Discharging a capacitor
Section: Capacitance | Syllabus: Cambridge AS 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 Circuit A 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 Curves Graphs 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 (). The Time Constant () Key Point: The time constant = 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] × [] = CV × VA = A sA = s. Logarithmic Analysis To verify the exponential relationship experimentally, we take natural logarithms: V = V_0 - 1/RC t A graph of V against t should be a straight line with gradient -1/RC .
Figure 19.20: Logarithmic Plot for Discharge Graph 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 Timeline Question: A 1000~μF capacitor charged to 1.5 V discharges through a 25~k resistor.
Calculate the current after 20 s. Solution Initial Current: I_0 = V / R = 1.5 / 25000 = 6.0 × 10^-5 A. Time Constant: = 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.
Smoothing Circuits 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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