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