Interpret the expression

The given quantity is $\sqrt{\frac{\varepsilon_0}{\mu_0}}$ where:

• $\varepsilon_0$ = permittivity of free space
• $\mu_0$ = permeability of free space

Use a standard electromagnetic relation

We know that $c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}}$ Hence, $\varepsilon_0 = \frac{1}{\mu_0 c^2}$

Therefore, $\frac{\varepsilon_0}{\mu_0} = \frac{1}{\mu_0^2 c^2}$ and so $\sqrt{\frac{\varepsilon_0}{\mu_0}} = \frac{1}{\mu_0 c}$

Find dimensions of $\mu_0$

From magnetic force law or standard SI dimensions, $[\mu_0] = MLT^{-2}A^{-2}$

Also, $[c] = LT^{-1}$

Compute dimensions of $\mu_0 c$

$[\mu_0 c] = (MLT^{-2}A^{-2})(LT^{-1}) = ML^2T^{-3}A^{-2}$

Take reciprocal

$\left[\sqrt{\frac{\varepsilon_0}{\mu_0}}\right] = [\mu_0 c]^{-1} = M^{-1}L^{-2}T^3A^2$

Match with options

This is $A^2 T^3 M^{-1} L^{-2}$

which matches Option B.

Conclusion

The dimensions of $\sqrt{\frac{\varepsilon_0}{\mu_0}}$ are $A^2T^3M^{-1}L^{-2}$ so the correct option is B.