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Question: If L, C and R denote the inductance, capacitance and resistance respectively, the dimensional formul...

If L, C and R denote the inductance, capacitance and resistance respectively, the dimensional formula for C2LRC^{2}LR is

A

[ML−2T−1I0]\lbrack ML^{- 2}T^{- 1}I^{0}\rbrack

B

[M0L0T3I0]\lbrack M^{0}L^{0}T^{3}I^{0}\rbrack

C

[M−1L−2T6I2]\lbrack M^{- 1}L^{- 2}T^{6}I^{2}\rbrack

D

[M0L0T2I0]\lbrack M^{0}L^{0}T^{2}I^{0}\rbrack

Answer

[M0L0T3I0]\lbrack M^{0}L^{0}T^{3}I^{0}\rbrack

Explanation

Solution

[C2LR]\lbrack C^{2}LR\rbrack = [C2L2RL]\left\lbrack C^{2}L^{2}\frac{R}{L} \right\rbrack = [(LC)2(RL)]\left\lbrack (LC)^{2}\left( \frac{R}{L} \right) \right\rbrack

and we know that frequency of LC circuits is given by f=12π1LCf = \frac{1}{2\pi}\frac{1}{\sqrt{LC}}i.e., the dimension of LC is equal to [T2]\lbrack T^{2}\rbrack

and [LR]\left\lbrack \frac{L}{R} \right\rbrack gives the time constant of L−RL - R circuit so the dimension of LR\frac{L}{R} is equal to [T].

By substituting the above dimensions in the given formula [(LC)2(RL)]=[T2]2[T−1]=[T3]\left\lbrack (LC)^{2}\left( \frac{R}{L} \right) \right\rbrack = \lbrack T^{2}\rbrack^{2}\lbrack T^{- 1}\rbrack = \lbrack T^{3}\rbrack .