Aseries circuit has a capacitor of 10−5 f, a resistor of 3 × 102 ω, and an inductor of 0.2 h. the initial charge on the capacitor is 10−6 c and there is no initial current. find the charge q on the capacitor at any time t.

Given the values to proceed to solve the exercise, we resort to the solution of the exercise through differential equations.

The problem can be modeled through a linear equation, in the form:

With the initial conditions as,

Where Q(t) is the charge.

The general solution of a linear equation is given as:

Applying this definiton in our differential equation we have that

To find b and a we use the first equation and find the roots:

Then we have

To find the values of the Constant we apply the initial conditions, then

And for the derivate:

We have a system of 2x2:

Solving we have:

The we can replace at the equation and we have that the Charge at any moment is given by,

If we obtain the derivate we find also the Current, then