An LCR curcuit is equivalent to a damped pendulum. In an LCR circuit the capacitor is charged to `(Q_0)` and then connected to the L ans R as shown below.

If a student plaots graphs of the square of maximum charge `(Q_(max)^(2))` on the capacitor with time (t) for two different values `L_(1) and L_(2) (L_(1)gtL_(2))` of L then which of the following represents this graph correclty? (plots are schematic and not drawn to scale).

Options

A.
B.
C.
D.

Answer

A

Explanation

at any time `t` apply `KVL`
`q/C-iR-L(di)/(dt)=0`
`i=-(dq)/(dt) rArr q/C+(dq)/(dt)R+(Ld^(2)q)/(dt^(2))=0`
`(d^(2)q)/(dt^(2))+R/L(dq)/(dt)+q/(LC)=0`
from damped harmonic oscillator, the amplitude is given by `A=A_(0^(e^(-(dt)/(2m))`,for general equation of double differential equation `(d^(2)x)/(dt^(2))+b/m(dx)/(dt)+k/mx=0`
`rArr Q_(max)^(t)=Q_(0)e^((-Rt)/(2L)) rArr Q_(max)^(2)=Q_(0)^(2)e^((-Rt)/(L))`
lesser the self inductance faster will be damping hence