19.3 Discharging a capacitor

A discharging capacitor has decreasing Q and V; current is related to the rate of charge change and initially has the greatest magnitude.

Use the circuit polarity and current direction consistently. The resistor receives energy as the capacitor’s electric field collapses.

A voltage-time trace falls steeply at first and then levels off, showing that the discharge rate slows.

Current is not constant during discharge, and a falling voltage does not mean the capacitor instantly becomes uncharged.

For a resistor R and capacitor C, τ=RC. After one time constant in a discharge, the relevant quantity has fallen to e⁻¹≈37% of its initial value.

Larger R or C makes the response slower. Identify the effective resistance seen by the capacitor, not every resistor in the diagram.

R=2.0 kΩ and C=100 μF gives τ=0.20 s.

τ is not the time to reach exactly zero; exponential decay approaches zero asymptotically.

A discharging quantity follows x=x₀e^(−t/RC); charging voltage follows V=V₀(1−e^(−t/RC)). Current and charge use the corresponding variable x.

At t=0 and after several τ, use limiting values to check the expression and identify whether the curve rises or falls.

After t=τ in a discharge, voltage and charge are 37% of their initial values; after about 5τ they are close to zero.

The exponent is dimensionless, so t and RC must have the same time units; exponential decay is not linear.