Learn Extracted exam questions AP Physics C: Electricity and Magnetism 2022 Free Response · Set 1
2022 Free Response · Set 1
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A nonconducting sphere of uniform volume charge density is surrounded by a thin concentric conducting spherical shell, as shown in the cutout view. The sphere has a charge of $-Q$ and the shell has a charge of $+3Q$. The radii of the inner sphere and spherical shell are $R$ and $4R$, respectively, as shown in the cross-section view.
[Two figures side by side, captioned "Note: Figures not drawn to scale." Left figure, labeled "Cutout View": a large shaded circular shell with a wedge cut out to reveal a smaller shaded sphere at the center. Right figure, labeled "Cross-Section View": a large circle (the shell) with a smaller circle at the center labeled $R$, and an arrow from the center to the outer circle labeled $4R$.]
Determine the charge on the outer surface of the shell.
Using Gauss's law, derive an expression for the electric field a distance $r$ from the center of the sphere for $r < R$. Express your answer in terms of $Q$, $R$, $r$, and physical constants, as appropriate.
The magnitude of the electric field at $r = R$ is $8\ \text{N/C}$. Calculate the value of the electric field at $r = 2R$.
Derive an expression for the absolute value of the potential difference between the outer surface of the sphere and the inner surface of the shell. Express your answer in terms of $Q$, $R$, and physical constants, as appropriate.
On the following axes that include regions I, II, and III, sketch a graph of the electric field $E$ as a function of the distance $r$ from the center of the sphere.
[Graph axes: horizontal axis labeled $r$, with vertical dashed reference lines at $r = R$ and $r = 4R$ dividing the plot into three labeled regions I, II, III (left to right). Vertical axis labeled $E$, origin marked $O$. Axes are blank, to be sketched on by the student.]
On the following axes that include regions I, II, and III, sketch a graph of the electric potential $V$ as a function of the distance $r$ from the center of the sphere.
[Graph axes: horizontal axis labeled $r$, with vertical dashed reference lines at $r = R$ and $r = 4R$ dividing the plot into three labeled regions I, II, III (left to right). Vertical axis labeled $V$, origin marked $O$. Axes are blank, to be sketched on by the student.]
The plates of a certain variable capacitor have an adjustable area. An experiment is performed to study the potential difference across the capacitor as it discharges through a resistor. A circuit is to be constructed with the following available equipment: a single ideal battery of potential difference $\Delta V_0$, a single voltmeter, a single resistor of resistance $R$, a single uncharged variable capacitor set to capacitance $C$, and one or more switches as needed.
[Diagram of circuit symbols with labels above each: "Battery" shown as a battery symbol; "Voltmeter" shown as a circle with $V$ inside; "Resistor" shown as a zigzag resistor symbol; "Variable Capacitor" shown as a capacitor symbol with an arrow through it; "Switch" shown as a switch symbol.]
Using the symbols shown, draw a schematic diagram of a circuit that can charge the capacitor and may also be used to study the potential difference across the capacitor as it discharges through the resistor.
The capacitor is fully charged by the battery. At time $t = 0$, the capacitor starts discharging through the resistor.
Show that the potential difference $\Delta V_C$ across the capacitor as a function of time $t$ is $\Delta V_C(t) = \Delta V_0 e^{-\tfrac{t}{RC}}$ as the capacitor discharges.
The experiment is performed using a resistor of $R = 150\ \text{k}\Omega$. Data for the potential difference $\Delta V_C$ across the capacitor as a function of $t$ are recorded and a plot of $\ln\!\left(\dfrac{\Delta V_C}{\Delta V_0}\right)$ as a function of $t$ is created on the graph below.
[Scatter plot on graph paper. Vertical axis labeled $\ln\!\left(\dfrac{\Delta V_C}{\Delta V_0}\right)$, ranging from $0.0$ to $-0.8$ in increments of $0.2$. Horizontal axis labeled $t\ (\text{s})$, ranging from $0.0$ to $0.6$ in increments of $0.1$. Data points approximately at: $(0.1, -0.10)$, $(0.2, -0.27)$, $(0.3, -0.38)$, $(0.4, -0.45)$, $(0.5, -0.63)$, $(0.6, -0.72)$.]
Draw the best-fit line for the data.
Using the best-fit line, calculate a value for the unknown capacitance $C$.
The capacitor is adjusted so that the surface area of the plates is increased, and the experiment is repeated. Would the slope of the best-fit line in the second experiment be more steep, less steep, or unchanged compared to the slope of the best-fit line in part (c)?
___ More steep ___ Less steep ___ Unchanged
Briefly justify your answer.
The ideal battery is then replaced with a non-ideal battery with internal resistance $r$, and the experiment is repeated.
Would the slope of the graph in this final experiment change compared to the graph in part (c)?
___ Yes ___ No
Briefly justify your answer.
Would the vertical intercept of the graph in this final experiment change compared to the graph in part (c)?
___ Yes ___ No
Briefly justify your answer.
A lightbulb of resistance $R = 10.0\ \Omega$ is connected to a rectangular loop of wire of negligible resistance near a very long current-carrying wire. The rectangular loop has a length $L = 4.0\ \text{cm}$ and a width $W = 2.0\ \text{cm}$ and is positioned so one of the longer sides of the loop is a distance $d = 1.0\ \text{cm}$ above and parallel to the long wire, as shown. The current in the long wire is initially flowing to the right and is given by $I(t) = C - Dt$, where $C = 10.0\ \text{A}$ and $D = 2.0\ \text{A/s}$. At time $t = 5.0\ \text{s}$, the current in the long wire is instantaneously zero as the current changes direction.
[Diagram: a rectangular loop of wire with a lightbulb (circle with X inside) in the top wire of the loop, labeled $R = 10.0\ \Omega$. The loop has width $W = 2.0\ \text{cm}$ (vertical side) and length $L = 4.0\ \text{cm}$ (horizontal side). Below the loop, a long horizontal wire carries current $I(t)$ to the right (arrow labeled $I(t)$ at the right end), separated from the bottom of the loop by a distance $d = 1.0\ \text{cm}$.]
What is the direction, if any, of the magnetic field produced by the induced current in the rectangular loop as the current in the long wire changes direction?
___ Into the page ___ Out of the page ___ No direction, because the field is zero
Justify your answer.
Calculate the magnetic flux through the loop due to only the long wire at time $t = 3.0\ \text{s}$.
Calculate the current through the lightbulb at time $t = 3.0\ \text{s}$.
A group of students attempts to experimentally verify whether the current through the lightbulb is consistent with the current calculation from part (c). The current in the rectangular loop is measured to be greater than the current calculated in part (c). Which of the following could explain this discrepancy? Select one answer.
___ The students did not account for Earth's magnetic field. ___ The rectangular loop is tilted and is not in the same plane as the wire. ___ The resistance of the lightbulb is greater than the recorded value. ___ The long side of the rectangular loop is shorter than the recorded value. ___ The current in the long wire changes at a faster rate than expected.
Briefly justify your answer.
Later, the same rectangular loop with lightbulb is rotated such that a short side of the loop is $1.0\ \text{cm}$ above and parallel to the long current-carrying wire, as shown. The current in the wire is again initially flowing from left to right and given by $I(t) = C - Dt$, where $C = 10.0\ \text{A}$ and $D = 2.0\ \text{A/s}$. The current through the lightbulb in the loop's new orientation at time $t = 3.0\ \text{s}$ is $I_2$. Which of the following correctly relates the $I_2$ to $I_1$, the current through the lightbulb in part (c)?
[Diagram: a tall rectangular loop of wire oriented vertically, with a lightbulb (circle with X inside) partway up the loop. Below the loop, a long horizontal wire carries current to the right (arrow labeled $I$ at the right end), with the short side of the loop positioned a distance $d = 1.0\ \text{cm}$ above and parallel to the wire.]
___ $I_2 < I_1$ ___ $I_2 = I_1$ ___ $I_2 > I_1$
Justify your answer.