If electronic charge e , electron mass m , speed of light in vacuum c and Planck's constant h are taken as fundamental quantities, the permeability of vacuum μ 0 can be expressed in units of:

Question

Aicharya · Accepted Answer

Magnetic potential energy inside a solenoid is

U = \frac{1}{2} L I^{2}

Where

L =

self-inductor

\Rightarrow U = \frac{1}{2} (μ_{0} n^{2} A l) I^{2}

∵ n =

number of turns per unit length

= \frac{N}{l}

U = \frac{1}{2} μ_{0} {(\frac{N}{l})}^{2} A l I^{2}

U = \frac{1}{2} μ_{0} \frac{N^{2}}{l^{2}} A l I^{2}

μ_{0} = \frac{2 U l}{N^{2} A I^{2}}

Dimensionally,

[U] = [\frac{h c}{λ}]

[I] = [\frac{e}{T}]

[A] = {[l]}^{2}

[l] = [l^{'}]

[μ_{0}] = [\frac{h c}{λ}] \frac{[l]}{[l^{2}]} \frac{[t^{2}]}{{[e]}^{2}}

= [\frac{h}{λ} \frac{t}{e^{2}}] = [\frac{h}{c e^{2}}]

μ_{0} = \frac{h}{c e^{2}}

If electronic charge e , electron mass m , speed of light in vacuum c and Planck's constant h are taken as…

Solution