Question

Let $I$ = current through a conductor, $R$ its resistance, and $V$ = potential difference across its ends. According to Ohm's law, the product of two of these quantities equals the third. Obtain Ohm's law from dimensional analysis. Dimensional formulae for $R$ and $V$ are $M{L}^2{I}^{-2}{T}^{-3}$ and $M{L}^2{T}^{-3}{I}^{-1}$ respectively.

Answer 1

Step 1. Given data:

Dimensional formula of $R$ = $\left[M{L}^2{I}^{-2}{T}^{-3}\right]$

Dimensional formula of $V=\left[M{L}^2{T}^{-3}{I}^{-1}\right]$

Dimensional formula of $I=\left[I\right]$

Step 2. Ohm's law:

This law states that $V=IR$, Where $V,IandR$ are voltage, current and respectively.

So $$\begin{gathered} \left[M{L}^2{T}^{-3}{I}^{-1}\right]=\left[I\right]\left[M{L}^2{T}^{-3}{I}^{-2}\right]=\left[M{L}^2{T}^{-3}{I}^{-1}\right] \\ L.H.S=R.H.S \end{gathered}$$

Hence, Ohm's law can also be derived in terms of dimensional formula.