Learn Extracted exam questions AP Physics C: Electricity and Magnetism 2014 Free Response
2014 Free Response
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[Circuit diagram: A variable DC power supply connects to an ammeter (A) in series, which connects to a node. From that node, resistor $R_2 = 50\ \Omega$ runs across the top back to the power supply, an unknown resistor $R_1$ runs down the middle in parallel, and resistor $R_3 = 50\ \Omega$ runs across the bottom, also back to the power supply. So $R_1$, $R_2$, and $R_3$ are all in parallel with each other, and this parallel combination is in series with the ammeter and the DC power supply.]
Physics students are analyzing the circuit above. A variable DC power supply is connected to an ammeter and three resistors. The resistances of two of the resistors are known to be $R_2 = R_3 = 50\ \Omega$, but the resistance of the third resistor is unknown. The students collect data on the potential difference across the power supply and the current measured by the ammeter, as follows.
| Potential Difference (V) | 2 | 4 | 6 | 8 | 10 |
|---|---|---|---|---|---|
| Current (mA) | 40 | 55 | 97 | 138 | 155 |
On the grid below, plot the data points for the current as a function of the potential difference. Clearly scale and label all axes, including units if appropriate. Draw a straight line that best represents the data.
[Grid: a blank dashed-gridline graph with a vertical axis (arrow pointing up, unlabeled) and a horizontal axis (arrow pointing right, unlabeled), for the student to plot Current (mA) vs. Potential Difference (V) and draw a best-fit line.]
Using the straight line from part (a), calculate the total resistance of the three-resistor combination.
Calculate the value of $R_1$.
The power supply is now fixed at 12 V.
Calculate the current through $R_2$.
Resistor 3 is now removed and replaced by an open switch in series with an uncharged 4 nF capacitor, as shown below. The power supply is still fixed at 12 V.
[Circuit diagram: A 12 V DC power supply connects through an ammeter (A) to a node; from that node $R_2 = 50\ \Omega$ runs across the top back to the supply, $R_1$ runs down the middle in parallel, and a branch containing an open switch in series with a 4 nF capacitor runs across the bottom back to the supply.]
i. Calculate the current in $R_2$ immediately after the switch is closed.
ii. A long time after the switch is closed, will the magnitude of the current in $R_2$ be greater than, less than, or equal to the current through $R_2$ found in part (d)?
__ Greater than __ Less than __ Equal to
Justify your answer.
The 4 nF capacitor is replaced with an uncharged 10 nF capacitor. Will the magnitude of the current in $R_2$ immediately after the switch is closed be greater than, less than, or equal to the current in part (e)i?
__ Greater than __ Less than __ Equal to
Justify your answer.
[Diagram: A number line along the x-axis marked at $0$, $L$, $2L$, $3L$, $4L$, $5L$. A rectangular wire loop of length $L$ and width $L/4$ starts to the left of $x=0$, moving right with velocity $v_0$ (shown as an arrow labeled $v_0$ pointing right, above the loop). A region from $x=L$ to $x=3L$ (marked by dashed vertical lines at $L$ and $3L$) contains a uniform magnetic field $B$ directed into the page, shown by rows of "×" symbols filling that region. A second, identical rectangular loop is shown to the right of $x=4L$, moving right with velocity $v_f$ (arrow labeled $v_f$ pointing right, above that loop), representing the same loop after it has emerged from the field.]
The rectangular loop of wire shown on the left in the figure above has mass $M$, length $L$, width $L/4$, and resistance $R$. It is initially moving to the right at constant speed $v_0$, with no net force acting on it. At time $t = 0$ the loop enters a region of length $2L$ that contains a uniform magnetic field of magnitude $B$ directed into the page. The loop emerges from the field at time $t_f$ with final speed $v_f$. Express all algebraic answers to the following in terms of $M$, $L$, $R$, $B$, $v_0$, and fundamental constants, as appropriate.
Let $x$ represent the position of the right end of the loop. Place a check mark in the appropriate box in each column in the table below to indicate whether the speed of the loop increases, decreases, or stays the same as the loop moves to the right.
| Speed of Loop | $L < x < 2L$ | $2L < x < 3L$ | $3L < x < 4L$ | $4L < x < 5L$ |
|---|---|---|---|---|
| Increases | ||||
| Decreases | ||||
| Stays the same |
Derive an expression for the magnitude of the current induced in the loop as its right edge enters the field.
What is the direction of the induced current determined in part (b)?
__ Clockwise __ Counterclockwise
Justify your answer.
Write, but do not solve, a differential equation for the speed $v$ as a function of time as the loop enters the field.
What is the direction of the acceleration of the loop just before its left edge leaves the field?
__ Left __ Right __ Up __ Down
Justify your answer.
A scientist describes an electrically neutral atom with a model that consists of a nucleus that is a point particle with positive charge $+Q$ at the center of the atom and an electron volume charge density of the form
where $\alpha$ and $\beta$ are positive constants and $r$ is the distance from the center of the atom.
On the axes below, let $r$ stand for the radius of a Gaussian sphere. Sketch the graph for each of the following charges enclosed by the Gaussian sphere as a function of $r$. Explicitly label any intercepts, asymptotes, maxima, or minima with numerical values or algebraic expressions, as appropriate.
i. The nuclear charge only
[Graph: axes with vertical axis $q$ (labeled $+Q$ and $-Q$ as tick marks above and below the origin) and horizontal axis $r$ (with a tick mark labeled $\alpha$); blank, for the student to sketch.]
ii. The electron charge only
[Graph: axes with vertical axis $q$ (labeled $+Q$ and $-Q$ as tick marks above and below the origin) and horizontal axis $r$ (with a tick mark labeled $\alpha$); blank, for the student to sketch.]
The dashed curve on the graph below represents the electric field as a function of distance $r$ due to the positive nucleus of the atom without any electrons. The nucleus is modeled as a point particle of charge $+Q$. On the same graph, sketch the electric field as a function of distance $r$ for the neutral atom as defined by the scientist's model, which includes the nucleus and the negative electrons surrounding it.
[Graph: axes with vertical axis $E$ and horizontal axis $r$ (with a tick mark labeled $\alpha$); a dashed curve is shown starting high near $r=0$ and decreasing smoothly (following an inverse-square-like shape) toward the $r$-axis as $r$ increases past $\alpha$, approaching zero for large $r$. The student is to add their own solid curve for the neutral-atom field on the same axes.]
Use Gauss's law to derive an expression for the electric field strength due to the neutral atom for the following positions in terms of $Q$, $\alpha$, $\beta$, $r$, and fundamental constants, as appropriate.
i. $r > \alpha$
ii. $r < \alpha$
Based on the model proposed by the scientist, what is the physical meaning of the constant $\alpha$?