Capacitor Charge and Discharge

The time-dependent behaviour of capacitors charging and discharging through resistors is one of the most important practical and theoretical topics in AQA A-Level Physics (specification 3.7.4). The exponential functions governing these processes appear frequently in exam questions.

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The RC Circuit

An RC circuit consists of a resistor (R) and a capacitor (C) connected in series. The product RC is called the time constant and determines how quickly the capacitor charges or discharges.

τ = RC

where τ is the time constant (s), R is the resistance (Ω), and C is the capacitance (F).

Dimensional check: [Ω][F] = [V A⁻¹][C V⁻¹] = [C A⁻¹] = [A s A⁻¹] = [s] ✓

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Discharging a Capacitor

When a charged capacitor (initial charge Q₀, initial voltage V₀) is disconnected from the supply and connected across a resistor R, it discharges exponentially.

Derivation of the Discharge Equation

At any instant during discharge, Kirchhoff's voltage law gives:

V_C = IR → Q/C = IR

Since I = −dQ/dt (the negative sign indicates that Q is decreasing):

Q/C = −R(dQ/dt)

Separating variables: dQ/Q = −dt/(RC)

Integrating: ln Q = −t/(RC) + constant

At t = 0, Q = Q₀: constant = ln Q₀

ln(Q/Q₀) = −t/(RC)

Q = Q₀ e^(−t/RC)

Since V = Q/C: V = V₀ e^(−t/RC)

Since I = V/R: I = I₀ e^(−t/RC) where I₀ = V₀/R

All three quantities — charge, voltage, and current — decay exponentially with the same time constant τ = RC.

Meaning of the Time Constant

After one time constant (t = RC):

• Q = Q₀ e⁻¹ = Q₀ × 0.368 ≈ 37% of Q₀
• V = V₀ × 0.368
• I = I₀ × 0.368

After 5 time constants (t = 5RC):

• Q = Q₀ e⁻⁵ = Q₀ × 0.0067 ≈ 0.67% of Q₀

The capacitor is considered fully discharged after about 5RC.

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Charging a Capacitor

When an uncharged capacitor is connected to a supply of EMF ε through a resistor R, it charges exponentially.

Charging Equations

Q = Q₀(1 − e^(−t/RC)) where Q₀ = Cε V = V₀(1 − e^(−t/RC)) where V₀ = ε I = I₀ e^(−t/RC) where I₀ = ε/R

During charging:

• Charge and voltage rise exponentially towards their maximum values.
• Current decreases exponentially from its initial maximum value I₀ = ε/R (when the capacitor is uncharged and the full EMF appears across R) to zero (when the capacitor is fully charged and V_C = ε).

After one time constant during charging:

• Q = Q₀(1 − e⁻¹) = Q₀ × 0.632 ≈ 63% of Q₀
• V = 63% of V₀

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Graphical Representations

Described diagram — Discharge graphs: Three graphs are shown. (1) Q-t: Curve starts at Q₀ and decays exponentially towards zero. At t = RC, Q = 0.37Q₀. (2) V-t: Same shape as Q-t, starting at V₀ and decaying to zero. (3) I-t: Starts at I₀ = V₀/R and decays to zero. For discharge, the current flows in one direction only.

Described diagram — Charging graphs: Three graphs. (1) Q-t: Starts at zero and rises towards Q₀ = Cε, following Q = Q₀(1 − e^(−t/RC)). At t = RC, Q ≈ 0.63Q₀. (2) V-t: Same shape, rising from 0 to ε. (3) I-t: Starts at I₀ = ε/R and decays exponentially to zero — identical in shape to the discharge current graph.

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Worked Examples

Worked Example 1 — Discharge Calculation

Question: 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 discharge current; (c) the voltage after 8.0 s; (d) the time for the voltage to fall to 1.0 V.

Solution:

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

(b) I₀ = V₀/R = 12 / (47 × 10³) = 2.55 × 10⁻⁴ A = 0.255 mA

(c) V = V₀ e^(−t/RC) = 12 × e^(−8.0/4.7) = 12 × e^(−1.702) = 12 × 0.1822 = 2.19 V