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

C

Explanation

(c) From KVL at anytime t

`(q)/(c)-iR-L(di)/dt)=0`
`i=-(dq)/(dt)implies(q)/(c)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_(o)e-(dt)/(2m)`
Double differential equation `(d^(2)x)/(dt^(2))+b/m(dx)/(dt)+(k)/(m)x=0` `Q_(max)=(Q_o)e-(Rt)/(2L) implies Q_(o)^(2)e-(Rt)/(L)`
Hence damping will be faster for leser inductance.