An LC circuit contains a 20 mH inductor and a 50 μF capacitor with an initial charge of 10 mC. The resistance of the circuit is negligible. Let the instant the circuit is closed be t = 0.
(a) What is the total energy stored initially? Is it conserved during LC oscillations?
(b) What is the natural frequency of the circuit?
(c) At what time is the energy stored
(i) completely electrical (i.e., stored in the capacitor)?
(ii) completely magnetic (i.e., stored in the inductor)?
(d) At what times is the total energy shared equally between the inductor and the capacitor?
(e) If a resistor is inserted in the circuit, how much energy is eventually dissipated as heat?
Given, an LC circuit.
Inductance, L = 20 mH = 20
Capacitance, C = 50 = 50
Initial charge, Q = 10 mC = 10
(a) Total initial energy stored in the circuit,
This energy stored shall remain conserved in the absence of resistance.
(b) Angular frequency,
(c) Let, at any instant the energy stored in the circuit is completely the electrical charge on the capacitor,
then,
Q is maximum only when,
.... (1)
Hence,
Energy stored is completely electrical at t = 0, T/2, T, 3T/2,.. and so on.
Now, let the energy stored be completely magnetic at any instant when electrical charge = 0.
i.e., q = 0.
From equation (1)
where, n =1,2,3,...
Thus, energy stored is completely magnetic at
(d) Energy shared between inductor and the capacitor is equal means the energy shared is half times the maximum energy of the circuit.
Electrical energy = which is half of the total energy.
This implies Q=
Using equation (1) we have,
i.e.,
t =
During these values of t, total energy will be shared equally between the inductor and the capacitor.
(e) Resistor damps out the LC oscillations. The whole of the initial energy 1.0 J, is eventually dissipated as heat.
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