First slide

Dimensions of physical quantities

Question

Let $[ε_{0}]$ denote the dimensional formula of the permittivity of the vacuum and $[ε_{0}]$ that of the permeability of the vacuum. If $M$ = mass, $L$ =length, $T$ = time and $I$ = electric current, then

Moderate

A

$[ε_{0}] = [M^{- 1} L^{- 3} T^{2} I]$
B

$[ε_{0}] = [M^{- 1} L^{- 3} T^{4} I^{2}]$
C

$[μ_{0}] = [{MLT}^{- 2} I^{- 1}]$
D

$[μ_{0}] = [{ML}^{2} T^{- 1} I]$

Solution

$F = \frac{1}{4 π ε_{0}} \cdot \frac{q_{1} q_{2}}{r^{2}} [ϵ_{0}] = \frac{[q_{1}] [q_{2}]}{[F] [r^{2}]} = \frac{[1 T]^{2}}{[{MLT}^{- 2}] [L^{2}]} = [M^{- 1} L^{- 3} T^{4} I^{2}]$
Speed of light,
$c = \frac{1}{\sqrt{ε_{0} μ_{0}}} ∴ [μ_{0}] = \frac{1}{[ε_{0}] [c]^{2}} = \frac{1}{[M^{- 1} L^{- 3} T^{4} I^{2}] {[L T^{- 1}]}^{2}} = [{MLT}^{- 2} I^{- 2}]$

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