If the charge of electron \(e\), mass of electron \(m\), speed of light in vacuum \(c\) and Planck's…
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
\(\frac{h}{m c^2}\)
\(\frac{h c}{m e^2}\)
\(\frac{h}{c e^2}\)
\(\frac{m c^2}{h e^2}\)
Solution
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 dimensional constant.
As we know that, the dimensions 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, $\frac{h}{c e^{2}}$.