# 21.6 Dc circuits containing resistors and capacitors  (Page 5/9)

 Page 5 / 9

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.

## 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, $\Omega \cdot F=s$ .

The $\text{RC}$ time constant in heart defibrillation is crucial to limiting the time the current flows. If the capacitance in the defibrillation unit is fixed, how would you manipulate resistance in the circuit to adjust the $\text{RC}$ constant $\tau$ ? Would an adjustment of the applied voltage also be needed to ensure that the current delivered has an appropriate value?

When making an ECG measurement, it is important to measure voltage variations over small time intervals. The time is limited by the $\text{RC}$ constant of the circuit—it is not possible to measure time variations shorter than $\text{RC}$ . How would you manipulate $R$ and $C$ in the circuit to allow the necessary measurements?

Draw two graphs of charge versus time on a capacitor. Draw one for charging an initially uncharged capacitor in series with a resistor, as in the circuit in [link] , starting from $\text{t}=0$ . Draw the other for discharging a capacitor through a resistor, as in the circuit in [link] , starting at $\text{t}=0$ , with an initial charge ${Q}_{0}$ . Show at least two intervals of $\tau$ .

When charging a capacitor, as discussed in conjunction with [link] , how long does it take for the voltage on the capacitor to reach emf? Is this a problem?

When discharging a capacitor, as discussed in conjunction with [link] , how long does it take for the voltage on the capacitor to reach zero? Is this a problem?

Referring to [link] , draw a graph of potential difference across the resistor versus time, showing at least two intervals of $\tau$ . Also draw a graph of current versus time for this situation.

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