An LC circuit contains a 20 mH inductor and a `50 mu F` 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 the oscillalions?
(b) What is the natural frequency of the circuit?
(c ) At what time is the energy stored? (i) Completely electrical ? (ii) Completely magnetic ?
(d) At what time 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) L = 20 mH = 20 `xx 10^(-3)` H
C = 50 `mu f = 50 xx 10^(-6)` F
`q_(0) = 10 mC = 10^(-2) ` C
Total initial energy
`U_(0) = (1)/(2) (q_(0)^(2))/(C) = (1)/(2) xx (10^(-2) xx 10^(-2))/(50 xx 10^(-6)) J` = 1.0 J
Yes , sum of energies stored in L and C is conserved If R = 0
(b) Natural frequency of the circuit is
f = `(1)/(2pi sqrt(LC))`
f = `(1)/( 2xx 3.14 sqrt(20 xx 10^(-3) xx 50 xx 10^(-6))`
(C) (i) charge on the capacitor at any instant is
q = `q_(0) ` cos `omega t `
or ` " " q = q_(0) cos (2pi)/(T)` t
The energy stored will be completely electrical when the charge will have maximum value.
when t = 0 , `(T)/(2 ) , T , (3T)/(2)` .....
(ii) Energy stored is completely magnetic when electirc energy as well as charge on the capacitor will be zero , ie.for
t `= (T)/(4), (3T)/(4) , (5T)/(4)` .....
(d) Maximum energy stored in capacitor = `(1)/(2) xx (q_(0)^(2))/(C)`
Half the total energy =`(1)/(2) ((1)/(2)) xx (q_(0^(2))/(C))`
= ` ((1)/(2)x((q_(0))/(sqrt(2)))^(2))/(C)`
energy will be half when charge becomes `(q_(0))/(sqrt(2))`
q = `q_(0) " cos " omega`t
`rArr (q_(0))/(sqrt(2)) = q_(0) cos (2pi)/(T) ` t
`cos (2 pi)/(T) t = (1)/(sqrt(2)) = cos (2n + 1) (pi)/(4)`
t = (2n + 1) `(T)/(8)`
energy will be half when
t = ` (T)/(8), (3T)/(8), (5T)/(8) `.....
here, T = `(1)/(f)` = 6.3 ms
(e) The resistance damps out the LC oscillations eventuall .
the whole of the initial energy equal to 1.0 J is eventually dissipated as heat.