A capacitance C charged to a potential difference V is discharged by connecting its plates through a resistance R. Find the heat dissipated in one time constant after the connections are made. Do this by calculating $\int\text{i}^2\text{R}$ dt and also by finding the decrease in the energy stored in the capacitor.
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Energy stored at a part time in discharging $=\frac{1}{2}\text{CV}^2\Big(\text{e}^{\frac{-\text{t}}{\text{RC}}}\Big)^2$
Heat dissipated at any time = (Energy stored at t = 0) - (Energy stored at time t)
$=\frac{1}{2}\text{CV}^2-\frac{1}{2}\text{CV}^2\big(-\text{e}^{-1}\big)^2$ $=\frac{1}{2}\text{CV}^2\big(1-\text{e}^2\big)$
Heat dissipated at any time = (Energy stored at t = 0) - (Energy stored at time t)
$=\frac{1}{2}\text{CV}^2-\frac{1}{2}\text{CV}^2\big(-\text{e}^{-1}\big)^2$ $=\frac{1}{2}\text{CV}^2\big(1-\text{e}^2\big)$
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