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(sqrt 2)` with the magnetic axis

Answer 1

Given : 

Angle made by observation point P with the axis of the dipole, `θ = tan^-1 (sqrt 2)`

⇒ `tan θ = sqrt 2`

⇒ `2 = tan^2 θ`

⇒ `tan θ = cot θ`

⇒ `tan θ /2 = cot θ`         .....(1)

we know , 

`tan θ /2 = tan α`      ....(2)

On comparing (1) and (2), we get

`tan α = cot θ`

⇒ `tan α = tan (90 - θ)`

⇒ `α = 90 - θ`

⇒ `θ + α = 90^circ`

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