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(C) We can express the permeability of vacuum as $\mu_0 \propto e^a m^b c^c h^d$ or $\mu_0 = k e^a m^b c^c h^d$ ...$(i)$,where $k$ is a dimensionless

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$\frac{h}{c e^2}$

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(C) We can express the permeability of vacuum as $\mu_0 \propto e^a m^b c^c h^d$ or $\mu_0 = k e^a m^b c^c h^d$ ...$(i)$,where $k$ is a dimensionless constant.Dimensions of the quantities are:$\mu_0 = [M L T^{-2} A^{-2}]$$e = [A T]$$m = [M]$$c = [L T^{-1}]$$h = [M L^2 T^{-1}]$Substituting these into Eq. $(i)$:$[M L T^{-2} A^{-2}] = [A T]^a [M]^b [L T^{-1}]^c [M L^2 T^{-1}]^d$$[M L T^{-2} A^{-2}] = [M]^{b+d} [L]^{c+2d} [T]^{a-c-d} [A]^a$Comparing powers on both sides:$a = -2$$b + d = 1$$c + 2d = 1$$a - c - d = -2$From $a = -2$,substituting into $a - c - d = -2$ gives $-2 - c - d = -2$,so $c + d = 0$,which means $c = -d$.Substituting $c = -d$ into $c + 2d = 1$ gives $-d + 2d = 1$,so $d = 1$.Then $c = -1$ and $b = 1 - d = 1 - 1 = 0$.Thus,$\mu_0 = k e^{-2} m^0 c^{-1} h^1 = k \frac{h}{c e^2}$.Therefore,$\mu_0$ is proportional to $\frac{h}{c e^2}$.

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

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