Learn Extracted exam questions AP Physics C: Electricity and Magnetism 2018 Free Response
2018 Free Response
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A solid plastic sphere of radius $a$ and a conducting spherical shell of inner radius $b$ and outer radius $c$ are shown in the figure above. The shell has an unknown charge. The solid plastic sphere has a charge per unit volume given by $\rho(r) = \beta r$, where $\beta$ is a positive constant and $r$ is the distance from the center of the sphere. Express your answers to parts (a), (b), and (c) in terms of $\beta$, $r$, $a$, and physical constants, as appropriate.
[Figure: A cross-sectional diagram showing a solid gray "Plastic Sphere" of radius $a$ at the center, surrounded by a gray shaded ring labeled "Conducting Spherical Shell" with inner radius $b$ and outer radius $c$. Arrows from the center label the radius $a$ to the edge of the plastic sphere, radius $b$ to the inner edge of the shell, and radius $c$ to the outer edge of the shell.]
Consider a Gaussian sphere of radius $r$ concentric with the plastic sphere. Derive an expression for the charge enclosed by the Gaussian sphere for the following region.
$r < a$
Consider a Gaussian sphere of radius $r$ concentric with the plastic sphere. Derive an expression for the charge enclosed by the Gaussian sphere for the following region.
$a < r < b$
Use Gauss's law to derive an expression for the magnitude of the electric field in the following region.
$r < a$
Use Gauss's law to derive an expression for the magnitude of the electric field in the following region.
$a < r < b$
At any point outside of the conducting shell, it is observed that the magnitude of the electric field is zero.
Determine the charge on the inner surface of the conducting shell.
Justify your answer.
Determine the charge on the outer surface of the conducting shell.
On the axes below, sketch the electric field $E$ as a function of distance $r$ from the center of the sphere. Sketch the graph for the range $r = 0$ at the center of the sphere to $r = c$ at the outside of the conducting shell.
[Graph: Axes with vertical axis labeled $E$ and horizontal axis labeled $r$, origin at $O$. Horizontal axis shows tick marks at $a$, $b$, $c$ (with $b$ and $c$ close together near the right side, and a dashed segment continuing past $c$). No data plotted — blank axes for the student to sketch on.]
The figure below shows the sphere and shell with four points labeled W, X, Y, and Z. Point W is at the center of the sphere, point X is on the surface of the sphere, and points Y and Z are on the inner and outer surface of the shell, respectively. Rank the points according to the electric potential at that point, with 1 indicating the largest electric potential. If two points have the same electric potential, give them the same numerical ranking.
_____ W _____ X _____ Y _____ Z
[Figure: A repeat of the cross-sectional diagram (plastic sphere inside conducting spherical shell), with four labeled points marked: W at the center of the sphere, X on the surface of the sphere (just outside W), Y on the inner surface of the shell, and Z on the outer surface of the shell.]
[Figure: A sketch of the experimental setup showing a Multimeter connected by two wires to a parallel-plate arrangement made of Aluminum Foil, with Paper sandwiched between the foil sheets.]
An experiment is designed to measure the dielectric constant of paper that has an area $A = 0.060\text{ m}^2$. Using aluminum foil, two parallel plates are created with the same area as the paper. Five hundred sheets of paper are placed between the aluminum foil plates to create a parallel plate capacitor, as shown in the figure above. Using a multimeter, the capacitance $C$ of the capacitor is measured. The number of sheets and the total thickness $d$ of the stack of paper are recorded. The experiment is repeated, reducing the number of sheets of paper each time. The data are recorded in the table below.
| Sheets of Paper | $d$ (m) | $C$ (F) |
|---|---|---|
| 500 | 0.045 | $6.5 \times 10^{-11}$ |
| 400 | 0.036 | $7.4 \times 10^{-11}$ |
| 300 | 0.027 | $8.9 \times 10^{-11}$ |
| 200 | 0.018 | $11.9 \times 10^{-11}$ |
| 100 | 0.010 | $21.0 \times 10^{-11}$ |
Indicate below which quantities should be graphed to yield a straight line whose slope could be used to calculate a numerical value for the dielectric constant of the paper.
Vertical axis: _____________
Horizontal axis: _____________
Use the remaining columns in the table above, as needed, to record any quantities that you indicated that are not given. Label each column you use and include units.
Plot the data points for the quantities indicated in part (a) on the graph below. Clearly scale and label all axes, including units if appropriate. Draw a straight line that best represents the data.
[Graph: A blank grid (large square grid of fine gridlines with heavier gridlines every 5 squares) for the student to plot data points and draw a best-fit straight line.]
Using the straight line, calculate a dielectric constant for the paper.
[Circuit diagram: A 36 V battery (with $+$ terminal marked) in series with an open switch, connected to a circuit containing three identical $80\ \Omega$ resistors and an $18\text{ nF}$ capacitor. Two $80\ \Omega$ resistors are in the top branch (one in series with the main loop, one in a parallel branch also carrying the label $80\ \Omega$), in series with the $18\text{ nF}$ capacitor; a third $80\ \Omega$ resistor is in the bottom branch of the circuit.]
The student now makes a capacitor using the same aluminum foil plates and just one sheet of paper. Using the experimentally determined dielectric constant, the student calculates the capacitance to be 18 nF. The student uses this uncharged capacitor to build a circuit using wire, a 36 V battery, 3 identical $80\ \Omega$ resistors, and an open switch, as shown in the figure above.
Calculate the current in the battery immediately after the switch is closed.
Determine the time constant for this circuit.
Students A and B measure the time it takes after the switch is closed for the voltage across the capacitor to reach half its maximum value and find that it is longer than expected.
Student A assumes that the capacitance value is correct. Would Student A conclude that the resistance value is larger or smaller than measured?
_____ Larger than measured _____ Smaller than measured
Explain experimentally what could account for this.
Student B assumes that the resistance value is correct. Would Student B conclude that the capacitance value is larger or smaller than measured?
_____ Larger than measured _____ Smaller than measured
Explain experimentally what could account for this.
[Figure 1. Side view: Wire 1 is shown end-on (circle with a dot, $\odot I_1$) with current $I_1$ directed out of the page; point P is a horizontal distance $d$ from Wire 1. Wire 1 is a vertical distance $d$ above Wire 2, which is drawn as a long horizontal wire (hatched double line) carrying current $I_2$ directed to the left.]
[Figure 2. Top view: Wire 1 is drawn as a vertical wire (with dashed extension above), Wire 2 is a horizontal wire to the left of Wire 1 carrying current $I_2$ directed to the left, and point P is a horizontal distance $d$ to the right of Wire 1 along the same horizontal line as Wire 2's continuation. A note states: "Wire 1 and point P are a distance $d$ above Wire 2." The current $I_1$ in Wire 1 is shown directed downward (out of the top-view plane arrangement).]
The figures above represent different views of two long, straight, horizontal wires, 1 and 2, carrying currents $I_1 = I$ and $I_2 = 2I$, respectively, in the directions shown. The wires are held in place. In Figure 1, the current in wire 1 is directed out of the page, and wire 1 is a distance $d$ above wire 2. Point P is a horizontal distance $d$ from wire 1 and a distance $d$ directly above wire 2. Express your answers to parts (a) and (b) in terms of $I$, $d$, and physical constants, as appropriate.
Use Ampere's law to derive an expression for the magnitude of the magnetic field at point P due to wire 1.
Derive an expression for the magnitude of the net magnetic field at point P.
Calculate the numerical value of the angle to the horizontal for the direction of the net magnetic field at point P.
Wire 1 is now released. Which of the following best describes the initial motion of wire 1 due to the magnetic field of wire 2? Assume gravitational effects are negligible.
_____ Wire 1 will not move.
_____ Wire 1 will move upward as viewed in Figure 1.
_____ Wire 1 will move downward as viewed in Figure 1.
_____ Wire 1 will rotate clockwise as viewed in Figure 2.
_____ Wire 1 will rotate counterclockwise as viewed in Figure 2.
Justify your answer.
[Figure 3. Side view: A rectangular loop of length $\ell$ (horizontal) and width $w$ (vertical) is positioned above Wire 2, at a distance $d$ from Wire 2 (measured from the bottom side of the loop down to the wire). Wire 2 (hatched double line) carries current $I_2$ directed to the left. The long side of the loop is parallel to the wire.]
Wire 1 is now replaced by a conducting rectangular loop of length $\ell$, width $w$, and resistance $R$. The loop is placed a distance $d$ from wire 2, as shown. The loop, wire, and distance $d$ are all in the plane of the page. The long side of the loop is parallel to the wire. The current $I_2$ for wire 2 is decreasing linearly as a function of time $t$ according to the equation $I_2 = 2I_0(1-kt)$, where $k$ is a positive constant with units of $s^{-1}$.
Of the following, select the integration that will give an expression for the flux $\Phi$ as a function of time $t$.
_____ $\Phi = \displaystyle\int_{r=d}^{r=d+w} \dfrac{\mu_0 (2I_0)(1-kt)\ell w}{2\pi}\,dr$
_____ $\Phi = \displaystyle\int_{r=d}^{r=w} \dfrac{\mu_0 (2I_0)(1-kt)\ell w}{2\pi}\,dr$
_____ $\Phi = \displaystyle\int_{r=d}^{r=d+w} \dfrac{\mu_0 (2I_0)(1-kt)}{2\pi r}\,\ell\,dr$
_____ $\Phi = \displaystyle\int_{r=d}^{r=w} \dfrac{\mu_0 (2I_0)(1-kt)}{2\pi r}\,\ell\,dr$
Given that the flux through the rectangular loop as a function of time $t$ is given by the equation
derive an expression for the magnitude of the current, if any, induced in the loop. Express your answers in terms of $I_0$, $d$, $r$, $R$, $w$, $k$, $\ell$, and physical constants, as appropriate.
What is the direction of the current, if any, induced in the loop as seen in Figure 3?
_____ Clockwise _____ Counterclockwise _____ Undefined, because there is no current induced in the loop
Justify your answer.