Q: The charge $$q(t) on a $$capacitor in a charging RC circuit is given by $$q(t) = C\varepsilon (1 - e$$^{-t/\tau}$$). $$What is the charge after one time constant $$\tau$$?

$$q(\tau) = 0.632 C\varepsilon$$

Q: A circuit consists of a battery $$\varepsilon = 3.00 V$$, resistors $$R_1 = 126$$\ \Omega$$ and R_2$ = 275\ \Omega$$, and a capacitor $$C = 182\ \muF in $$series. What is the charge on the capacitor a long time after the switch is closed?

$$Q = (182\ \mu\text{F})(3.00\ \text{V}) = 546\ \mu\text{C}$$

Q: A circuit contains a resistor $$R = 120\ \Omega$$ and a capacitor $$C = 3.5\ \muF. $$What is the time constant $$\tau$$?

$$\tau = (120\ \Omega)(3.5 \times 10$$^{-6}$$\ \text{F}) = 4.2 \times 10$$^{-4}$$\ \text{s}$$

Question 1

In the circuit shown with three capacitors $$C_1$$, $$C_2$$, and $$C_3$$ connected in parallel to a battery of emf $$\varepsilon$$, what is the expression for the total charge $$Q$$ stored in the system?

Accepted Answer

$$Q = (C_1$ + C_2$ + C_3$)\varepsilon$$

Question 2

Which of the following statements about capacitors in parallel is correct?

Accepted Answer

The equivalent capacitance is greater than the greatest individual capacitance.

Question 3

For three capacitors $$C_1$$, $$C_2$$, and $$C_3$$ connected in series, what is the correct formula for the equivalent capacitance $$C_{eq}$$?

Accepted Answer

$$\frac{1}{C_{eq}} = \frac{1}{C_1$} + \frac{1}{C_2$} + \frac{1}{C_3$}$$

Question 4

A touch-sensitive lamp uses the principle of capacitance. When you touch the lamp, what happens to the circuit capacitance?

Accepted Answer

It increases because your body acts as a capacitor plate, increasing plate area.

Question 5

In a circuit with a 12.0 V battery, a 10.0 $$\muF $$and a 5.00 $$\muF $$capacitor in series, and a 20.0 $$\muF $$capacitor in parallel with them, what is the equivalent capacitance?

Accepted Answer

$$C_{eq} = 23.3\ \mu	ext{F}$$

Question 6

In a simple RC circuit, what is the time constant $$	au$$ and what are its units?

Accepted Answer

$$	au = RC$$, units: seconds

Question 7

The charge $$q(t) on a $$capacitor in a charging RC circuit is given by $$q(t) = C\varepsilon (1 - e$$^{-t/	au}$$). $$What is the charge after one time constant $$	au$$?

Accepted Answer

$$q(	au) = 0.632 C\varepsilon$$

Question 8

In a charging RC circuit, what happens to the current $$I(t) as $$time $$t$$ increases?

Accepted Answer

It decreases exponentially.

Question 9

A circuit consists of a battery $$\varepsilon = 3.00 V$$, resistors $$R_1 = 126$$\ \Omega$$ and R_2$ = 275\ \Omega$$, and a capacitor $$C = 182\ \muF in $$series. What is the charge on the capacitor a long time after the switch is closed?

Accepted Answer

$$Q = (182\ \mu	ext{F})(3.00\ 	ext{V}) = 546\ \mu	ext{C}$$

Question 10

In a discharging RC circuit, the charge remaining on the capacitor at time $$t is $$given by $$q(t) = Q e$$^{-t/	au}$$. If the initial charge is Q$, what is the charge after two time constants (t = 2$$	au$$)?

Accepted Answer

$$q(2	au) = Q e$$^{-2}$$

Question 11

In a charging RC circuit, what is the initial current $$I(0)$$ immediately after the switch is closed?

Accepted Answer

$$I(0) = \frac{\varepsilon}{R}$$

Question 12

In a circuit where a capacitor is being charged, what does the capacitor behave like at $$t = 0$$?

Accepted Answer

A short circuit

Question 13

In a circuit where a capacitor is being charged, what does the capacitor behave like as $$t \rightarrow \infty$$?

Accepted Answer

An open switch

Question 14

A student wants to design a circuit where the capacitor charges to 80% of its maximum value in 100 ms. If the time constant $$	au is $$required for this, what equation should they use to solve for $$	au$$?

Accepted Answer

$$0.800 = 1 - e$$^{-t/	au}$$

Question 15

In a circuit with two resistors $$R in $$parallel and a capacitor, what is the initial current through the battery immediately after the switch is closed?

Accepted Answer

$$I = \frac{2\varepsilon}{R}$$

Question 16

Which of the following best describes the behavior of the current in a charging RC circuit as the capacitor becomes fully charged?

Accepted Answer

The current approaches zero.

Question 17

If a capacitor is connected directly to a battery, how quickly does it acquire charge?

Accepted Answer

Almost instantaneously

Question 18

In a series capacitor circuit, what is true about the charge on each capacitor?

Accepted Answer

Each capacitor has the same magnitude of charge $$Q.$$

Question 19

A circuit contains a resistor $$R = 120\ \Omega$$ and a capacitor $$C = 3.5\ \muF. $$What is the time constant $$	au$$?

Accepted Answer

$$	au = (120\ \Omega)(3.5 	imes 10$$^{-6}$$\ 	ext{F}) = 4.2 	imes 10$$^{-4}$$\ 	ext{s}$$

Capacitors and RC Circuits in Electric Circuits

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