### Introduction

The charge q on a capacitor’s plate is proportional to the potential difference V across the capacitor. We express this relationship with

$V = \frac{q}{C}$

where C is a proportionality constant known as the capacitance. C is measured in the unit of the farad, F, (1 farad = 1 coulomb/volt).

If a capacitor of capacitance C (in farads), initially charged to a potential V0 (volts) is connected across a resistor R (in ohms), a time-dependent current will flow according to Ohm’s law. This situation is shown by the RC (resistor-capacitor) circuit below when the switch is connecting terminals 33 and 34.

As the charge flows, the charge q on the capacitor is depleted, reducing the potential across the capacitor, which in turn reduces the current. This process creates an exponentially decreasing current, modeled by

$V(t) = V_{0}e^{-\frac{t}{RC}}$

The rate of the decrease is determined by the product RC, known as the time constant of the circuit. A large time constant means that the capacitor will discharge slowly.

In contrast, when the capacitor is charged, the potential across it approaches the final value exponentially, modeled by

$V(t) = V_{0} \left( 1-e^{-\frac{t}{RC}} \right)$

The same time constant, RC, describes the rate of charging as well as discharging.

### Objectives

• Measure an experimental time constant of a resistor-capacitor circuit.
• Compare the time constant to the value predicted from the component values of the resistance and capacitance.
• Measure the potential across a capacitor as a function of time as it discharges and as it charges.
• Fit an exponential function to the data. One of the fit parameters corresponds to an experimental time constant.