Question

Find the dimensions of
(a) electric field E.
(b) magnetic field B and
(c) magnetic permeability ${\mu }_0$.
The relevant equation are
$F=qE,F=qvB,\mathrm{and}B=\frac{{\mu }_{0}I}{2\pi a};$
where F is force, q is charge, v is speed, I is current, and a is distance.

Answer 1

(a) Electric field is defined as electric force per unit charge.
i.e., $\mathrm{E}=\frac{F}{q}$
$$\begin{gathered} \mathrm{Also},\left[F\right]=\left[{\mathrm{MLT}}^{-2}\right]\mathrm{and}\left[q\right]=\left[\mathrm{A}T\right] \\ \mathrm{So},\mathrm{dimension}\mathrm{of}\mathrm{electric}\mathrm{field},\left[E\right]=\left[{\mathrm{MLT}}^{-3}{\mathrm{A}}^{-1}\right] \end{gathered}$$

(b) Magnetic field, $B=\frac{\mathit{F}}{\mathit{q}\mathit{v}}$
$$\begin{gathered} \mathrm{Here},\left[F\right]=\left[{\mathrm{MLT}}^{-2}\right],\left[q\right]=\left[\mathrm{AT}\right]\mathrm{and}\left[v\right]=\left[{\mathrm{LT}}^{-1}\right] \\ \mathrm{So},\mathrm{dimension}\mathrm{of}\mathrm{magnetic}\mathrm{field},\left[B\right]=\frac{\left[{\mathrm{MLT}}^{-2}\right]}{\left[\mathrm{A}T\right]\left[{\mathrm{LT}}^{-1}\right]}=\left[M{\mathrm{L}}^0{T}^{-2}{\mathrm{A}}^{-1}\right] \end{gathered}$$

(c) Magnetic permeability, ${\mathrm{\mu }}_0=\frac{B\times 2\pi r}{I}$
$$\begin{gathered} \mathrm{Here},\left[B\right]=\left[{\mathrm{MT}}^{-2}{\mathrm{A}}^{-1}\right]\mathrm{and}\left[r\right]=\left[\mathrm{L}\right] \\ \mathrm{So},\mathrm{dimension}\mathrm{of}\mathrm{magnetic}\mathrm{permeability},\left[{\mathrm{\mu }}_0\right]=\frac{\left[{\mathrm{MT}}^{-2}{\mathrm{A}}^{-1}\right]\times \left[\mathrm{L}\right]}{\left[\mathrm{A}\right]}=\left[{\mathrm{MLT}}^{-2}{\mathrm{A}}^{-2}\right] \end{gathered}$$