Charging and Discharging Capacitors

When a capacitor charges or discharges through a resistor, the process follows exponential curves. Understanding these curves — and the mathematics behind them — is essential for the Edexcel A-Level specification and has wide practical applications in timing circuits, sensors, and signal processing.

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Discharging a Capacitor

Consider a capacitor of capacitance C, initially charged to voltage V₀ (storing charge Q₀ = CV₀), connected across a resistor R. When the switch is closed, charge flows through the resistor and the capacitor discharges.

The charge, voltage, and current all decay exponentially:

Q = Q₀e^(−t/RC)

V = V₀e^(−t/RC)

I = I₀e^(−t/RC)

where I₀ = V₀/R is the initial current.

Why Exponential?

At any instant, the current through the resistor is I = V/R, and V = Q/C. As charge flows off the plates, Q decreases, so V decreases, so I decreases. The rate of discharge is proportional to the amount remaining — this is the defining property of exponential decay.

Mathematically: dQ/dt = −Q/RC, which has the solution Q = Q₀e^(−t/RC).

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The Time Constant

The time constant τ (tau) is defined as:

τ = RC