACTICE GRAPH - Use the graph below to practice your sketch for part (f). Any work shown on the graph below will NOT be graded."]