## Question

A series circuit has a capacitor of  F and an inductor of H. If the initial charge on the capacitor is  C and there is no initial current, find the charge on the capacitor Q(t).

#### Solution

Capacitance, C =

Inductance, L = H

Initial Charge, =  C

Initial Current, or = = 0 A

Since a resistor was not mentioned to be a part of the circuit, we can assume that .

Also, there’s no impressed voltage mentioned, therefore, we can assume that .

Using the form

The differential equation for this problem will be

The characteristic equation will then be

Using the quadratic formula, the roots of the characteristic equation is

Since the roots are complex imaginary numbers, the general equation for the amount of charge in the circuit will then be

Also note that, since the roots are imaginary, the oscillation of the circuit is underdamped.

To find the function for current, differentiate the previous equation, yielding

Next, substitute the initial conditions to solve for the undetermined coefficients

Finally, the equation for the amount of charge in the given circuit is