Torque on a Magnetic Dipole of a Bar Magnet

The magnetic dipole can be defined as small magnets of subatomic or microscopic dimensions that show the characteristics and properties of an electric charge around a loop. The magnetic dipole can be called an analogue of the electric dipole. This means the magnetic dipole behaves like an electric dipole. But the analogy is not flawless. In nature, positive and negative electric charges can exist separately, but the north and south poles cannot exist separately.

To summarise, a bar magnet shows magnetic dipole and magnetic dipole moments. So, in a magnetic field, torque is created. We will discuss the formula of torque on a magnetic dipole. 

Magnetic dipole and magnetic dipole moment 

The magnetic dipole is used for an electric current loop behaving like a tiny magnet. In electromagnetism, a current-carrying loop of any shape gives rise to magnetic fields. Thus, the current loop acts like a tiny magnet. We can say that these current loops behave like a magnetic dipole. We can control the strength of this magnetic dipole, which is not possible in the case of a permanent magnet. 

Magnetic dipole moment

The strength of a magnetic dipole is the magnetic dipole moment. The magnetic dipole moment depends on factors such as:

• The current of the magnetic dipole (I)
• The number of coils in the loop (n)
• The area of the current loop (A)

Therefore, the magnetic dipole moment (M) formula can be represented as 

M= nIA. 

The unit of the magnetic dipole moment is ampere metre-square [ Am² ].

The magnetic dipole moment is a vector quantity. Besides having the magnitude of nIA, it has a direction that can be found by applying the Right-hand Rule following the direction of the current flow. 

The magnetic dipole moment of a bar magnet has the formula of pole strength x the distance between the pole [ side length ], i.e., M= m2l [ side length of the bar magnet is 2l].

Formulae of torque on magnetic dipole 

When we place a current loop in a uniform magnetic field, it acts as a magnetic dipole and experiences force. If the magnetic dipole is suspended at an angle, it will experience torque. The torque will make the magnetic dipole align with the field lines of the uniform magnetic field. We will understand this better with a simple experiment.

Experiment

Let us perform a simple experiment. We assume the following:

1. We have suspended a bar magnet as the magnetic dipole in a uniform magnetic field. 
2. The length of the bar magnet is 2l, and the magnitude of the uniform magnetic field is B.
3. The magnetic field lines run from left to right as B is a vector quantity. 
4. The bar magnet is suspended so that its north pole is in an upward direction and the south pole is downward. 
5. The slant of the bar magnet makes an angle θ with the lines of the uniform magnetic field.

Observation

• The bar magnet will face a force in each of its poles. 
• The north pole will experience a force in the direction of the uniform field lines.
• The south pole will experience a force in the direction opposite to the uniform field lines.
• This will make the bar magnet rotate, and it will experience torque. 

Conclusion of the experiment

The force on the north and south pole has a magnitude of the product of the pole strength (m) and the magnitude of the uniform magnetic field (B). F=mB. The torque can be calculated by the formula force x perpendicular distance. 

The perpendicular distance of the bar magnet in the uniform magnetic field will be 2lsinθ.

Therefore the formula of the torque is 

T = mB2lsinθ, or T = m(2l)Bsinθ. 

Thus we can also write T = MBsinθ. [ as M = m2l ]

The torque is the product of the magnetic moment, the magnitude of the uniform magnetic field, and the sine of the angle θ.

This experiment answers two questions:

1. What is the formula of torque on a magnetic dipole?
2. How does torque relate to the magnetic field?

Conclusion

We will conclude this topic by highlighting the main points. The formula of magnetic dipole moments is different for a current loop and a bar magnet. The magnetic dipole moment of a current loop can be increased instead of the bar magnet. The bar magnet and the current loop experience torque when kept in a uniform magnetic field. They behave in the same manner within the magnetic field. To gain more knowledge on this topic, one should read about the potential energy and forces that act on a mar magnet inside a magnetic field.