Discharging a capacitor
Section: Capacitance | Syllabus: Cambridge A Level Physics (9702)
Exponential Decay
When a capacitor discharges through a resistor, the current I, potential difference V, and charge Q all decrease exponentially over time.
Figure 19.18: Capacitor Discharge CircuitA charged capacitor C connected in series with a resistor R and a switch. When the switch closes, charge flows through the resistor.
Figure 19.19: Exponential Decay CurvesGraphs showing how charge Q, current I, and potential difference V decrease exponentially with time during discharge. After one time constant τ = RC, each quantity falls to 37% of its initial value.
x = x_0 e^-(t/RC)Where x represents Q, V, or I, and RC is the time constant (τ).
The Time Constant (τ)Key Point: The time constant τ = RC is the time for the current, potential difference, and charge for a discharging capacitor to fall to 1/e ≈ 36.8\% of their original values.
Unit Check: [RC] = [F] × [Ω] = (C/V) × (V/A) = (A s/A) = s.
Logarithmic Analysis
To verify the exponential relationship experimentally, we take natural logarithms:
ln V = ln V_0 - (1/RC) tA graph of ln V against t should be a straight line with gradient -1/RC.
Figure 19.20: Logarithmic Plot for DischargeGraph of ln V against time t showing a straight line with negative gradient equal to -1/RC. The y-intercept gives ln V₀.
Worked Examples
Worked Example: Discharge TimelineQuestion: A 1000~ capacitor charged to 1.5 V discharges through a 25~kΩ resistor. Calculate the current after 20 s.
Solution- Initial Current: I_0 = V / R = 1.5 / 25000 = 6.0 × 10^-5 A.
- Time Constant: τ = RC = (1000 × 10^-6) × 25000 = 25 s.
- Current after 20s: I = I_0 e^-t/RC = 6.0 × 10^-5 × e^-20/25 ≈ 2.7 × 10^-5 A.
Capacitors are used in power supplies to "smooth" rectified D.C. by discharging slowly between peaks, maintaining a more constant output voltage.
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