An oscillating circuuit incorporates a leaking capacitor. Its capacitance is equal `C` and active resistance to `R`. The coil inductance is is `L`. The resistance of the coil and the wires is negligible. Find `:`
`(a)` the damped oscillation frequecy of such a circuit ,
`(b)` its quality factor.

Explanation

In leaky condenser
`(dq)/( dt)=I-I^(')` where` I^(') = (V)/( R)=` leak current
Now `V=(q)/(C)=-L(dI)/( dt)=-L(d)/(dt)((dq)/(dt)+(V)/(R))`
`=-L(d^(2)q)/(dt^(2))-(L)/(RC)(dq)/(dt)`
or `ddot(q)+(1)/(RC) (dQ)/( dt)+(1)/(LC)q=0`
Then `q=q_(m)e^(-betat)sin ( omegat+ alpha)`
(a) `beta=(1)/( 2RC) , omega_(0)^(2)=(1)/(LC), omega=sqrt(omega_(0)^(2)-beta^(2))`
`=sqrt((1)/( LC)-(1)/( 4R^(2)C^(2)))`
(b) `Q=(omega)/(2 beta)=RCsqrt((1)/(LC)-(1)/(4R^(2)C^(2)))`
`=(1)/(2) sqrt((4CR^(2))/(L)-1)`