Question

Show that the magnetic field at a point due to a magnetic dipole is perpendicular to the magnetic axis if the line joining the point with the centre of the dipole makes an an of tan ${}^{-1}\left(\sqrt{2}\right)$with the magnetic axis.

Answer 1

Given $\theta =ta{n}^{-1}\sqrt{2}$

$\Rightarrow tan\theta =\sqrt{2}\Rightarrow 2=ta{n}^2\theta$

$\Rightarrow tan\theta =2cot\theta$

$\Rightarrow \frac{tan\theta }{2}=cot\theta$ ...(i)

we know $\frac{tan\theta }{2}=tan\alpha$ ...(ii)

Comparing (i) and (ii) we get,

$tan\alpha =cot\theta$

or, $tan\alpha =-tan(90-\theta )$

or, $\alpha =90-\theta$

or, $\theta +\alpha ={90}^0$

Hence magnetic field due to the dipole is perpendicular to the magnetic axis.