Dipole in Uniform Magnetic field

Most people are familiar with magnets primarily as toys or as decorative items on a refrigerator door

Magnetism has a much broader range of applications

Considering two opposite poles (i.e., North and South poles) as a system is known as a magnetic dipole

Therefore, based on the definition we can say that a bar magnet forms a magnetic dipole

Let us see what happens when we place a Dipole in Magnetic field

When a bar magnet or magnetic dipole is placed in a uniform Magnetic field, each pole will experience a force

Due to this Force, the Dipole is bound to experience a Torque

Let us see the Force and Torque acting on a Magnetic Pole in a field

When an electric charge is placed in an Electric field, Force acts on the charge due to the Electric Field Intensity

Similarly, a Magnetic pole experiences a Force due to a Magnetic Field

The Force acting on the Magnetic Pole is given by

Where, =The Magnetic pole strength =Magnetic induction

The pole strength acting on the North Pole is , so the Force acting on it will be

The direction of the Force acting on the North magnetic Pole will be in the direction of the Magnetic Field

The pole strength of the South magnetic Pole will be , so the Force acting on it will be

The direction of the Force acting on the South magnetic Pole will be opposite to the direction of Magnetic Field

Now, let us consider a Dipole placed in a Uniform Magnetic Field at an angle

The Force acting on the dipole is zero since the forces acting on the individual poles are equal and opposite

The Torque acting on a dipole can be mathematically expressed as, perpendicular distance between North Pole N and South Pole P

The vertical distance between the North Pole and the South Pole is

In ,

Replacing the value of ,

Rearranging the terms,

Where, is the Magnetic Moment on the Dipole

The equation of Torque can be written in vector form as,

In the equation , if or , the Dipole is lying Parallel or Anti-Parallel to the Magnetic Field

In this case, the Torque acting on the Dipole will be Zero

This is because the Torques acting on the North Pole and South Pole will be balanced

Now, if ,

In this case, the Torque acting on the Dipole will be

Additionally, if , the Torque acting on the Dipole will be equal to Magnetic Moment

The Torque acting on a Dipole placed in a Unit Magnetic field perpendicular to the direction of Magnetic field is known as the Magnetic Dipole Moment

Let us calculate the Work done on the Dipole

Work is done whenever a Torque is used to rotate a Dipole by an angle

Work done can be mathematically expressed as Work done = Torque x Angular Displacement in the direction of the Force

The Work done to rotate a bar magnet or Dipole by a small angle is given by,

Now, further Work done is required to move the Dipole from an initial angle to final angle

This Work done is calculated by,

If ,

If , Since,

This Work done will be stored in the form of Potential Energy. Therefore,

Revision

A magnet in which opposite poles (i.e., North and South poles) are on opposite sides of the magnet is called a Dipole magnet

When a bar magnet or magnetic dipole is placed in a uniform Magnetic field, it will experience a torque

The Force acting on a Dipole multiplied by the perpendicular distance between the Poles is the Torque acting on the Dipole

The work done on the dipole if , will be

The End