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

4

Explanation

If `q` is charge on capacitor at any time `'t`' and current is `'I'`, then applying Kirchoff's law
`(q)/(C ) - ie - (Ldi)/(dt) = 0`
Putting `i = - (dq)/(dt)` and `(di)/(dt) = (-d^(2) q)/(dt^(2))`
In the above equation
`(d^(2) q)/(dt^(2)) + (R )/(L) (dQ)/(dt) + (q)/(LC) = 0`
Which has general solution of amplitude
`q_("max") = Q_(0) e^(-(RT)/(2L))`
`Q_("max")^(2) = Q_(0)^(2) e^(-(Rt)/(L))`