If the charge of electron \(e\), mass of electron \(m\), speed of light in vacuum \(c\) and Planck's constant \(h\) are taken as fundamental quantities, then the permeability of vacuum \(\mu_0\) can be expressed as

We can expressed the permeability of vacuum, \(\mu_0 \propto e^a m^b c^c h^d\) or \(\mu_0=k e^a m^b c^c h^d\) ...(i) Where, \(k\) is a dimensional constant. As we know that, Dimension of \(\begin{aligned} \mu_0 & =\left[\mathrm{M} \mathrm{L} \mathrm{T}^{-2} \mathrm{~A}^{-2}\right], \\ e & =[\mathrm{AT}], \\ m & =[\mathrm{M}], \\ c & =\left[\mathrm{L} \mathrm{T}^{-1}\right], \\ h & =\left[\mathrm{M} \mathrm{L}^2 \mathrm{~T}^{-1}\right], \end{aligned}\) and Putting the dimension of various physical quantities in Eq. (i), we get \(\left[\mathrm{M} \mathrm{L} \mathrm{T}^{-2} \mathrm{~A}^{-2}\right]=k[\mathrm{~A} \mathrm{~T}]^a\left[\mathrm{M}^b\left[\mathrm{~L} \mathrm{~T}^{-1}\right]^c\right]\left[\mathrm{M} \mathrm{L}^2 \mathrm{~T}^{-1}\right]^d\) \(\left[\mathrm{M} \mathrm{L} \mathrm{T}^{-2} \mathrm{~A}^{-2}\right]=k[\mathrm{M}]^{b+d}[\mathrm{~L}]^{c+2 d}[\mathrm{~T}]^{a-c-d}[\mathrm{~A}]^a\) Comparing the powers of \(\mathrm{M}, \mathrm{L}, \mathrm{T}\) and \(\mathrm{A}\) on the both sides, we get \(\begin{aligned} b+d & =1 \quad \ldots (ii) \\ c+2 d & =1 \quad \ldots (iii) \\ a-c-d & =-2 \quad \ldots (iv) \\ a & =-2 \quad \ldots (v) \end{aligned}\) After solving the Eqs. (ii), (iii), (iv) and (v), we get \(\begin{aligned} a & =-2, \\ b & =0, \\ c & =-1, \\ \text{and } d & =1, \\ \therefore \mu_0 & =\frac{h}{c e^2} \end{aligned}\) So, the permeability of vacuum \(\mu_0\) can be expressed as, \(\frac{h}{c e^2}\).