Question

Consider an electric circuit (see figure below) consisting of an inductor with an inductance of L = 0.5 Henry and a capacitor with a capacitance of C = 0.1 Farad connected in series with a voltage source of V (t) = 50sin (wt) Volts. (a) Show that the differential equation satisfied by the charge q (t) in this circuit is q + 20 q = l00 sin(wt). (b) Determine the value (s) w_res for which resonance occurs. (c) Solve the initial value problem to find the charge q_NR (t) as a function of time in the case whore there IS NO resonance. (d) Solve the initial value problem to find the charge q_R (t) as a function of time for one of the values w_res where there IS resonance (you can choose which w_res value to consider).

Answer 1

v - L di/dt - q/e = 0 L di/dt + q/e = 50 sin omega t Rightarrow 1/2 d/dt (dq/dt) + q/0.1 = 50 sin omega t Rightarrow d^2q/dt^2 + 20 q = 100 sin omega t Rightarrow q = 20q = 100 sin omega t ---(1) b) For resources to occurs omega = (1/squareroot (LC)) = 1/Squareroot(0.5 times 0.1) = 1/Squareroot(5/100) = 10/Squareroot(5) omega = 10/Squareroot(5) = 4.472 radls