7.4.4

# Capacitor Discharge

Test yourself

## Discharging a Capacitor

When a charged capacitor with capacitance C is connected to a resistor with resistance R, then the charge stored on the capacitor decreases exponentially.

### Discharge graph

• $Q=Q_0e^{-\frac{t}{RC}}$
• Where $Q_0$ is the initial charge on the capacitor.

### Time to halve

• The time for the charge to reach half of the initial charge, $T_{1/2}$, can be found:
• $Q=\frac{Q_0}{2}=Q_0e^{-\frac{T_{1/2}}{RC}}$
• $0.5=e^{-\frac{T_{1/2}}{RC}}$
• $\ln (0.5)=\ln (e^{-\frac{T_{1/2}}{RC}})$
• $-\ln(2)=-\frac{T_{1/2}}{RC}$
• $T_{1/2}=RC\ln 2$
• Because $\ln(2) = 0.69,\space T_{1/2}=0.69RC$.

### Time constant

• The product RC is known as the time constant.
• It is a property of exponential decay graphs that the curve will decrease by a constant fraction with each time constant.

## Potential Difference and Current in a Discharging Capacitor

The potential difference and the current in a discharging capacitor have similar forms.

### Potential difference

• The potential difference across a capacitor with time constant RC, initial potential difference V0, and having been discharged for a time t is:
• $V=V_0e^{-\frac{t}{RC}}$

### Current

• The current passing through a capacitor with time constant RC, initial current I0, and having been discharged for a time t is:
• $I=I_0e^{-\frac{t}{RC}}$
• R is the resistance across the capacitor
• $I_0=\frac{V_0}{R}$