Question

The dimensions of $\mu_{0} I / M B,$ where $\mu_{0}$ is the permeability of free space, $I$ is the moment of inertia, $M$ is the magnetic moment and $B$ is the magnetic induction respectively, are those of

• A. time
• B. (time) $^{2}$
• C. (time) $^{3}$
• D. $(\text { time })^{1 / 2}$

Answer 1

Answer: (B)

$B=\mu_{0} n i$ (For a solenoid) where $n$ is the number of turns per unit length, $i$ is the current and $\mu_{0}$ is the permeability of free space.
or $ \frac{\mu_{0}}{B}=\frac{1}{n i}$
$ \therefore \left[\frac{\mu_{0}}{B}\right]=\left[\frac{1}{n i}\right]$
$=\frac{1}{\left[ L ^{-1}\right][ A ]}=\left[ LA ^{-1}\right]$
$[M]=[$ Magnetic moment $]=[m \times 2 l]$
=[ Pole strength $\times$ magnetic length ]
=[ weber $\times$ L]$=\left[M L^{2} T^{-2} A^{-1} \times L\right]=\left[M L^{3} T^{-2} A^{-1}\right]$
$[I]=[$ Moment of inertia $]=\left[ ML ^{2}\right]$
$\therefore \left[\frac{\mu_{0} I}{B M}\right]$
$=\left[\frac{\mu_{0}}{B}\right] \times\left[\frac{I}{M}\right]$
$=\left[ LA ^{-1}\right] \times \frac{\left[ ML ^{2}\right]}{\left[ ML ^{3} T ^{-2} A ^{-1}\right]}$
$=\left[ T ^{2}\right]=(\text { time })^{2}$