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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.
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.
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).
The time constant τ (tau) is defined as:
τ = RC
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