Learn Extracted exam questions AP Physics C: Electricity and Magnetism 2017 Free Response
2017 Free Response
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A very large nonconducting slab with a uniform positive volume charge density $\rho_0$ is fixed with the origin of the $xyz$-axes at its center, as shown in the figure above. The thickness of the slab is $d$, the length is $L$, and the width is $W$, where $L \gg d$ and $W \gg d$. The large faces of the slab are parallel to the $xy$-plane.
Consider a Gaussian cylinder with a cross-sectional area $A$ and height $h$ that is positioned with its axis along the $z$-axis, as shown in the figure below.
[Perspective view and side view of the nonconducting slab (thickness $d$, length $L$, width $W$) centered at the origin, with a Gaussian cylinder of cross-sectional area $A$ and height $h$ positioned along the $z$-axis through the slab. In the side view, the slab extends from $z = -d/2$ to $z = +d/2$; the Gaussian cylinder has its top circular cap labeled $R$ at height $h/2$ above the $z=0$ plane and its bottom circular cap labeled $S$ at height $h/2$ below the $z=0$ plane, so the cylinder has total height $h$ centered on $z=0$.]
Draw a single vector on each of the dots below representing the direction of the electric field at the given points. If the electric field at either point is zero, write "$E = 0$" next to the point.
[Two blank diagrams, each showing a single dot: the first labeled i. with a dot marked $\bullet R$, the second labeled ii. with a dot marked $\bullet S$. The student is to draw a field-direction vector at each dot or write $E=0$.]
Use Gauss's law to derive expressions for the following. Express your answers in terms of $\rho_0$, $A$, $d$, $h$, $z$, and physical constants, as appropriate.
Derive an expression for the total flux $\Phi$ through the Gaussian surface shown.
Derive an expression for the magnitude of the electric field as a function of $z$ for any position inside the slab, and show that it is equal to $E = \dfrac{\rho_0 z}{\varepsilon_0}$.
[Figure: two large horizontal metal plates separated by a distance of 0.010 m, approximately the thickness of the slab, with the charged nonconducting slab centered between them (not touching either plate); $z = 0$ marks the mid-plane. The top plate carries surface charge density $\sigma = -2.0 \times 10^{-6}\ \text{C/m}^2$; the bottom plate carries surface charge density $\sigma = +2.0 \times 10^{-6}\ \text{C/m}^2$.]
The charged slab is now placed between two large metal plates separated by a distance of 0.010 m, which is approximately the thickness of the slab, but the slab does not contact either metal plate. The metal plates are charged, resulting in the surface charge densities $\sigma = \pm 2.0 \times 10^{-6}\ \text{C/m}^2$, as shown in the figure above. Assume the charge distribution inside the slab remains unchanged by the presence of the charged plates and that the slab's volume charge density is $\rho_0 = 1.00 \times 10^{-3}\ \text{C/m}^3$.
The magnitude of the electric field inside the slab is zero on the $z$-axis at position $z_0$. Which of the following correctly indicates the value for $z_0$?
____ $z_0 > 0$ ____ $z_0 = 0$ ____ $z_0 < 0$
Justify your answer.
Calculate the value $z_0$.
Calculate the magnitude of the electric potential difference from the center of the slab to the top of the slab.
[Circuit diagram: an ideal battery of voltage $V_0$ is connected across a parallel combination of two branches, with an open switch $S$ at the top of the circuit. One branch contains capacitor $C_0$ in series/parallel with resistor $R_1$ (capacitor $C_0$ on top, resistor $R_1$ below it, forming a middle branch), and a third branch on the right contains resistor $R_2$. The switch $S$ connects the battery/$C_0$-$R_1$ branch node to the top rail that also connects to $R_2$.]
In the circuit above, an ideal battery of voltage $V_0$ is connected to a capacitor with capacitance $C_0$ and resistors with resistances $R_1$ and $R_2$, with $R_1 > R_2$. The switch $S$ is open, and the capacitor is initially uncharged.
The switch is closed at time $t = 0$. On the axes below, sketch the charge $q$ on the capacitor as a function of time $t$. Explicitly label any intercepts, asymptotes, maxima, or minima with numerical values or algebraic expressions, as appropriate.
[Blank axes: vertical axis labeled $q$, horizontal axis labeled $t$, origin labeled $O$, dashed guide lines from the origin.]
On the axes below, sketch the current $I$ through each resistor as a function of time $t$. Clearly label the two curves as $I_1$ and $I_2$, the currents through resistors $R_1$ and $R_2$, respectively. Explicitly label any intercepts, asymptotes, maxima, or minima with numerical values or algebraic expressions, as appropriate.
[Blank axes: vertical axis labeled $I$, horizontal axis labeled $t$, origin labeled $O$, dashed guide lines from the origin.]
The circuit is constructed using an ideal 1.5 V battery, an 80 $\mu\text{F}$ capacitor, and resistors $R_1 = 150\ \Omega$ and $R_2 = 100\ \Omega$. The switch is closed, allowing the capacitor to fully charge. The switch is then opened, allowing the capacitor to discharge.
The time it takes to charge the capacitor to 50% of its maximum charge is $\Delta t_C$. The time it takes for the capacitor to discharge to 50% of its maximum charge is $\Delta t_D$. Which of the following correctly relates the two time intervals?
____ $\Delta t_C > \Delta t_D$ ____ $\Delta t_C = \Delta t_D$ ____ $\Delta t_C < \Delta t_D$
Justify your answer.
Calculate the current through resistor $R_2$ immediately after the switch is opened.
Is the current through resistor $R_2$ increasing, decreasing, or constant immediately after the switch is opened?
____ Increasing ____ Decreasing ____ Constant
Justify your answer.
Calculate the energy stored in the capacitor immediately after the switch is opened.
Calculate the energy dissipated by resistor $R_1$ as the capacitor completely discharges.
[Figure: a trapezoidal table setup showing a solenoid of length $\ell$ (a coiled wire wound on a hollow tube) connected in series with a resistor, and a power supply (with a dial/gauge, terminals $+$ and $-$) and a multimeter (with a digital display and a dial selector) sitting on the table nearby, not yet wired together.]
When studying Ampere's law, students collect data on the magnetic field of two different solenoids in order to determine the magnetic permeability of free space $\mu_0$. The solenoids are created by wrapping wire around a hollow plastic tube. The solenoids of length $\ell$ with $N$ turns of wire will be connected in series to a power supply and resistor. A multimeter will be used as an ammeter to measure the magnitude of the current $I$ through the solenoids. The main components for the setup with one of the solenoids are shown in the figure above.
On the figure above, draw wire connections between the solenoid, power supply, resistor, and multimeter that will complete the circuit and allow students to measure the magnitude of the current through the solenoid.
Using the connections you made in part (a)i above, what will be the direction of the magnetic field inside the solenoid?
____ Toward the top of the page ____ To the left ____ Out of the page ____ Toward the bottom of the page ____ To the right ____ Into the page
The rectangle shown below represents the solenoid (the loops of wire are not shown). Points $A$, $B$, and $C$ are along the central axis of the solenoid with point $B$ at the middle of the solenoid. Point $D$ is directly above point $B$.
[Diagram: a rectangle representing the solenoid, with dot $D$ above the rectangle at its midpoint, and inside the rectangle three dots along the central axis labeled $\bullet A$, $\bullet B$ (both toward the left-middle), and $\bullet C$ (toward the right edge, outside the main cluster of $A$ and $B$, near the right end).]
From the choices below, select the point where you would place a magnetic field probe (a probe that can measure the magnitude of the magnetic field) to best measure the strength of the magnetic field of the solenoid in order to determine the magnetic permeability of free space $\mu_0$.
____ A ____ B ____ C ____ D
Justify your answer based on the model for a simple solenoid.
The figures below show two different solenoids that will be connected in the circuit above. Solenoid 1 has a length $\ell = 25$ cm with $N = 100$ turns. Solenoid 2 has a length $\ell = 5.0$ cm with $N = 5$ turns.
[Two diagrams of coiled solenoids: "Solenoid 1" shown as a long tightly-wound coil, and "Solenoid 2" shown as a short coil with few turns. Note: Figures not drawn to scale.]
A graph of the magnitude of the magnetic field $B$ as a function of $NI/\ell$ is shown below. The best-fit lines for the data are shown as a solid line for solenoid 1 and as a dashed line for solenoid 2.
[Graph: vertical axis $B\ (\text{T} \times 10^{-5})$ ranging 0 to 12 in increments of 2; horizontal axis $\dfrac{NI}{\ell}\ \left(\dfrac{\text{A}}{\text{m}}\right)$ ranging 0 to 100 in increments of 20. Solid line "Solenoid 1" is a best-fit line through filled data points, rising from about $(12, 2.3)$ to about $(95, 11)$, roughly linear with a positive $y$-intercept near 0.5–1. Dashed line "Solenoid 2" is a best-fit line through open-circle data points, rising from about $(15, 1.7)$ to about $(85, 9)$, also roughly linear, lying slightly below Solenoid 1's line at low $NI/\ell$ and crossing to be below it at high $NI/\ell$ as well, with more scatter in its data points.]
Which solenoid's best-fit line would give the best results for determining a value for the magnetic permeability of free space $\mu_0$?
____ Solenoid 1 ____ Solenoid 2
Justify your answer.
Use the slope of the best-fit line for the solenoid chosen in part (b) to calculate the magnetic permeability of free space $\mu_0$.
Calculate the percent error for the experimental value of the magnetic permeability of free space $\mu_0$ determined in part (c)i.
What is a reasonable physical explanation for a best-fit line that does not pass through the origin?
Suppose a student connects the solenoid in a closed circuit similar to the circuit in part (a)i but without the resistor. The student notices the multimeter stops functioning after the power supply is turned on. Explain what causes the failure of the multimeter.