A series RLC circuit has resistance \(R\), inductance \(L\), and capacitance \(C\). At what time does the energy in the circuit reach half of its initial value?

Inductive energy, \(W_L\), is given by the formula: \(W_L = \frac{1}{2}LI^2\), where \(I\) is the current flowing through the inductor. Similarly, capacitive energy, \(W_C\), can be given by the formula: \(W_C = \frac{1}{2}C{V_C}^2\), where \(V_C\) is the voltage across the capacitor. The total energy in the circuit, \(W\), is the sum of the inductive and capacitive energies: \(W = W_L + W_C\).

Kirchoff's loop law tells us that the sum of voltage changes around a closed circuit must equal zero. In our RLC circuit, we have: \(V_R + V_L + V_C = 0\). From Ohm's law, we know the voltage across the resistor is \(V_R = IR\), and the voltage across the inductor is \(V_L = L\frac{dI}{dt}\), and from the definition of capacitance, \(V_C = \frac{Q}{C}\), where \(Q\) is the charge stored in the capacitor. So, the Kirchoff's loop law equation becomes: \(IR + L\frac{dI}{dt} + \frac{Q}{C} = 0\). Now, we know that \(I = \frac{dQ}{dt}\), so we can rewrite the above equation as: \(R\frac{dQ}{dt} + L\frac{d^2Q}{dt^2} + \frac{Q}{C} = 0\). This is a second-order linear differential equation for charge \(Q\).

In general, the solution for the differential equation can take the form of damped oscillations. But the precise form of the solution depends on the damping factor defined by \(\zeta = \frac{R}{2\sqrt{LC}}\). For this problem, let's assume that the solution looks like: \(Q(t) = Q_0e^{-\alpha t}\cos(\omega t + \phi)\), where \(Q_0\), \(\alpha\), \(\omega\), and \(\phi\) are constants. Substitute this expression into the differential equation and solve for the constants. This step can be quite involved and depends on the specific values of \(R\), \(L\), and \(C\).

Now, we calculate \(I(t) = \frac{dQ}{dt}\) using the expression for \(Q(t)\). Substitute these expressions into \(W_L = \frac{1}{2}LI^2\) and \(W_C = \frac{1}{2}C{V_C}^2\), and compute the total energy \(W = W_L + W_C\) as a function of time.

We finally need to determine when the energy reaches half of its initial value, i.e., \(W\left(\frac{1}{2}\right) = \frac{1}{2}W(0)\). This equation involves the constants \(\alpha\), \(\omega\), and \(\phi\) found earlier, and solving for the time. The exact solution depends on the specific values of \(R\), \(L\), and \(C\). To summarize, the steps to find the time at which the energy in a series RLC circuit reaches half of its initial value are: 1. Write expressions for energy stored in inductor and capacitor 2. Write expressions for current and voltage across the capacitor using Kirchoff's law and Ohm's law 3. Solve the second-order linear differential equation for charge \(Q\) 4. Calculate the total energy in terms of charge and current expressions 5. Find the time at which energy is half the initial energy by setting \(W\left(\frac{1}{2}\right) = \frac{1}{2}W(0)\) and solving for time