First slide

Capacitors and capacitance

Question

The plates of a capacitor are charged to a potential difference of 100 V and then connected across a resistor. The potential difference across the capacitor decays exponentially with respect to time. After one second the potential difference between the plates of the capacitor is 80 V. What is the fraction of the stored energy which has been dissipated?

Moderate

Solution

Energy loss $ΔU = \frac{1}{2} {CV}_{0}^{2} - \frac{1}{2} {CV}^{2}$
Fractional energy loss
$\frac{ΔU}{U_{0}} = \frac{\frac{1}{2} {CV}_{0}^{2} - \frac{1}{2} {CV}^{2}}{\frac{1}{2} {CV}_{0}^{2}} = \frac{V_{0}^{2} - V^{2}}{V_{0}^{2}}$
$= \frac{(100)^{2} - (80)^{2}}{(100)^{2}} = \frac{20 \times 180}{(100)^{2}} = \frac{9}{25} = 0 .36$

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