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 oscillations?
(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

Inductance of the induc tor, `L=20mH=20xx10^(-3)H`
Capacitance of the capacitor, `C=50muF=50xx10^(-6)F`
Initial charge on the capacitor , `Q=10mC=10xx10^(-3)C`
(a) Total energy stored initially in the circuit is given as:
`E=(1)/(2)(Q^(2))/(C)`
`=((10xx10^(-3))^(2))/(2xx50xx10^(-6))=1J`
Hence , the toal energy stored in the LC circuit will be conserved because there is no resistor connected in the circuit.
(b) Natural frequency of the circuit is giv en by the relation,
`v=(1)/(2pisqrt(LC))`
`=(1)/(2pisqrt(20xx10^(-1)xx50xx 10^(-6))`
`=(10^(3))/(2pi)=159.24Hz`
`(10^(3))/(2pi)=159.25Hz`
Natural angular frequency,
`omega_(r)=(1)/sqrt(LC)`
`=(1)/(sqrt(20xx10^(-30)xx50xx10^(-6))=(1)/sqrt(10^(-6))=10^(3)` rad/s
Hence, the natural frequecy of the circuit is `10^(3)` rad/s.