The Dipole in a Uniform Magnetic Field | Torque On Bar Magnet
The Dipole in a Uniform Magnetic Field | Torque On Bar Magnet :- Let the pole strength of a bar magnet of effective length 2l be m. This bar magnet is placed in a uniform magnetic field B in such a way that the angle between the magnetic dipole moment (M) of the magnet and the magnetic field (B) is θ.
The force mB on the north pole of the bar magnet will act in the direction of the magnetic field B and the force mB on the south pole will act in the opposite direction to the magnetic field B. Both the forces are equal in magnitude but opposite in direction, hence the resultant force on the bar magnet is zero due to which it will not make translational motion, but the lines of action of these forces are not along a single line but there is a perpendicular distance of 2l sinθ between them. Due to this, a torque will act on the bar magnet and it will rotate.
This torque tries to bring the bar magnet to stable equilibrium position (M and B in the same direction), hence it is called restoring torque.
Torque (τ) = Force (mB) × perpendicular distance between lines of action of forces (2l sinθ)
⇒ τ = (m2l ) B sinθ
here m2l = M (magnetic dipole moment)
∴ τ = MB sinθ …..(1)
In vector form,
The Dipole in a Uniform Magnetic Field – Special conditions of Torque On Bar Magnet
- If θ = 0°(stable equilibrium), then sinθ = 0 ⇒ τ = 0, therefore, in the case of stable equilibrium the torque on the bar magnet will be zero.
- If θ = 90°, then sinθ = 1 ⇒ τ = MB (Maximum), In this situation the value of torque will be maximum.
- If θ = 180°(unstable equilibrium), then sinθ = 0 ⇒ τ = 0Therefore, in the case of unstable equilibrium, the torque on the bar magnet will be zero. If you deviate even slightly from this position, the bar magnet will rotate towards the stable equilibrium position and will start oscillating and will finally come to rest.
Definition of magnetic dipole moment (M) from the formula of Torque On Bar Magnet
(The Dipole in a Uniform Magnetic Field)
In equation (1) if B = 1T and θ = 90° then, M = τ
Hence, “The magnitude of the torque acting on a bar magnet placed perpendicular to a unit magnetic field is equal to the magnetic dipole moment of the bar magnet.”