Question

6) Capacitor Potential Energy: (a) At time t = 0 a capacitor in an RC circuit is fully charged, so that a voltage V. resides across the capacitor of capacitance C. What is the potential energy stored in the electric field of the capacitor? (b) The capacitor begins to discharge at timet > 0. What is the potential energy of the capacitor as a function of time? (e) Sketch the potential energy of the discharging capacitor as a function of time on the graph below. U

electromagnetic

Answer 1

(a) $\mathcal{E} = \frac12 CV_0^2.$

(b) In a RC circuit the time constant is $\tau = RC$ . This is easy to see since RC has the dimension of time. When the capacitor starts discharging the charge stored decayes exponentially, so does the potentia. So, $V(t) = V_0 \left(\exp\left(\frac{t}{RC}\right)\right).$ So, $\mathcal{E} = \frac12 C V(t)^2 = \frac12 C V_0^2 \left(\exp\left(\frac{t}{RC}\right)\right)^2.$

(c)

1.0 0.8 8 0.6 0.4 0.2 0.2 0.4 0.6 0.8 1.0 1.2 1.4