Question

Sol. [Key Concept: Substitute the dimensions of $H$ (magnetic field), $E$, permittivity and $x$ into $H = \frac{x^{p}\varepsilon^{q}E^{r}}{t^{s}}$ and match powers of $M,L,T,I$ to solve for $p,q,r,s$.]

As per the question, considering H as magnetic field

$${\lbrack H\rbrack = \frac{{MLT}^{- 2}}{{LT}^{- 1}IT} = {MI}^{- 1}{\text{ }T}^{- 2} }{\lbrack E\rbrack = \frac{{MLT}^{- 2}}{IT} = {MLI}^{- 1}{\text{ }T}^{- 3} }{\lbrack\epsilon\rbrack = \frac{I^{2}{\text{ }T}^{2}}{{MLT}^{- 2}{\text{ }L}^{2}} = M^{- 1}{\text{ }L}^{- 3}I^{2}{\text{ }T}^{4} }{{MI}^{- 1}{\text{ }T}^{- 2} = M^{- q + r}L^{p - 3q + r}I^{2q - r}T^{4q - 3r - s} }{- q + r = 1 }{P - 3q + r = 0 }{2q - r = - 1 }{4q - 3r - S = - 2}$$

Solving we get

$${q = 0,r = 1 }{p = - 1,\text{ }S = - 1 }$$No option matching

Considering H as magnetic field intensity

$H = \frac{B}{\mu_{0}} = Ni \Rightarrow$ current per unit length $\Rightarrow {IL}^{- 1}$

Option (1) is matching

Note : As per the question if we consider H as magnetic field, no option is matching but if we consider H as magnetic field intensity then option $(1)$ is matching.