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A LCR circuit behaves like a damped harm...

A LCR circuit behaves like a damped harmonic oscillator. Comparing it with a physical spring mass damped oscillator having damping constant ‘b’. If the amount of initial charge on the capacitor be `Q_(0).` then the amplitude of the amount of charge on the capacitor as a function of time t will be:

A

`Q= Q_(0)e^(-(Rt)/(2l))`

B

`Q= Q_(0) (1- e^(-(2Rt)/(L)))`

C

`Q= Q_(0)e^(-(Rt)/(L))`

D

`Q= Q_(0) (1- e^(-(Rt)/(L)))`

Text Solution

Verified by Experts

The correct Answer is:
A
For damped oscillation : `ma + bv + kx= 0`
`m(d^(2)x)/(dt^(2))+ b(dx)/(dt) + kx= 0` ….(i)
For LCR series circuit
`-iR -L (di)/(dt)- (q)/(C )= 0`
`L(d^(2)q)/(dt^(2))+ R (dq)/(dt) + (1)/(C )q= 0` …(ii)
Comparing (i) & (ii), `L harr m, C harr (1)/(k), R harr b`
Amplitude `A= A_(0)e^(-(bt)/(2m)) therefore` Amplitude charge `Q= Q_(0) e^(-(Rt)/(2l))`
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