AP Physics C E and M 13.5 LR Circuits Overview

[Maximum number: 15]

Students are asked to determine the resistance R of two identical resistors. The resistors are in parallel with each other and are connected in series to a battery of known emf $\mathcal{E}$ , an inductor of known inductance L, and a switch, as shown in Figure 1. The students have access to a voltmeter that can measure potential difference as a function of time. The students are required to measure a quantity that decreases with time to determine R.

(a)

On the circuit diagram shown in Figure 1, draw the voltmeter, using the following symbol, with connections that would allow the students to correctly measure a potential difference that decreases with time.

Figure for Question (a) — AP Physics C: Electricity & Magnetism

Voltmeter Symbol
ii. Describe a procedure for collecting data that would allow the students to graphically determine the experimental value for R using the measured quantity that decreases with time. Provide enough detail so that another student could replicate the experiment.

[ 3 ]

(b)

On the axes shown in Figure 2, produce a graph that represents the expected trend of the data by completing the following tasks.

- Label the quantities graphed on the vertical and horizontal axes.

- Sketch a line or curve that represents the expected trend of the collected data.

- Label any appropriate intercepts and/or asymptotes in terms of the quantities provided.

Figure 2

ii. Describe how the information from the graph in part (b)(i) would be used to determine the experimental value for R.

[ 6 ]

(i)

For correctly labeling potential difference on the vertical axis and time on the horizontal axis

1 point

For a concave-up and decreasing curve

1 point

For a curve that asymptotically approaches zero, or a curve that starts at the origin and approaches a horizontal asymptote that is consistent with the placement of the voltmeter in the response in part (a)(i)

1 point

For including a vertical intercept or a horizontal asymptote that is consistent with the
placement of the voltmeter in the response in part (a)(i) that is correctly labeled as $\mathcal{E}$

1 point

Example Response

Figure for Answer to Question (b) — AP Physics C: Electricity & Magnetism

(ii) For indicating that a curve fit to the graph is that of an exponential function or for correctly relating the area under the curve to the current in the circuit
Alternate Solution
For indicating the time on the graph where the potential difference is approximately $0.37 \mathcal{E}$ , or $0.63 \mathcal{E}$ for a curve that starts at the origin and approaches a horizontal asymptote that is consistent with the placement of the voltmeter in the response in part (a)(i)
For indicating that the coefficient in front of the t of the curve-fit equation is equal to $\frac{R}{2 L}$ or for correctly relating R to L and the current
Alternate Solution
For indicating that the time constant is equal to $\frac{2 L}{R}$

Example Response
The data in the graph should be fit with an exponential function for the equation $V_{L}=\mathcal{E}\left(e^{-t \frac{R}{2 L}}\right)$ . Because $\mathcal{E}$ and L are known, R can be calculated.

Alternate Example Response
The potential difference at $0.37 \mathcal{E}$ along the vertical axis corresponds to the time constant along the horizontal axis. Because the time constant is equal to $\frac{2 L}{R}$ , and L is known, R can be calculated.

Total for part (b) 6 points

(c)

Starting with an appropriate application of Kirchhoff's loop rule, derive, but do NOT solve, a differential equation that can be used to determine the current I in the inductor at time t after the switch is closed. Express your answer in terms of $R, \mathcal{E}, L, t$ , and physical constants, as appropriate.

After reaching steady state, the absolute value of the potential difference across the inductor is $\left|\Delta V_{1}\right|$ . The students replace the original inductor with a new inductor that has nonnegligible resistance. The experiment is repeated. After a long time, the absolute value of the potential difference across the new inductor is $\left|\Delta V_{2}\right|$ .

[ 3 ]

For a multi-step derivation that begins with an attempt at using Kirchhoff's loop rule 1 point
Example Response

\mathcal{E}-\Delta V_{R}-\Delta V_{L}=0

For indicating that the potential difference across the parallel combination of resistors is 1 point

I\left(\frac{R}{2}\right)

For indicating that the absolute value of the potential difference across the inductor is $L \frac{d I}{d t}$ 1 point
Example Solution

\begin{aligned} & \mathcal{E}-\Delta V_{R}-\Delta V_{L}=0 \\ & \mathcal{E}-I\left(\frac{R}{2}\right)-L \frac{d I}{d t}=0 \\ & \frac{\mathcal{E}}{L}-\frac{I R}{2 L}=\frac{d I}{d t} \\ & -\frac{R}{L}\left(\frac{I}{2}-\frac{\mathcal{E}}{R}\right)=\frac{d I}{d t} \end{aligned}

Total for part (c)

3 points

(d)

Indicate whether $\left|\Delta V_{2}\right|$ is greater than, less than, or equal to $\left|\Delta V_{1}\right|$ .

\left|\Delta V_{2}\right|>\left|\Delta V_{1}\right|

\left|\Delta V_{2}\right|<\left|\Delta V_{1}\right|

\left|\Delta V_{2}\right|=\left|\Delta V_{1}\right|

Justify your answer.

Figure 1

[ 3 ]

For selecting $\left|\Delta V_{2}\right|>\left|\Delta V_{1}\right|$ with an attempt at a relevant justification

1 point

For indicating that the potential difference across the original inductor is zero when steady- state conditions are established

1 point

For indicating that the potential difference across the inductor of nonnegligible resistance is nonzero when steady-state conditions are established

1 point

Example Response
$\left|\Delta V_{2}\right|>\left|\Delta V_{1}\right|$ . The potential difference $\Delta V_{1}$ across the original, ideal inductor is zero at steady state. The potential difference across the new inductor would be nonzero due to the inductor's nonnegligible resistance. The value of $\left|\Delta V_{2}\right|$ would be the product of the steady

state current and the inductor's resistance.

Total for part (d)

for question 2

15 points

Covered syllabus points

13.5 Circuits with Resistors and Inductors (LR Circuits) question 1

Question (a)

Question (b)

Question (c)

Question (d)