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 angle of ${\tan }^{-1}\left(\sqrt{2}\right)$with the magnetic axis.

Answer 1

Given:
Angle made by observation point P with the axis of the dipole, $\theta$ = ${\tan }^{-1}\left(\sqrt{2}\right)$
$$\begin{gathered} \Rightarrow \tan \theta =\sqrt{2} \\ \Rightarrow 2={\tan }^2\theta \\ \Rightarrow \tan \theta =\cot \theta \\ \Rightarrow \frac{\tan \theta }{2}=\cot \theta ....\left(1\right) \\ \mathrm{We}\mathrm{know}, \\ \frac{\tan \theta }{2}=\tan \alpha ....\left(2\right) \end{gathered}$$

On comparing (1) and (2), we get
tan α = cot θ
$\Rightarrow$tan α = tan (90 − θ)
$\Rightarrow$α = 90 − θ
$\Rightarrow$θ + α = 90°
Hence, the magnetic field due to the dipole is perpendicular to the magnetic axis.