Discharging Capacitors

A charged capacitor connected to a resistor is one of the purest examples of exponential decay in physics. Left alone, a capacitor on its own would stay charged indefinitely — the plates are insulated and have nowhere to lose their charge. But connect the plates through a resistor and the charge drains away with a beautifully simple mathematical form: an exponential. This lesson introduces the time constant τ = RC and the exponential decay laws for charge, voltage and current.

This lesson continues OCR H556 Module 6.1.

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The Discharge Circuit

Consider a capacitor C charged to some initial voltage V₀, and then connected through a switch to a resistor R:

graph LR
    Cap[Capacitor C] --> S{Switch}
    S --> R1[Resistor R]
    R1 --> Cap

When the switch closes, current flows from the positive plate, through the resistor, to the negative plate, neutralising the charge bit by bit.

At any instant:

• The voltage across the capacitor is v = q/C
• The voltage across the resistor must also be v (it is in parallel with the capacitor)
• The current is i = v / R = q / (RC)

But the current is the rate at which charge leaves the capacitor: i = −dq/dt. Combining,

dq/dt = − q / (RC)

This is the defining equation of exponential decay.

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Solving the Decay Equation

The solution to dq/dt = −q/(RC) with initial charge Q₀ is

q(t) = Q₀ e^(−t/RC)

The voltage is proportional to the charge, so

v(t) = V₀ e^(−t/RC)

and the current is proportional to the voltage through Ohm's law:

i(t) = I₀ e^(−t/RC) with I₀ = V₀ / R

Every quantity — charge, voltage, current — decays with the same exponential and the same time constant.

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The Time Constant τ = RC

The quantity RC has units of seconds. You can prove this with dimensional analysis: [R] = V/A, [C] = A·s/V, so [RC] = s. It is given a special name, the time constant, symbol τ (Greek tau):

τ = RC

The time constant has a clear physical meaning: after one time constant, the charge has dropped to e⁻¹ of its initial value — about 37%. After two time constants, it is e⁻² ≈ 13.5%. After five time constants, it is e⁻⁵ ≈ 0.67% — the capacitor is essentially fully discharged.

Time	Fraction of initial charge remaining
0	100.0 %
τ	36.79 % (1/e)
2τ	13.53 %
3τ	4.98 %
5τ	0.67 %
10τ	0.0045 %

graph LR
    A[t=0] -->|"Q₀"| B[t=τ]
    B -->|"0.37 Q₀"| C[t=2τ]
    C -->|"0.14 Q₀"| D[t=5τ]
    D -->|"0.007 Q₀"| E[t=10τ]

The decay curve for charge or voltage looks like this (time on x-axis, fraction of initial value on y-axis):

 1.0 |*
     | *
 0.8 |  *
     |   *
 0.6 |    *
     |     **
 0.4 |       **
     |         **
 0.2 |           ***
     |              ****
 0.0 |------------------*****---------
       0   τ   2τ  3τ  4τ  5τ   t

The tangent to the curve at t = 0 meets the time axis at exactly t = τ. This is a graphical way to measure τ from an experimental discharge trace.

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Worked Example — Discharge Through a Resistor

A 100 μF capacitor is charged to 12 V and then discharged through a 47 kΩ resistor. Calculate (a) the time constant, (b) the initial current, (c) the voltage after 10 s, and (d) the time for the voltage to fall to 1 V.

(a) Time constant

τ = RC = 47 × 10³ × 100 × 10⁻⁶ = 4.7 s

(b) Initial current