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

The correct option is B $\left[M^0{L}^0{T}^3{I}^0\right]$

Given $\left[C^{2}LR\right]=\left[C^2{L}^2\frac{R}{L}\right]=\left[{\left(LC\right)}^2\frac{R}{L}\right]$

ad 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 $\left[\frac{L}{R}\right]$ is equal to $\left[T\right]$

Hence the required dimensions

$\left[\frac{L}{R}\right]\left[{\left(LC\right)}^2\frac{R}{L}\right]={\left[T^2\right]}^2\left[T^{-1}\right]=\left[T^3\right]$