Learn Extracted exam questions AP Physics C: Electricity and Magnetism 2015 Free Response
2015 Free Response
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A parallel-plate capacitor is constructed of two parallel metal plates, each with area $A$ and separated by a distance $D$. The plates of the capacitor are each given a charge of magnitude $Q$, as shown in the figure above. Ignore edge effects.
[Figure: Two parallel horizontal plates, separated by distance $D$ (shown with a vertical double arrow between them on the left). The top plate is labeled $+Q$, the bottom plate is labeled $-Q$.]
On the figure above, draw an arrow to indicate the direction of the electric field between the plates.
On the figure above, draw an appropriate Gaussian surface that will be used to derive an expression for the magnitude of the electric field $E$ between the plates.
Using Gauss's law and the Gaussian surface from part (a)-ii, derive an expression for the magnitude of the electric field $E$ between the plates. Express your answer in terms of $A$, $D$, $Q$, and physical constants, as appropriate.
[Figure: A vertical $x$-axis points upward. The bottom plate is at $x = 0$, labeled $-Q$; the top plate is at $x = D$, labeled $+Q$. A dashed vertical center line runs between the plates.]
The space between the plates is now filled with a dielectric material that is engineered so that its dielectric constant varies with the distance from the bottom plate to the top plate, defined by the $x$-axis indicated in the diagram above. As a result, the electric field between the plates is given by $\vec{E} = -\dfrac{Q}{\varepsilon_0 \kappa_0 e^{-x/D} A}\hat{i}$, where $\kappa_0$ is a positive constant. Express all algebraic answers to the remaining parts in terms of $A$, $D$, $Q$, $\kappa_0$, $x$, and physical constants, as appropriate.
Determine an expression for the dielectric constant $\kappa$ as a function of $x$.
Write, but do NOT solve, an equation that could be used to determine the potential difference $V$ between the plates of the capacitor.
Using the equation from part (c)-i, derive an expression for the potential difference $V_D - V_0$, where $V_D$ is the potential of the top plate and $V_0$ is the potential of the bottom plate.
Determine the capacitance of the capacitor.
The energy stored in the capacitor that has a varying dielectric is $U_V$. A second capacitor that has a constant dielectric of value $\kappa_0$ is also given a charge $Q$. The energy stored in the second capacitor is $U_C$. How do the values of $U_V$ and $U_C$ compare?
____ $U_V < U_C$ ____ $U_V > U_C$ ____ $U_V = U_C$
Justify your answer.
[Figure: A circuit diagram. At the top, a voltmeter (circle labeled $V$) is connected across a variable resistor (labeled $R$, drawn as a resistor symbol with a diagonal arrow through it, indicating it is adjustable). Below, in a dashed box, a battery is shown as an internal resistance $r$ (resistor symbol) in series with an emf source $\mathcal{E}$. The battery (dashed box) is connected in series with the variable resistor $R$, and the voltmeter is connected across $R$.]
A student performs an experiment to determine the emf $\mathcal{E}$ and internal resistance $r$ of a given battery. The student connects the battery in series to a variable resistance $R$, with a voltmeter across the variable resistor, as shown in the figure above, and measures the voltmeter reading $V$ as a function of the resistance $R$. The data are shown in the table below.
| Trial # | Resistance ($\Omega$) | Voltage (V) | $1/R$ (1/$\Omega$) | $1/V$ (1/V) |
|---|---|---|---|---|
| 1 | 0.50 | 5.6 | 2.00 | 0.179 |
| 2 | 1.0 | 7.4 | 1.00 | 0.135 |
| 3 | 2.0 | 9.4 | 0.50 | 0.106 |
| 4 | 3.0 | 10.6 | 0.33 | 0.094 |
| 5 | 5.0 | 10.9 | 0.20 | 0.092 |
| 6 | 10 | 11.4 | 0.10 | 0.088 |
Derive an expression for the measured voltage $V$. Express your answer in terms of $R$, $\mathcal{E}$, $r$, and physical constants, as appropriate.
Rewrite your expression from part (a)-i to express $1/V$ as a function of $1/R$.
On the grid below, plot data points for the graph of $1/V$ as a function of $1/R$. Clearly scale and label all axes, including units as appropriate. Draw a straight line that best represents the data.
[Figure: A blank grid (approximately 30 x 30 gridlines with heavier gridlines every 6 squares) for plotting $1/V$ vs. $1/R$.]
Use the straight line from part (b) to obtain a value for $\mathcal{E}$.
Use the straight line from part (b) to obtain a value for $r$.
Using the results of the experiment, calculate the maximum current that the battery can provide.
A voltmeter is to be used to determine the emf of the battery after removing the battery from the circuit. Two voltmeters are available to take this measurement—one with low internal resistance and one with high internal resistance. Indicate which voltmeter will provide the most accurate measurement.
____ The voltmeter with low resistance will provide the most accurate measurement.
____ The voltmeter with high resistance will provide the most accurate measurement.
____ The two voltmeters will provide equal accuracy.
Justify your answer.
[Figure, "Edge View": A point $P$ is shown above a horizontal rod/loop segment (drawn edge-on as a thin shaded bar). Several parallel arrows labeled $\vec{B}$ point up and to the right, representing the magnetic field. A dashed vertical line from $P$ down to the bar makes a $60°$ angle with the direction of $\vec{B}$.]
[Figure, "Perspective View": A point $P$ is shown above a circular loop (drawn as a tilted ellipse/ring to indicate perspective), with the same parallel arrows labeled $\vec{B}$ pointing up and to the right through the loop, and a dashed vertical line from $P$ down to the loop.]
A circular wire loop with radius 0.10 m and resistance 50 $\Omega$ is held in place horizontally in a magnetic field $\vec{B}$ directed upward at an angle of $60°$ with the vertical, as shown in the figure above. The magnetic field in the direction shown is given as a function of time $t$ by $B(t) = a(1-bt)$, where $a = 4.0$ T and $b = 0.20\ \text{s}^{-1}$.
Derive an expression for the magnetic flux through the loop as a function of time $t$.
Calculate the numerical value of the induced emf in the loop.
Calculate the numerical value of the induced current in the loop.
What is the direction of the induced current in the loop as viewed from point $P$?
____ Clockwise ____ Counterclockwise
Justify your answer.
Assuming the loop stays in its current position, calculate the energy dissipated in the loop in 4.0 seconds.
Indicate whether the net magnetic force and net magnetic torque on the loop are zero or nonzero while the loop is in the magnetic field.
Net magnetic force: ____ Zero ____ Nonzero
Net magnetic torque: ____ Zero ____ Nonzero
Justify both of your answers.