​ A small magnetic needle NS having magnetic dipole moment is kept in two uniform ​

Question

A small magnetic needle NS having magnetic dipole moment \({\overrightarrow P_m}\) is kept in two uniform and perpendicular magnetic fields \({\overrightarrow B_f}\) and \({\overrightarrow B_H}\) as shown below

(i) What is the effect of each of the magnetic fields \({\overrightarrow B_f}\) and \({\overrightarrow B_H}\) on the needle ?

(ii) When the needle is in equilibrium, obtain an expression for angle θ made by the needle with \({\overrightarrow B_H}\) in terms of \({\overrightarrow B_f}\) and \({\overrightarrow B_H}\) only.

Answer 1

(i) We know that whenever a magnetic needle is placed in a uniform magnetic field it experience a torque, which tries to set the dipole in the direction of the magnetic field. This torque is given by MB sinθ, where M is the magnetic dipole moment and θ the angle between M and B. Here there, are two fields, therefore the magnetic needle experiences two torques

1. MB_H sin θ due to the field B_H and 

2. MB_F sin (90° – θ) = MB_H cosθ due to the magnetic field BF trying to set the dipole in the direction of BF.

(ii) The magnetic needle will point neither in the direction of B_H, nor in the direction of B_F. It will come to equilibrium in a position making an angle θ with the direction of B_H such that the two torques are equal and opposite i.e. when.