The charge in a series RLC circuit can be modeled using a differential equation similar to that of the spring-mass oscillator. A resistor in a circuit acts like a damper, an inductor acts like the mass, and the capacitor acts like the spring. An applied voltage (e.g. a battery or an AC power source) acts like a forcing function.
The differential equation that models the charge, Q, in the RLC circuit is
LQ'' + RQ' + 1/C Q = E(t)
where L is the inductance in henries, R is the resistance in ohms, C is the capacitance in farads, and E(t) is the applied voltage.
(a) A series circuit has a capacitor of 0.25 x 10^-6 farad and an inductor of 1 henry. If the initial charge on the capacitor is 10^-6 coulomb and there is no initial current (i.e. Q'(0) = 0), find the charge Q on the capacitor at any time t.
(b) If a series circuit has a capacitor of C = 0.8 x 10^-6 farad and an inductor of L = 0.2 henry, find the resistance R so that the circuit is critically damped.