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An LC circuit contains a 20 mH inductor and a 50 uF 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 ? |
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Answer» Solution :Here, L = 20 mH `=20 xx 10^(-3) H, C = 50 muC = 50 xx 10^(-6)` F and `q_(m) = 10 mC = 10 xx 10^(-3) C = 0.01 C`, when t=0. (a) `therefore` Total energy stored initially `U = 1/2 q_(0)^(2)/C = (0.01)^(2)/(2 xx 50 xx 10^(-6)) = 1J` Yes, the energy remains conserved during LC oscillations provided that resistance R of the circuit is zero. (b) Natural frequency of the circuit. `v_(0) =1/(2pi sqrt(LC)) = 1/(2 xx 3.14 xx sqrt(20 xx 10^(-3) xx 50 xx 10^(-6))) = 159 Hz` and natural frequency `omega_(0) = 2pi v_(0) = 2pi xx 159 = 1000 rad s^(-1)` ( c)Let time period of oscillations be T, where T=`1/v_(0) =1/159 = 6.3 xx 10^(-3)s = 6.3 rms` then, (i) at times t=0, `T/2, T , (3t)/2, 2T` .......... the energy stored is completely electrical and (ii) at times t =`T/4 , (3T)/4, (5T)/4, (7T)/4`, ......... the energy stored is completely MAGNETIC. (d) Let at time t the electrical and magnetic energies are equal i.e., `q^(2)/(2C) =1/2 LI^(2) = 1/2 [(q_(0)^(2)/(2C))]` i.e. at time t, q =`+- q_(0)/sqrt(2)` `therefore q = q_(0) cos omega` `RARR +- q_(0)/sqrt(2) = q_(0) cos omega t` or `cos omega t = +- 1/sqrt(2)` or `omega t = pi/4, (3pi)/4, (5pi)/4`..... substituting `omega = (2pi)/T`, we have `t=T/8, (3T)/8, (5T)/8, (7T)/8`,......... e) If a resistor is inserted in the circuit, continuously some of the INITIAL energy (U=1) will bedissipated as heat energy. As a result, amplitude of LC oscillations gradually decreases with time and finally the oscillations stop altogether. |
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