Question

A series RLC circuit with R = 6 ohm, C = 0.02 Farad and L = 0.1 has no applied voltage. Find the subsequent current in the circuit if the initial charge, on the capacitor is q0 and the initial current is zero.

Answer 1

The magnitude of the subsequent current can be obtained from the equation of damped oscillations : $\ddot{q}+2\beta\dot{q}+\omega_0^2q=0$ , where $\beta=\frac{R}{2L},\omega_0=\frac{1}{\sqrt{LC}}$

In our case $\omega_0^2-\beta^2<0$ then is a case of aperiodic damping. The equation solution is

$q=c_1e^{-\alpha_1t}+c_2e^{-\alpha_2t}$ where $\alpha_1=\beta+\sqrt{\beta^2-\omega_0^2}$ , $\alpha_2=\beta-\sqrt{\beta^2-\omega_0^2}$

From the conditions $q(0)=q_0=c_1+c_2$ and $\dot{q}(0)=I(0)=0=\alpha_1c_1+\alpha_2c_2$ we find

$c_1=-\frac{\alpha_2q_0}{\alpha_1-\alpha_2}$ ,$c_2=\frac{\alpha_1q_0}{\alpha_1-\alpha_2}$ then $\dot{q}(t)=I(t)=q_0\frac{\alpha_1\alpha_2}{\alpha_1-\alpha_2}(e^{-\alpha_1t}-e^{-\alpha_2t})$ . As $\alpha_1=50, \alpha_2=10$ then

$I(t)=12.5q_0(e^{-50t}-e^{-10t})$ A