Question
The dimensional formula for permeability of free space, $${\mu _0}$$ is
A.
$$\left[ {ML{T^{ - 2}}{A^{ - 2}}} \right]$$
B.
$$\left[ {M{L^{ - 1}}{T^2}{A^{ - 2}}} \right]$$
C.
$$\left[ {M{L^{ - 1}}{T^{ - 2}}{A^2}} \right]$$
D.
$$\left[ {ML{T^{ - 2}}{A^{ - 1}}} \right]$$
Answer :
$$\left[ {ML{T^{ - 2}}{A^{ - 2}}} \right]$$
Solution :
From Biot-Savart law
$$dB = \frac{{{\mu _0}}}{{4\pi }}\frac{{Idl\sin \theta }}{{{r^2}}}$$
$$Idl = $$ current element
$$r =$$ displacement vector
$${\mu _0} = \frac{{4\pi {r^2}\left( {dB} \right)}}{{Idl\sin \theta }} = \frac{{\left[ {{L^2}} \right]\left[ {M{T^{ - 2}}{A^{ - 1}}} \right]}}{{\left[ A \right]\left[ L \right]}} = \left[ {ML{T^{ - 2}}{A^{ - 2}}} \right]$$