6.1FieldsStretch

Charging and Discharging Capacitors

A capacitor discharging through a resistor never mathematically reaches zero charge — it just gets ever closer, at an ever-slowing rate. This exponential behaviour is entirely captured by a single number for any given circuit: the time constant.

40 min Video by Lascells Science Module 6.1 Capacitors

RC Circuit: Charging and Discharging capacitorsWatch the full walkthrough before the notes below.

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What you'll be able to do

Use the discharge equation $Q = Q_{0} e^{- t / R C}$
Define the time constant $τ = R C$
Use the charging equation $Q = Q_{0} (1 - e^{- t / R C})$
Find the time constant from a linearised (ln) graph of experimental data

Discharging

A charged capacitor discharging through a resistor loses charge exponentially — current, p.d. and charge all follow the same exponential decay shape, since they are all proportional to each other.

Q = Q_{0} e^{- t / R C}

Exponential discharge.

1Use

Q = Q_{0} e^{- t / R C}

Q = 5.0 \times 1 0^{- 4} \times e^{- 6.0/3.0} = 5.0 \times 1 0^{- 4} \times e^{- 2.0}

Answer

Q \approx 6.77 \times 1 0^{- 5}

The time constant

The time constant, $τ = R C$ , is the time for charge (or p.d., current) to fall to 37% of its initial value on discharge, or rise to 63% of its final value on charge.

τ = R C

Time constant.

1Use

τ = R C

τ = 4.0 \times 1 0^{3} \times 250 \times 1 0^{- 6}

Answer

τ = 1.0

Tip — After 5 time constants, a capacitor is conventionally treated as "fully" discharged (under 1% of charge remains).

Charging

Charging follows the mirror-image curve, with charge and p.d. rising toward their final values, while current starts at a maximum and decays exponentially.

Q = Q_{0} (1 - e^{- t / R C})

Exponential charging.

Finding the time constant from a graph

Taking natural logs of the discharge equation gives a straight line: plotting $ln Q$ against $t$ has gradient $- 1/ R C$ .

ln Q = ln Q_{0} - \frac{t}{R C}

Linearised discharge equation.

1Gradient

= - \frac{1}{R C}

, so

R C = \frac{1}{0.20} = 5.0

C = \frac{R C}{R} = \frac{5.0}{5.0 \times 1 0 ^{3}}

AnswerC = 1.0×10⁻³ F (1000 µF)

Equation recap

Q = Q_{0} e^{- t / R C}

Exponential discharge.

τ = R C

Time constant.

Common mistakes to avoid

Confusing the time constant (37% fall) with the time for charge to halve.

The time constant corresponds to a fall to 1/e (≈37%); halving time is a different value, RC ln2 ≈ 0.69RC.

Thinking a capacitor fully discharges in exactly one time constant.

Discharge is exponential and never mathematically reaches zero — after 5 time constants it is treated as "fully" discharged.

Key takeaways

Discharge: Q, V and I all fall exponentially with the same time constant RC.
Time constant τ=RC: time to fall to 37% (discharge) or rise to 63% (charge).
Plotting ln(Q) against t gives gradient −1/RC, the standard experimental method for finding RC.

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Charging and Discharging Capacitors

40 min Video by Lascells Science Module 6.1 Capacitors