An LC circuit contains a 20 mH inductor and a 50 `muF` 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?

Explanation

(a) 1.0 J. Yes, sum of the energies stored in L and C is conserved if R = 0.
(b) `omega=10^(3)rad s^(-1), v=159Hz`
(c ) `q=q_(0)cos omega t`
(i) Energy stored is completely electrical at `t=0, (T)/(2), T, (3T)/(2), ……`
(ii) Energy stored is completely magnetic (i.e., electrical energy is zero) at `t=(T)/(4), (3T)/(4), (5T)/(4).......`, where `T=(1)/(v)=6.3ms`.
(d) At `(T)/(8), (3T)/(8), (5T)/(8) ..........`, because `q=q_(0)"cos"(omegaT)/(8) =q_(0)"cos"(pi)/(4)=(q_(0))/(sqrt(2))`.
Therefore, electrical energy `=(q^(2))/(2C)=(1)/(2)((q_(0)^(2))/(2C))` which is half the total energy.
R damps out the LC oscillations eventually. The whole of the initial energy (= 1.0 J) is eventually dissipated as heat.