Units & Dimensions

JEE Main level, single correct, medium

131 of 149

Question 131 of 149

If electronic charge ee, electron mass mm, speed of light in vacuum cc and Planck's constant hh are taken as fundamental quantities, the permeability of vacuum μ0{\mu }_{0} can be expressed in units of:

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Answer: (D) (hce2)\left(\dfrac{h}{c{e}^{2}}\right)

Magnetic potential energy inside a solenoid is

U=12LI2U=\dfrac{1}{2}L{I}^{2} Where L=L=self-inductor⇒U=12(μ0n2Al)I2\Rightarrow U=\dfrac{1}{2}\left({\mu }_{0}{n}^{2}Al\right){I}^{2}

∵n=∵n= number of turns per unit length

=Nl=\dfrac{N}{l}

U=12μ0(Nl)2AlI2U=\dfrac{1}{2}{\mu }_{0}{\left(\dfrac{N}{l}\right)}^{2}Al{I}^{2}

U=12μ0N2l2AlI2U=\dfrac{1}{2}{\mu }_{0}\dfrac{{N}^{2}}{{l}^{2}}Al{I}^{2}

μ0=2UlN2AI2{\mu }_{0}=\dfrac{2Ul}{{N}^{2}A{I}^{2}}

Dimensionally, [U]=[hcλ]\left[U\right]=\left[\dfrac{hc}{\lambda }\right]

[I]=[eT]\left[I\right]=\left[\dfrac{e}{T}\right]

[A]=[l]2\left[A\right]={\left[l\right]}^{2}

[l]=[l′]\left[l\right]=\left[{l}^{'}\right]

[μ0]=[hcλ][l][l2][t2][e]2\left[{\mu }_{0}\right]=\left[\dfrac{hc}{\lambda }\right]\dfrac{[l]}{\left[{l}^{2}\right]}\dfrac{\left[{t}^{2}\right]}{{\left[e\right]}^{2}}

=[hλte2]=[hce2]=\left[\dfrac{h}{\lambda }\dfrac{t}{{e}^{2}}\right]=\left[\dfrac{h}{c{e}^{2}}\right]

μ0=hce2{\mu }_{0}=\dfrac{h}{c{e}^{2}}

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