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

Let at an instant of time `t`, during discharging,`q` be the charge on capacitor and `I` be the current in the circuit and `(dI)/(dt)` be the rate of change of current
The emf equation of the circuit is
`(q)/(C ) = IR + L (dI)/(dt) = (dq)/(dt) R + (d)/(dt) ((dq)/(dt))`
or `(d^(2) q)/(dt^(2)) + (R )/(L) (dq)/(dt) = (q)/(LC) = 0`
For damped harmonic oscillator, the general eqation of damped oscillation is
`(d^(2) x)/(dt^(2)) + (b)/(m) (dx)/(dt) = (k)/(m) x = 0`
The amplitude of damped oscillation,
`A = A_(0) e^(-(bt)/(2m))`
Comparing equations (i) and (ii), we note that the equation (i) represents damped oscillatory discharge of capacitor in the cicuit whose amplitude of oscillation,
` Q_(max) (t) = Q_(0) e^(-(R )/(2L))` or `Q_(max)^(2) = Q_(0)^(2) e^(-(Rt)/(L))`
It shows that the lesser self-inductance `L`, faster will be the damping force. Thus. option (a) is ture.