# 21.6 Dc circuits containing resistors and capacitors  (Page 5/10)

 Page 5 / 10

When is the potential difference across a capacitor an emf?

Only when the current being drawn from or put into the capacitor is zero. Capacitors, like batteries, have internal resistance, so their output voltage is not an emf unless current is zero. This is difficult to measure in practice so we refer to a capacitor’s voltage rather than its emf. But the source of potential difference in a capacitor is fundamental and it is an emf.

## Phet explorations: circuit construction kit (dc only)

An electronics kit in your computer! Build circuits with resistors, light bulbs, batteries, and switches. Take measurements with the realistic ammeter and voltmeter. View the circuit as a schematic diagram, or switch to a life-like view.

## Test prep for ap courses

A battery is connected to a resistor and an uncharged capacitor. The switch for the circuit is closed at t = 0 s.

1. While the capacitor is being charged, which of the following is true?

1. Current through and voltage across the resistor increase.
2. Current through and voltage across the resistor decrease.
3. Current through and voltage across the resistor first increase and then decrease.
4. Current through and voltage across the resistor first decrease and then increase.
2. When the capacitor is fully charged, which of the following is NOT zero?

1. Current in the resistor.
2. Voltage across the resistor.
3. Current in the capacitor.
4. None of the above.

An uncharged capacitor C is connected in series (with a switch) to a resistor R 1 and a voltage source E . Assume E = 24 V, R 1 = 1.2 kΩ and C = 1 mF.

1. What will be the current through the circuit as the switch is closed? Draw a circuit diagram and show the direction of current after the switch is closed. How long will it take for the capacitor to be 99% charged?
2. After full charging, this capacitor is connected in series to another resistor, R 2 = 1 kΩ. What will be the current in the circuit as soon as it’s connected? Draw a circuit diagram and show the direction of current. How long will it take for the capacitor voltage to reach 3.24 V?

(a) 20 mA, [link] , 5.5 s; (b) 24 mA, [link] , 2 s

## Section summary

• An $\text{RC}$ circuit is one that has both a resistor and a capacitor.
• The time constant $\tau$ for an $\text{RC}$ circuit is $\tau =\text{RC}$ .
• When an initially uncharged ( ${V}_{0}=0$ at $t=0$ ) capacitor in series with a resistor is charged by a DC voltage source, the voltage rises, asymptotically approaching the emf of the voltage source; as a function of time,
$V=\text{emf}\left(1-{e}^{-t/\text{RC}}\right)\text{(charging).}$
• Within the span of each time constant $\tau$ , the voltage rises by 0.632 of the remaining value, approaching the final voltage asymptotically.
• If a capacitor with an initial voltage ${V}_{0}$ is discharged through a resistor starting at $t=0$ , then its voltage decreases exponentially as given by
$V={V}_{0}{e}^{-t/\text{RC}}\text{(discharging).}$
• In each time constant $\tau$ , the voltage falls by 0.368 of its remaining initial value, approaching zero asymptotically.

## Conceptual questions

Regarding the units involved in the relationship $\tau =\text{RC}$ , verify that the units of resistance times capacitance are time, that is, $\Omega \cdot F=s$ .

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