Hands On Physics

Some Mathematical Results

When a constant voltage (perhaps from a battery) is suddenly applied to a resistor-capacitor circuit, the voltage rises steadily at first and then levels off.

Figure P11a
RC (resistor, capacitor) Circuit

The graph looks someting like the following:

Figure P11b
Charging a Capacitor

For a while, the voltage graph is a straight line. The dotted line shows what would happen if the voltage continued rising at its initial rate. Vmax is the voltage of applied to the circuit. The voltage across the capacitor can never get above Vmax. In fact, as it nears Vmax, the voltage veres away from the dotted line and increases more slowly. The voltage approaches Vmax asympotically, getting ever closer, but never quite reaching it.

The equation for the voltage across the capacitor that is approximately accurate for short times is

V = Vmax*t/(R*C)

Here t is the time the capacitor C has charge through the resistor R.

This equation can be solved for t so that you can determine the time from knowing the other variables:

t = R*C*V/Vmax

The product (R*C) occurs so often in these equations, it is given a special name: the time constant often symbolized by the Greek letter tau.

The time constant has an important meaning. For times much less than one time constant, the linear equation is quite accurate. For longer times, you have to use the complete equation.