Q 11-01-227JEE MainJEE Main 2019 (12 Apr, Shift 1)Easy
Which of the following combinations has the dimension of electrical resistance ($\varepsilon_0$ is the permittivity of vacuum and $\mu_0$ is the permeability of vacuum)?
Answer: (D) $\sqrt{\dfrac{\mu_0}{\varepsilon_0}}$
$\mu_0$ has the units of inductance per metre (H/m) and $\varepsilon_0$ of capacitance per metre (F/m), so
$$\frac{\mu_0}{\varepsilon_0} \sim \frac{L}{C}$$
Since $\omega L$ and $\dfrac{1}{\omega C}$ are both resistances, $\dfrac LC$ has the dimensions of (resistance)$^2$. Hence $\sqrt{\dfrac{\mu_0}{\varepsilon_0}}$ has the dimension of resistance (it is the impedance of free space, about $377\ \Omega$).
Solution by Sreeraj P, M.Sc Physics