Learn Extracted exam questions AP Physics C: Electricity and Magnetism 2019 Free Response · Set 1
2019 Free Response · Set 1
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Note: Figure not drawn to scale.
[Figure: A long horizontal cylinder lies along the x-axis, centered on the y-axis, carrying uniform linear charge density $+\lambda$. Point P is on the y-axis at height $y = c$ above the cylinder's center. The cylinder extends from $x = -L/2$ to $x = +L/2$, so its total length is $L$.]
A very long, thin, nonconducting cylinder of length $L$ is centered on the y-axis, as shown above. The cylinder has a uniform linear charge density $+\lambda$. Point P is located on the y-axis at $y = c$, where $L \gg c$.
On the figure shown below, draw an arrow to indicate the direction of the electric field at point P due to the long cylinder. The arrow should start on and point away from the dot.
[Figure: Point P shown as a dot on a vertical dashed line above a horizontal dashed line representing the cylinder's axis, with a small $+$ segment marked at the axis below P.]
Describe the shape and location of a Gaussian surface that can be used to determine the electric field at point P due to the long cylinder.
Use your Gaussian surface to derive an expression for the magnitude of the electric field at point P. Express your answer in terms of $\lambda$, $c$, $L$, and physical constants, as appropriate.
A proton is released from rest at point P. On the axes below, sketch the velocity $v$ as a function of position $y$ and the acceleration $a$ as a function of position $y$ for the proton.
[Two blank graphs, side by side. Left graph: vertical axis $v$, horizontal axis $y$, origin $O$, with a tick mark labeled $c$ on the $y$-axis. Right graph: vertical axis $a$, horizontal axis $y$, origin $O$, with a tick mark labeled $c$ on the $y$-axis. Both axes have dashed guide lines at $y = c$; no curves are drawn — student sketches the curves.]
The original cylinder is now replaced with a much shorter thin, nonconducting cylinder with the same uniform linear charge density $+\lambda$, as shown in the figure below. The length of the cylinder to the right of the y-axis is $a$, and the length of the cylinder to the left of the y-axis is $b$, where $a < b$.
[Figure: A short horizontal cylinder centered near the y-axis, carrying charge density $+\lambda$. Point P is on the y-axis at height $y = c$ above the cylinder. The cylinder extends a distance $b$ to the left of the y-axis and a distance $a$ to the right of the y-axis, with $a < b$.]
On the figure shown below, draw an arrow to indicate the direction of the electric field at point P due to the shorter cylinder. The arrow should start on and point away from the dot.
[Figure: Point P shown as a dot on a vertical dashed line above a horizontal dashed line representing the shorter cylinder's axis, with a small $+$ segment marked at the axis below P.]
Is there a single Gaussian surface that can be used with Gauss's law to derive an expression for the electric field at point P?
____ Yes ____ No
If your answer to part (d)(i) is yes, explain how you can use Gauss's law to derive an expression for the field at point P. If your answer to part (d)(i) is no, explain why Gauss's law cannot be applied to derive an expression for the electric field in this case.
[Figure: The shorter-cylinder figure shown again for reference — point P on the y-axis at $y = c$ above a cylinder of length $b$ to the left of the y-axis and $a$ to the right of the y-axis, carrying charge density $+\lambda$. Note: This figure is shown again for reference.]
A student in class argues that using the integral shown below might be a useful approach for determining the electric field at point P.
The student uses this approach and writes the following two integrals for the magnitude of the horizontal and vertical components of the electric field at point P.
Horizontal component:
Vertical component:
One of the two expressions above is not correct. Which expression is not correct?
____ Horizontal component ____ Vertical component
Identify two mistakes in the incorrect expression, and explain how to correct the mistakes.
[Figure 1: A circuit with two 6.0 V batteries and three resistors. The left branch has a $150\ \Omega$ resistor in series carrying current $I_1$ (arrow pointing up) connected to a 6.0 V battery. The middle branch has a $200\ \Omega$ resistor carrying current $I_2$ (arrow pointing up). The right branch has a $100\ \Omega$ resistor in series carrying current $I_3$ (arrow pointing up) connected to a 6.0 V battery. The $150\ \Omega$ and $100\ \Omega$ resistors are along the top, joining at a node above the $200\ \Omega$ resistor.]
The circuit shown above is constructed with two 6.0 V batteries and three resistors with the values shown. The currents $I_1$, $I_2$, and $I_3$ in each branch of the circuit are indicated.
Using Kirchhoff's rules, write, but DO NOT SOLVE, equations that can be used to solve for the current in each resistor.
Calculate the current in the $200\ \Omega$ resistor.
Calculate the power dissipated by the $200\ \Omega$ resistor.
[Figure 2: A circuit with a battery of emf $\mathcal{E}$, a $150\ \Omega$ resistor and $100\ \Omega$ resistor along the top in series from the battery's positive terminal, a $200\ \Omega$ resistor in the middle branch, a voltmeter V connected across the $200\ \Omega$ resistor, and a $50\ \Omega$ resistor in the right branch below the $100\ \Omega$ resistor.]
The two 6.0 V batteries are replaced with a battery with voltage $\mathcal{E}$ and a resistor of resistance $50\ \Omega$, as shown above. The voltmeter V shows that the voltage across the $200\ \Omega$ resistor is 4.4 V.
Calculate the current through the $50\ \Omega$ resistor.
Calculate the voltage $\mathcal{E}$ of the battery.
[Figure: Same circuit as Figure 2 but the $200\ \Omega$ resistor is replaced with a $200\ \mu\text{F}$ capacitor in the middle branch, battery $\mathcal{E}$, $150\ \Omega$ and $100\ \Omega$ resistors along the top, $50\ \Omega$ resistor in the right branch.]
The $200\ \Omega$ resistor in the circuit in Figure 2 is replaced with a $200\ \mu\text{F}$ capacitor, as shown on the right, and the circuit is allowed to reach steady state. Calculate the current through the $50\ \Omega$ resistor.
[Figure: Same circuit as Figure 2 but the $200\ \Omega$ resistor is replaced with an ideal 50 mH inductor in the middle branch, battery $\mathcal{E}$, $150\ \Omega$ and $100\ \Omega$ resistors along the top, $50\ \Omega$ resistor in the right branch.]
The $200\ \Omega$ resistor in the circuit in Figure 2 is replaced with an ideal 50 mH inductor, as shown on the right, and the circuit is allowed to reach steady state. Is the current in the $50\ \Omega$ resistor greater than, less than, or equal to the current calculated in part (b)?
____ Greater than ____ Less than ____ Equal to
Justify your answer.
Note: Figures not drawn to scale.
[Figure, left ("Solenoid"): A solenoid of inner radius $a$ and length $\ell$ with current $I$ entering, wound with wire, oriented along its axis.]
[Figure, right ("Cutaway View"): A cutaway view of the solenoid showing the xyz-coordinate system, with the x-axis along the axis of the solenoid, current $I$ shown entering at the near end, point P at the origin in the middle of the solenoid, and the y- and z-axes shown perpendicular to the x-axis at P.]
A solenoid is used to generate a magnetic field. The solenoid has an inner radius $a$, length $\ell$, and $N$ total turns of wire. A power supply, not shown, is connected to the solenoid and generates current $I$, as shown in the figure on the left above. The x-axis runs along the axis of the solenoid, as shown in the cutaway view on the right above. Point P is in the middle of the solenoid at the origin of the xyz-coordinate system, as shown in the cutaway view. Assume $\ell \gg a$.
Select the correct direction of the magnetic field at point P.
____ +x-direction ____ +y-direction ____ +z-direction ____ -x-direction ____ -y-direction ____ -z-direction
Justify your selection.
On the cutaway view below, clearly draw an Amperian loop that can be used to determine the magnetic field at point P at the center of the solenoid.
[Figure: Cutaway view of the solenoid with current $I$ entering along the "Axis of Solenoid" dashed line, and point P marked at the center of the solenoid; blank for the student to draw an Amperian loop.]
Use Ampere's law to derive an expression for the magnetic field strength at point P. Express your answer in terms of $I$, $\ell$, $N$, $a$, and physical constants, as appropriate.
Some physics students conduct an experiment to determine the resistance $R_S$ of a solenoid with radius $a = 0.015\ \text{m}$, total turns $N = 100$, and total length $\ell = 0.40\ \text{m}$. The students connect the solenoid to a variable power supply. A magnetic field sensor is used to measure the magnetic field strength along the central axis at the center of the solenoid. The plot of the magnetic field strength $B$ as a function of the emf $\mathcal{E}$ of the power supply is shown below.
[Graph: Scatter plot of $B\ (\text{T}\times10^{-4})$ on the vertical axis, ranging 0 to 3.0 in increments of 0.5, versus $\mathcal{E}\ (\text{V})$ on the horizontal axis, ranging 0 to 8 in increments of 2. Data points approximately at: $(1.6, 0.65)$, $(3.0, 1.4)$, $(4.2, 1.55)$, $(5.4, 2.35)$, $(6.8, 2.85)$. Grid lines shown throughout; no line drawn yet — student draws a best-fit line.]
On the graph above, draw a best-fit line for the data.
Use the straight line to determine the resistance $R_S$ of the solenoid used in the experiment.
One of the students notes that the horizontal component of the magnetic field of Earth is $2.5\times10^{-5}\ \text{T}$. Is there evidence from the graph that the horizontal orientation of the solenoid affects the measured values for $B$?
____ Yes ____ No
Justify your answer.
Would the horizontal orientation of the solenoid affect the calculated value for $R_S$?
____ Yes ____ No
Justify your answer.
[Figure: A solenoid with current $I$ entering, inner radius $a$, length $\ell$, labeled "Solenoid." A thin conducting loop of radius $b$ is shown concentric with and around the solenoid, with curved arrows labeled "Counterclockwise" and "Clockwise" indicating the two possible current directions in the loop, labeled "Loop."]
A thin conducting loop of radius $b$ and resistance $R_L$ is placed concentric with the solenoid, as shown above. The current in the solenoid is decreased from $I$ to zero over time $\Delta t$.
Is the direction of the induced current in the loop clockwise or counterclockwise during the time period that the current in the solenoid is decreasing?
____ Clockwise ____ Counterclockwise
Justify your answer.
Derive an equation for the average induced current $i_{\text{IND}}$ in the loop during the time period that the current in the solenoid is decreasing. Express your answer in terms of $I$, $\ell$, $N$, $a$, $b$, $R_L$, $R_S$, $\Delta t$, and physical constants, as appropriate.