- #1

Potatochip911

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## Homework Statement

I'm trying to derive the voltage waveform across the capacitor for a discharging capacitor that has been fully charged by a DC power supply ##v_0##, i.e. ##v_c(t=0)=v_0## and then at ##t=0## the switch is flipped and the capacitor starts to discharge.

## Homework Equations

## The Attempt at a Solution

From KVL we obtain

$$v_c=\frac{q}{c}=iR$$

Taking the time derivative of this and since capacitor is discharging ##-\frac{dq}{dt}=i##... $$-\frac{i}{RC}=\frac{di}{dt}\\ -\int_0^t \frac{dt}{RC}=\int_{i_0}^{i}\frac{di}{i}\\ -\frac{t}{RC}=\ln (\frac{i}{i_0})\\ i=i_0e^{-\frac{t}{RC}}$$

Now to solve for ##i_0## at t=0 ##\frac{q}{c}=i_0Re^{-\frac{t}{RC}}## becomes $$\frac{v_0}{c}=i_0R\\ i_0=\frac{v_0}{RC}\Longrightarrow i=\frac{v_0}{RC}e^{-\frac{t}{RC}}$$

Finally from KVL $$v_c=iR=\frac{v_0}{C}e^{-\frac{t}{RC}}$$

which is not the correct answer.