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

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Solution

Given:
$θ = {tan}^{- 1} \sqrt{2} \Rightarrow tan θ = \sqrt{2} \Rightarrow 2 = {tan}^{2} θ$

$\Rightarrow tan θ = 2 cot θ \Rightarrow \frac{tan θ}{2} = cot θ$

We know $\frac{tan θ}{2} = tan α$

Comparing we get, $tan α = cot θ$
or, $tan α - tan (90 - θ)$ or
$α = 90 - θ$ or
$θ + α = 90$
Hence magnetic field due to the dipole is $⊥ r$ to the magnetic axis.

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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(√2) with the magnetic axis.

Given:θ=tan−1√2⇒tanθ=√2⇒2=tan2θ⇒tanθ=2cotθ⇒tanθ2=cotθWe know tanθ2=tanαComparing we get, tanα=cotθor, tanα−tan(90−θ) orα=90−θ orθ+α=90Hence magnetic field due to the dipole is ⊥r to the magnetic axis.

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