Question

Let $[ \varepsilon_0 ]$ denote the dimensional formula of the permittivity of the vacuum and $[ \mu_0 ]$ that of the permeability of the vacuum. If $M$ = mass, $L$ = length, $T$ = time and $I$ = electric current.

• A. $[ \varepsilon_0 ] = [ M^{ - 1} L^{ - 3} T^2 I ]$
• B. $[ \varepsilon_0 ] = [ M^{ - 1} \, L^{ - 3} \, T^4 \, I^2 ]$
• C. $[ \mu_0 ] = [ ML \, T^{ - 2} I^{ - 2} ]$
• D. $[ \mu_0 ] = [ ML^2 \, T^{ - 1} \, I ]$

Answer 1

Answer: (C)

$[ \mu_0 ] = [ ML \, T^{ - 2} I^{ - 2} ]$

Answer 2

F = $\frac{1}{ 4 \pi \varepsilon_0} . \frac{ q_1 q_2 }{ r^2 }$
$[ \varepsilon_0 ] = \frac{ [ q_1 ] [ q_2 ] }{ [ F] [ r^2 ] } = \frac{ [ IT^2 ] }{ [ MLT^{ - 2} ] \, [ L^2 ] } = [ M^{ - 1} \, L^{ - 3} \, T^4 \, I^2 ]$
Speed of light, c = $\frac{1}{ \sqrt{ \varepsilon_0 \, \mu_0 }}$
$\therefore [ \mu_0 ] = \frac{1}{ [ \varepsilon_0 ] \, [ c]^2 } = \frac{ 1}{ [ M^{ - 1} \, L^{ - 3} \, T^4 I^2 ] \, [ LT^{ - 1} ]^2 }$
= $[ MLT^{ - 2} \, I^{ - 2} ]$