The series circuit consists of a 10^-5 F capacitor, a 300 Ω resistor, and a 0.2 H inductor. With an initial capacitor charge of 10^-6 C and an initial current of zero, calculate the capacitor charge Q at any time t.

To find the charge Q on the capacitor at any time t, we need to analyze the circuit as an RLC series circuit with an initial charge on the capacitor but no initial current.

An RLC series circuit is governed by the second-order differential equation: v(t) = L * (d²i/dt²) + R * (di/dt) + (1/C) * q(t)

In the above equation, v(t) is the voltage across the circuit, L is the inductance, R is the resistance, C is the capacitance, i(t) is the current, and q(t) is the charge on the capacitor as a function of time t.

Because there is no external voltage being applied to this circuit (v(t) = 0), the equation simplifies to the homogeneous second-order differential equation: 0 = L * (d²i/dt²) + R * (di/dt) + (1/C) * q(t)

We can solve this differential equation for the charge q(t) on the capacitor. However, the nature of the solution depends on the damping factor, which is determined by the resistance value in the circuit.

The general solution of the second-order differential equation for an underdamped system (which happens if the resistance is relatively small compared to the reactance of the inductor and capacitor) is: q(t) = Qe^(-αt) * cos(ω_d * t + φ)

Here, α represents the damping factor, ω_d is the damped natural frequency, and φ is the phase constant determined by the initial conditions. α and ω_d are given by: α = R/(2L), ω_d = sqrt(1/(L*C) - α^2)