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A L-C-R circuit is equivalent to a dampe...

A L-C-R circuit is equivalent to a damped pendulum . In an L-C-R circuit the capacitor is charged to `underset(o)(Q)` and then connected to the L and R as shown below.
If a student plots graph of the square of maximum charge on the capacitor with time (t) for two different values `underset(1)(L)` and `underset(2)(L)` `(underset(1)(L)`gt`underset(2)(L)`of L), then which of the following represents this graph correctly (plots are schematic and not drawn to scale)

A

B

C

D

Text Solution

Verified by Experts

The correct Answer is:
A
At any time t apply KVL , `q/C - iR - L di/dt = 0`
`i. = -dq/dt`
`implies` `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)`, for general equation of double differential equation `d^(2)x/dt^(2)` + `b/m dx/dt + k/m x = 0`
`implies` `Q_max^(f)` = `Q_(o)e^(Rt/2L)` `implies` `Q_max^(2)` = `Q_o^(2)``e^(Rt/L)`
Lesser the self-inductance , faster will be damping hence.
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