An initially charged undriven LCR circuit having inductance L, capacitance C and resistance R will be

Options

A. oscillate with frequency `(1)/(sqrt(LC))`
B. oscillate without damping, if `R^(2) lt (4L)/(C )`
C. oscillate with damping, if `R^(2) gt (4L)/(C )`
D. oscillate with damping, if `R^(2) lt (4L)/(C )`

Answer

D

Explanation

In an L-C-R circuit (without any AC input), when `(R^(2))/(4L^(2)) = (1)/(LC)`, charge decays exponentially and no oscillations occurs.
When `(R^(2))/(4L^(2)) gt (1)/(LC.)` then also charge decays exponentially and no oscillations occurs.
But when `(R^(2))/(4L^(2)) lt (1)/(LC)` or when `R^(2) lt (4L)/(C)`, then oscillations occurs with frequency,
`f = (1)/(2pi) sqrt((1)/(LC) - (R^(2))/(4L^(2)))`