Magnetic Dipole - Physics Class 12 - NEET PDF Download

A magnetic dipole is an arrangement of two unlike magnetic poles of equal pole strength separated by a small distance. Examples of magnetic dipoles are a small bar magnet, a magnetic needle, and a current-carrying circular loop when observed from distances large compared with the size of the source.

Magnetic dipole moment

For a small bar magnet (two magnetic poles of strength m separated by a distance 2l) the magnitude of the magnetic dipole moment is

M = m(2l)

For a planar current loop of steady current I and area A, the magnetic dipole moment (vector) is

M = I A n̂

where n̂ is a unit vector normal to the plane of the loop determined by the right-hand rule: curl the fingers in the direction of current, the thumb gives the direction of M.

The magnetic dipole moment is commonly denoted by M in this text. It is also frequently written as the symbol μ in other references.

SI unit: ampere-metre² (A·m²). This unit is equivalent to joule per tesla (J·T^-1).

Direction: For a bar magnet the direction of M is from the south pole towards the north pole. For a current loop the direction is given by the right-hand rule described above.

Magnetic field due to a magnetic dipole

B(r) = (μ₀ / 4π) · (1 / r³) · [3 (M · r̂) r̂ - M]

This vector expression yields the standard magnitudes on the axial and equatorial lines given below.

On the axial line

Consider a point on the axis of the dipole at distance r from its centre (r ≫ dipole size). The magnetic field magnitude on the axis is

B_axial = (μ₀ / 4π) · (2M / r³)

The field at such a point lies along the axis. The direction depends on whether the point is on the north side or south side of the dipole; the vector formula above gives the correct sign and direction in each case.

On the equatorial line (perpendicular bisector)

At a point on the equatorial line (the line perpendicular to the dipole axis through its centre) at distance r (r ≫ dipole size), the magnitude of the field is

B_equatorial = (μ₀ / 4π) · (M / r³)

The direction of the field on the equatorial line is perpendicular to the axis and given by the vector expression; in terms of the dipole moment it is opposite to the component of M along the field point direction.

Torque on a magnetic dipole in a uniform magnetic field

When a magnetic dipole of moment M is placed in a uniform magnetic field B, a torque acts on the dipole which tends to align M with B. The torque is a vector given by

τ = M × B

The magnitude of the torque is

|τ| = MB sin θ

where θ is the angle between the vectors M and B. The direction of the torque follows the right-hand rule for the vector cross product.

If the dipole is free to rotate, the torque produces rotation until M is aligned with B, which is the condition for stable equilibrium.

Potential energy of a magnetic dipole in a uniform magnetic field

U = -M · B = -MB cos θ

When M is aligned with B (θ = 0), the energy is minimum (U = -MB) and the equilibrium is stable. When M is anti-aligned with B (θ = π), the energy is maximum (U = +MB) and the equilibrium is unstable.

Important points

• Relation between representations: A small bar magnet can be idealised as two equal and opposite magnetic poles separated by a small distance, or as a magnetic dipole moment M placed at its centre. Both representations give the same far-field expressions for r ≫ size of magnet.
• Equivalence of units: 1 A·m² = 1 J·T^-1.
• Direction conventions: For current loops, use the right-hand rule to get the direction of M. For a bar magnet, M points from the south pole to the north pole.
• Dipole approximation: The dipole field expressions are valid when the observation distance r is much larger than the physical dimensions of the source.