Question

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

• A. $\left[M{L}^{-2}{T}^{-1}{I}^0\right]$
• 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, $\left[C^{2}LR\right]=\left[C^2{L}^2\frac{R}{L}\right]=\left[(LC{)}^2\left(\frac{R}{L}\right)\right]$
and we know that frequency of $LC$ circuits
$f=\frac{1}{2\pi }\frac{1}{\sqrt{LC}}$. Here the dimension of $LC$ 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
$\left[(LC{)}^2\left(\frac{R}{L}\right)\right]={\left[T^2\right]}^2\left[T^{-1}\right]=\left[T^3\right].$