Question

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

• A. $\left[ ML ^{-2} T ^{-1}I ^{0}\right]$
• B. $\left[M^{0} L^{0} T^{3}I^{0}\right]$
• C. $\left[M^{-1} L^{-2} T^{6}I^{2}\right]$
• D. $\left[M^{0} L^{0} T^{2}I^{0}\right]$

Answer 1

Answer: (B)

Given, $\left[C^{2} L R\right]=\left[C^{2} L^{2} \frac{R}{L}\right]=\left[(L C)^{2}\left(\frac{R}{L}\right)\right]$
and we know that frequency of $L C$ circuits
$f=\frac{1}{2 \pi} \frac{1}{\sqrt{L C}}$. Here the dimension of $L C$ is equal to $\left[T^{2}\right]$.
$\left[\frac{L}{R}\right]$ gives the time constant of $L-R$ circuit,
so that the dimension of $\frac{L}{R}$ is equal to $[T]$.
Hence the required dimensions
$\left[(L C)^{2}\left(\frac{R}{L}\right)\right]=\left[T^{2}\right]^{2}\left[T^{-1}\right]=\left[T^{3}\right] .$