The dimensions of mutual inductance ($M$) can be determined using the formula for the energy stored in an inductor or the relation between magnetic flux and current.

Method 1: Using Energy Formula The potential energy ($U$) stored in an inductor is given by $U = \frac{1}{2}LI^2$ (where $L$ or $M$ represents inductance). Dimensions of Energy ($U$) = $[ML^2T^{-2}]$ Dimensions of Current ($I$) = $[A]$ Thus, dimensions of $M = \frac{[U]}{[I^2]} = \frac{[ML^2T^{-2}]}{[A^2]} = [ML^2T^{-2}A^{-2}]$.

Method 2: Using Flux Relation Magnetic flux $\Phi = MI \implies M = \frac{\Phi}{I}$. Dimensions of Magnetic Flux ($\Phi$): $[ML^2T^{-2}A^{-1}]$ Dimensions of $M = \frac{[ML^2T^{-2}A^{-1}]}{[A]} = [ML^2T^{-2}A^{-2}]$.