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A student determines a dimensionless quantity, $B = \frac{e^{n}}{2 ε_{0} h c}$ . Find the value of n. (here, e= electric charge, $ε_{0} =$ electric permittivity of vacuum, h=Planck’s constant and c= speed of light.)

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answer is 2.

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Detailed Solution

Dimension of B $= [M^{0} L^{0} T^{0}]$

Dimension of e $= [A T]$

Dimension of $ε_{0} = [A^{2} M^{- 1} L^{- 3} T^{4}]$

Dimension of $h = [M L^{2} T^{- 1}]$

Dimension of $c = [L T^{- 1}]$

$∴ [M^{0} L^{0} T^{0}] = \frac{{[A T]}^{n}}{[A^{2} M^{- 1} L^{- 3} T^{4}] [M L^{2} T^{- 1}] [L T^{- 1}]}$

$\Rightarrow [M^{0} L^{0} T^{0}] = [A^{n - 2} L^{0} T^{n - 2}]$

$\Rightarrow n - 2 = 0$

$\Rightarrow n = 2$

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