Question

An oscillating LC circuit has a current amplitude of 9.70 mA, a potential amplitude of 274 mV, and a capacitance of 233 nF. What are (a) the period of oscillation, (b) the maximum energy stored in the capacitor, (c)the maximum energy stored in the inductor, (d) the maximum rate at which the current changes, and (e) the maximum rate at which the inductor gains energy?

Answer 1

current ampitude I max= 9.70*10^-3A

potential amplitude Vmax = 274 *10^-3V

capacitance C = 220 nF = 233 *10^-9F

We know that Vmax = Imax*Xc

We knoqw that capacitance Xc = 1/?C

274 *10^-3V   = 9.70*10^-3A*1/?*233 *10^-9F

? = (9.70 *10^-3A/274*10^-3V)/233 *10^-9F = 1.51937*10⁵rad/sec

We know that period of oscillation T = 2?/? = 2?/1.5193*10⁵rad/sec =41.355*10^-6sec

                                                                     =  41.355?C

B) We know that maximum stored in the capacitor is Umax =(1/2)CVmax²

                                  Umax =  (1/2)*233 *10^-9F*( 274*10^-3V)²

                                            = 8.746 *10^-9J = 8.746 nJ

C) maximum energy stored in the inductor is equal to maximum energy stored in the capacitor Then maximum energystored in the inductor = 8.746 nJ

D) we know that maximum voltage across the inductorVmax = Ldi/dt

L = CVmax²/Imax²

Then   Vmax =(CVmax²/Imax²)di/dt

Then maximum rate of change of current (di/dt)max  = Imax²/CVmax

                                                                               = (9.7*10^-3A)²/233 *10^-9F *274 *10^-3V

                                                                               = 1.473 *10³A/sec

E) We know that energy gained by the inductor is U =(1/2)LI²

Then dU/dt =(1/2)LImax²?Sin(2?t)

Then (dU/dt )max =( 1/2)LI²? =(1/2)Vmax * Imax = (1/2)*9.7 *10^-3A*274*10^-3V

                                                                                  = 1.3289 mW