Final answer:
The second-order differential equation for the charge q in a series RLC circuit, which includes resistor R, inductor L, and capacitor C, is given by L(d^2q/dt^2) + R(dq/dt) + q/C = E(t), where E(t) is the voltage source as a function of time.
Step-by-step explanation:
The mathematical model that gives the change of charge (q) according to time in a series RLC circuit is derived from the Kirchhoff's voltage law applied to the circuit, which includes a resistor (R), an inductor (L), and a capacitor (C). The differential equation for an RLC circuit with a charge q on the capacitor is:
L\( \frac{d^2q}{dt^2} \) + R\( \frac{dq}{dt} \) + \( \frac{q}{C} \) = E(t)
This is a second-order linear differential equation where L is the inductance, R is the resistance, C is the capacitance, and E(t) represents the voltage source as a function of time. The equation describes how the charge on the capacitor varies with time due to the interplay between the stored energy in the inductor and capacitor and the energy dissipated as heat in the resistor.