Question

The dimensions of ${\mu }_{0}I/MB,$ 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)

$B={\mu }_{0}ni$ (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 $\frac{{\mu }_0}{B}=\frac{1}{ni}$
$\therefore \left[\frac{{\mu }_0}{B}\right]=\left[\frac{1}{ni}\right]$
$=\frac{1}{\left[L^{-1}\right][A]}=\left[L{A}^{-1}\right]$
$[M]=[$ Magnetic moment $]=[m\times 2l]$
=[ 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 $]=\left[M{L}^2\right]$
$\therefore \left[\frac{{\mu }_{0}I}{BM}\right]$
$=\left[\frac{{\mu }_0}{B}\right]\times \left[\frac{I}{M}\right]$
$=\left[L{A}^{-1}\right]\times \frac{\left[M{L}^2\right]}{\left[M{L}^3{T}^{-2}{A}^{-1}\right]}$
$=\left[T^2\right]=(\text{ time }{)}^2$