การปล่อยประจุของตัวเก็บประจุ

The decay that never quite finishes

• A discharging capacitor loses 63% of its charge in the first time constant, then 63% of what remains in the next, and so on for ever.
• After five time constants less than 1% is left, but the curve never reaches zero. It cannot, because the rate of loss is always proportional to how much is left.
• Every quantity that behaves that way, radioactive decay included, gives the same curve, so learning it once serves twice.
• This lesson is the discharge 放电 equations, the time constant 时间常数, and the log plot that turns the curve into a straight line.

Where the exponential comes from

• Close the switch and the capacitor's p.d. drives a current through the resistor. Kirchhoff's second law gives $V_C = V_R$, and with $V_C = Q/C$, $V_R = IR$ and $I = -dQ/dt$:

• The rate of loss is proportional to the charge remaining. That is the defining property of exponential decay 指数衰减, and it is why the curve flattens as it falls.

The three equations

• Charge, p.d. and current all decay together, with the same time constant:

• The current is largest at the instant the switch closes, $I_0 = V_0/R$, because the full initial p.d. is then across the resistor. It falls as the p.d. falls.
• The area under the current-time graph is the charge that has flowed. All three curves have the same shape, so measuring $\tau$ from any one of them gives $RC$.

Steep at first, then ever flatter, and never quite zero

The time constant

• The time constant is:

• Its unit is the second, since an ohm times a farad is a second.
• It is the time for the decaying quantity to fall to $\dfrac{1}{e} \approx 37\%$ of its starting value. After $2\tau$ about 13.5% remains, and after $5\tau$ less than 1%.
• A large $R$ or a large $C$ means a slow discharge: a big store emptying through a narrow pipe.

Worked example: a full discharge question

Worked example: reading $\tau$ off a graph

The log plot

• Take logarithms of $V = V_0 e^{-t/RC}$:

• So a graph of $\ln V$ against $t$ is a straight line with gradient $-1/RC$ and intercept $\ln V_0$.
• This is the standard experimental method: it turns a curve into a line, so $RC$ comes from a gradient rather than from a single reading, and any departure from a straight line shows the decay is not exponential.

Marks that slip away

You've got it