AP Physics C E and M 11.8 RC Circuits

[Maximum number: 13]

A non-ideal capacitor has internal resistance that can be modeled as an ideal capacitor in series with a small resistor of resistance $r_{C}$ . A group of students performs an experiment to determine the internal resistance of a capacitor. A circuit is to be constructed with the following available equipment: a single ideal battery of potential difference $\Delta V_{0}$ , a single ammeter, a single variable resistor of resistance R, a single uncharged non-ideal capacitor of capacitance C, and one or more switches as needed.

Figure for Question 11.8.B—Describe the behavior of a circuit containing combinations of resistors and capacitors question 1 — AP Physics C: Electricity & Magnetism

(a)

Using the symbols shown, draw a schematic diagram of a circuit that can charge the capacitor and may also be used to study the current through the capacitor as it discharges through the resistor.

The capacitor is fully charged by the battery. At time t=0, the capacitor starts discharging through the resistor.

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(b)

Show that the current I through the capacitor as a function of time t is $I(t)=I_{0} e^{\frac{-t}{\left(R+r_{C}\right) C}}$ as the capacitor discharges.

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For using an appropriate loop equation by substituting a correct expression for the potential difference across the capacitor in terms of I, C, and $r_{C}$ and IR for the potential difference across the variable resistor, if included

1 point Example Response

\begin{aligned} & V_{C}-V_{R}=0 \\ & \frac{q}{C}-I r_{C}=I R \therefore \frac{q}{C}=I\left(R+r_{C}\right) \end{aligned}

For substituting $R+r_{C}$ as the total resistance of the circuit 1 point

Example Response

R_{\text {total }}=R+r_{C}

Scoring Notes:

- This point is earned if the above substitution is made anywhere in part (b).

- If the variable resistor is not included in the expression, accept expressions without R throughout.

For a correct differential equation consistent with the first point that could be used to determine the current I through the capacitor as a function of time t

Example Response
$\frac{d q}{d t} \frac{1}{C}=\left(R+r_{C}\right) \frac{d I}{d t}$

-I \frac{1}{C}=\left(R+r_{C}\right) \frac{d I}{d t} ; \quad I=-\frac{d q}{d t}

Where q is the charge on one plate of the capacitor that decreases over time

\begin{aligned} & \frac{1}{I} \frac{d I}{d t}=\frac{-1}{C\left(R+r_{C}\right)} \\ & \ln \frac{I}{I_{0}}=\frac{-t}{C\left(R+r_{C}\right)} \\ & I=I_{0} e^{\frac{-t}{C\left(R+r_{C}\right)}} \end{aligned}

Total for part (b) 3 points

(c)

The students determine the time constant $\tau$ for the circuit as a function of the resistance R. The students' data are shown in the following graph.

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

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(i)

Draw the best-fit line for the data.

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(ii)

Using the best-fit line, calculate a value for the internal resistance $r_{C}$ of the capacitor.

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(ii) For correctly determining the slope of the line 1 point
Example Response

\text { slope }=\frac{(0.24-0.08) \mathrm{s}}{(1.0-0.2) \Omega}=0.2 \mathrm{~F}

For using the equation $\tau=R C$ to determine that the slope must be C 1 point
Example Response

\begin{aligned} & \tau=R C \\ & \tau=\left(R+r_{C}\right) C \\ & \tau=C R+r_{C} C \end{aligned}

For correctly relating the vertical intercept to the internal resistance of the capacitor 1 point
Example Response

\begin{aligned} & \text { Vertical intercept }=\tau_{0}=r_{\mathrm{C}} C \\ & \text { Vertical intercept }=0.08 \mathrm{~s} \end{aligned}

\begin{aligned} r_{\mathrm{C}} & =\frac{\text { vertical intercept }}{C} \\ & =\frac{0.08 \mathrm{~s}}{0.2 \mathrm{~F}} \end{aligned}

r_{\mathrm{C}}=0.4 \Omega

Total for part (c)

4 points

(d)

The values of the variable resistor in the original experiment ranged from $0.5 \Omega$ to $2.5 \Omega$ . The experiment is repeated with values ranging from $3.0 \Omega$ to $6.0 \Omega$ . Would the slope of the best-fit line be more steep, be less steep, or remain unchanged compared to the graph in part (c)?

More steep Less steep Remain unchanged
Briefly justify your answer.

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

Note: Figures not drawn to scale.

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11.8.B—Describe the behavior of a circuit containing combinations of resistors and capacitors question 1

Question (a)

Question (b)

Question (c)

Question (i)

Question (ii)

Question (d)