Question
Figure shows three oscillating $$LC$$ circuit with identical inductors and capacitors. If $${t_1},{t_2},{t_3}$$ are the time taken by the circuits $$I, II, III$$ for fully discharge, then
Figure shows three oscillating $$LC$$ circuit with identical inductors and capacitors. If $${t_1},{t_2},{t_3}$$ are the time taken by the circuits $$I, II, III$$ for fully discharge, then
A.
$${t_1} > {t_2} > {t_3}$$
B.
$${t_1} < {t_2} < {t_3}$$
C.
$${t_2} < {t_1} < {t_3}$$
D.
$${t_3} = \sqrt {{t_1}{t_2}} $$
Answer :
$${t_2} < {t_1} < {t_3}$$
Solution :
Time period,
$$\eqalign{ & {T_1} = 2\pi \sqrt {LC} , \cr & {T_2} = 2\pi \sqrt {\frac{{LC}}{2}} \cr & {T_3} = 2\pi \sqrt {2LC} \cr} $$
Clearly $${t_2} < {t_1} < {t_3}.$$
Time period,
$$\eqalign{ & {T_1} = 2\pi \sqrt {LC} , \cr & {T_2} = 2\pi \sqrt {\frac{{LC}}{2}} \cr & {T_3} = 2\pi \sqrt {2LC} \cr} $$
Clearly $${t_2} < {t_1} < {t_3}.$$
