First slide

Dimensions of physical quantities

Question

The dimension of $e^{2} / 4 π ε_{0} h c, where e, ε_{0},$ h and c are electric charge, electric permittivity of free space, Planck's constant and velocity of light in vacuum, respectively, is

Moderate

A

$[M^{0} L^{0} T^{0}]$
B

$[M^{1} L^{0} T^{0}]$
C

$[M^{0} L^{1} T^{0}]$
D

$[M^{0} L^{0} T^{1}]$

Solution

$[e] = [A T], ε_{0} = [M^{- 1} L^{- 3} T^{4} A^{2}], [h] = [M L^{2} T^{- 1}] a n d [c] = [L T^{- 1}]$
$\begin{matrix} [\frac{e^{2}}{4 π \in_{0} h c}] = [\frac{A^{2} T^{2}}{M^{- 1} L^{- 3} T^{4} A^{2} \times M L^{2} T^{- 1} \times L T^{- 1}}] \\ = [M^{0} L^{0} T^{0}] \end{matrix}$
To calculate $\in_{0}$ , we can use the formula $F = \frac{q_{1} q_{2}}{4 {πε}_{0} r^{2}}$

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