Learn Extracted exam questions AP Physics C: Electricity and Magnetism 2022 Free Response · Set 2
2022 Free Response · Set 2
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A very long nonconducting cylinder is surrounded by a thin concentric conducting cylindrical shell, as shown in the cutout view. A segment of length $L$ of the inner cylinder has a net charge of $+Q$ uniformly distributed throughout its volume. A segment of length $L$ of the outer shell has a net charge of $+4Q$. The radii of the inner cylinder and outer shell are $R$ and $3R$, respectively, as shown in the cross-section view.
[Cutout View: a long horizontal cylinder (the inner nonconducting cylinder) enclosed by a concentric thin outer cylindrical shell; a segment of length $L$ is marked at the left end of the assembly.] [Cross-Section View: two concentric circles; the inner circle has radius $R$, the outer circle has radius $3R$.] [Note: Figures not drawn to scale.]
Determine the charge on the outer surface of the cylindrical shell within length $L$.
Using Gauss's law, derive an expression for the electric field a distance $r$ from the center of the inner cylinder for $r < R$. Express your answers in terms of $Q$, $R$, $r$, $L$, and physical constants, as appropriate.
The magnitude of the electric field at $r = R$ is $12\ \text{N/C}$. Calculate the value of the electric field at $r = 2R$.
Derive an expression for the absolute value of the potential difference between the surface of the nonconducting cylinder and the inner surface of the cylindrical shell.
On the following axes that include regions I, II, and III, sketch the graph of the electric field $E$ as a function of the distance $r$ from the axis of the inner cylinder.
[Graph axes: vertical axis $E$, horizontal axis $r$; three regions labeled I, II, III are marked above the axis, with vertical dashed reference lines at $r = R$ (boundary between I and II) and $r = 3R$ (boundary between II and III); origin labeled $O$.]
On the following axes that include regions I, II, and III, sketch the graph of the electric potential $V$ as a function of the distance $r$ from the axis of the inner cylinder.
[Graph axes: vertical axis $V$, horizontal axis $r$; three regions labeled I, II, III are marked above the axis, with vertical dashed reference lines at $r = R$ (boundary between I and II) and $r = 3R$ (boundary between II and III); origin labeled $O$.]
A non-ideal capacitor has internal resistance that can be modeled as an ideal capacitor in series with a small resistor of resistance $r_C$. A group of students performs an experiment to determine the internal resistance of a capacitor. A circuit is to be constructed with the following available equipment: a single ideal battery of potential difference $\Delta V_0$, a single ammeter, a single variable resistor of resistance $R$, a single uncharged non-ideal capacitor of capacitance $C$, and one or more switches as needed.
[Circuit symbol key shown: Battery (given potential difference $\Delta V_0$), Ammeter (A), Variable Resistor (resistance $R$), Non-ideal Capacitor (capacitance $C$, drawn as a dashed box containing an ideal capacitor and a small resistor), Switch.]
Using the symbols shown, draw a schematic diagram of a circuit that can charge the capacitor and may also be used to study the current through 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 current $I$ through the capacitor as a function of time $t$ is $I(t) = I_0 e^{\frac{-t}{(R+r_C)C}}$ as the capacitor discharges.
The students determine the time constant $\tau$ for the circuit as a function of the resistance $R$. The students' data are shown in the following graph.
[Graph: vertical axis $\tau$ (s) from 0.0 to 0.8 in increments of 0.2 (gridlines every 0.1); horizontal axis $R$ ($\Omega$) from 0.0 to 3.0 in increments of 1.0 (gridlines every 0.25). Plotted data points approximately at: (0.5, 0.20), (1.0, 0.28), (1.5, 0.36), (2.0, 0.51), (2.5, 0.60).]
Draw the best-fit line for the data.
Using the best-fit line, calculate a value for the internal resistance $r_C$ of the capacitor.
The ammeter is found to be nonideal. Is the actual value for the internal resistance $r_C$ for the capacitor greater than, less than, or equal to the experimental internal resistance of the capacitor calculated in part (c)?
___ Greater than ___ Less than ___ Equal to
Briefly justify your answer using features of the graph in part (c).
The values of the variable resistor in the original experiment ranged from $0.5\ \Omega$ to $2.5\ \Omega$. The experiment is repeated with values ranging from $3.0\ \Omega$ to $6.0\ \Omega$. Would the slope of the best-fit line be more steep, be less steep, or remain unchanged compared to the graph in part (c)?
___ More steep ___ Less steep ___ Remain unchanged
Briefly justify your answer.
[Side View: a solenoid drawn as a coil of wire with current $I(t)$ entering at the left, labeled "Solenoid"; a single circular "Loop of Wire" is shown positioned partway along/around the solenoid's axis, oriented with its plane perpendicular to the solenoid's axis.] [End View: the solenoid seen end-on as a circle, with a smaller concentric circle labeled "Loop of Wire" carrying current $I(t)$ shown with a circular arrow indicating direction.] [Note: Figures not drawn to scale.]
A single loop of wire with resistance $3.0\ \Omega$ and radius $0.10\ \text{m}$ is placed inside a solenoid, with the normal to the loop parallel to the axis of the solenoid. The solenoid has 500 turns, is $0.25\ \text{m}$ long, and is connected to a power supply that is not shown. At time $t = 0$, the power supply is turned on, and the current $I$ in the solenoid as a function of $t$ is given by the equation $I(t) = \beta t$, where $\beta = 5.0\ \text{A/s}$. The direction of the current in the solenoid is clockwise, as shown in the end view.
At time $t = 2.0\ \text{s}$, is the induced current in the loop, as seen from the end view shown, clockwise, counterclockwise, or zero?
___ Clockwise ___ Counterclockwise ___ Zero
Justify your answer.
Calculate the current in the loop of wire at time $t = 2.0\ \text{s}$.
Calculate the total energy dissipated by the loop of wire from time $t = 0$ to time $t = 2.0\ \text{s}$.
A group of students attempts to verify experimentally the calculation of the current from part (b). The current in the inner circular loop at time $t = 2.0\ \text{s}$ is measured to be less than the current calculated in part (b). Which of the following could explain this discrepancy? Select one answer.
___ The experiment did not account for Earth's magnetic field. ___ The plane of the loop is not perpendicular to the axis of the solenoid. ___ The center of the loop is not on the axis of the solenoid. ___ The resistance of the loop is less than the given value. ___ The radius of the loop is actually larger than $0.10\ \text{m}$.
Justify your answer.
The power supply is now turned off. The original loop of wire is then replaced with a second loop made from wire that has the same thickness and is made from the same material as the original loop of wire. The second loop has radius $0.20\ \text{m}$, is placed in the same orientation as the original loop, and fits completely inside the solenoid. The power supply is turned on, and the current $I$ in the solenoid as a function of $t$ is again given by the equation $I(t) = \beta t$, where $\beta = 5.0\ \text{A/s}$. Which of the following expressions correctly indicates the ratio $\dfrac{I_2}{I_1}$, where $I_1$ represents the current induced in the original loop of wire in part (b) and $I_2$ represents the current induced in the second loop of wire?
___ $\dfrac{I_2}{I_1} = 1$ ___ $1 < \dfrac{I_2}{I_1} < 2$ ___ $\dfrac{I_2}{I_1} = 2$ ___ $\dfrac{I_2}{I_1} > 2$
Justify your answer.