The speed of light in vacuum, $c$, depends on two fundamental constants, the permeability of free space,…

The speed of light in vacuum, $c$, depends on two fundamental constants, the permeability of free space, $\mu_{0}$ and the permittivity of free space, $\varepsilon_{0}$. The speed of light is given by $c=\frac{1}{\sqrt{\mu_{0} \varepsilon_{0}}}$. The units of $\varepsilon_{0}$ are $\mathrm{N}^{-1} \mathrm{C}^{2} \mathrm{~m}^{-2}$. The units for $\mu_{0}$ are
  1. $\mathrm{kg}^{-1} \mathrm{~m}^{-1} \mathrm{C}^{2}$
  2. $\mathrm{kg} \mathrm{mC}^{-2}$
  3. $\mathrm{kg} \mathrm{ms}^{-4} \mathrm{C}^{-2}$
  4. $\mathrm{kg}^{-1} \mathrm{~s}^{-3} \mathrm{C}^{-2}$

Solution

We have, $c=\frac{1}{\sqrt{\mu_{0} \in_{0}}} \Rightarrow \mu_{0}=\frac{1}{\epsilon_{0} c^{2}}$
units of $\mu_{0}=\frac{1}{\text { units of } \in_{0} c^{2}}$
$=\frac{1}{N^{-1} C^{2} m^{-2}\left(m s^{-1}\right)^{2}}=N s^{2} C^{-2}$
$=\left(\mathrm{kgms}^{-2}\right) s^{2} C^{-2}=\mathrm{kgm} C^{-2}$ /

Asked in: JEE Mains - Units and Dimensions - Test 1

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