Magnetic Field of a Magnetic Dipole (Small Bar Magnet)

A bar magnet has two magnetic poles: North (N) and South (S). When the separation between the poles is very small compared with the distance of the observation point from the magnet, it behaves as a short magnetic dipole.

A magnetic dipole produces a magnetic field that decreases rapidly with distance:

M = m_p​(2l)

• SI unit: A m²
• Direction: from the South pole to the North pole.

Consider a bar magnet of pole strength m_p, magnetic length 2l, and magnetic dipole moment M. Let point P lie on the axial line at distance r from its centre.

Exact Expression

The magnetic field at P is:

For a Short Bar Magnet

• For r ≫ l: (r² − l²)² ≈ r⁴
• Therefore, B_axial = \[{\frac{\mu_0}{4\pi}\frac{2M}{r^3}}\]

Direction: \[\vec B_{axial}\] is along \[\vec M\], i.e. from S to N.

Let point P lie on the equatorial line at a distance rr from the centre of the bar magnet.

Exact Expression

For a Short Bar Magnet

• For r ≫ l: (r² + l²)^3/2 ≈ r³
• Hence, B_equatorial = \[{\frac{\mu_0}{4\pi}\frac{M}{r^3}}\]

Direction: \[\vec B_{equatorial}\] is opposite to \[\vec M\], i.e. from N to S.

A short bar magnet has magnetic dipole moment M = 0.48 A m². Find the magnetic field at a point 10 cm from its centre:

1. on its axial line;
2. on its equatorial line.

Take \[\frac {μ_0}{4π}\] = 10⁻⁷ T m A⁻¹.

Given: M = 0.48 A m² ,r = 10 cm = 0.10 m

(A) Axial Field

Direction: S to N along the magnetic axis.

(B) Equatorial Field

Direction: N to S, opposite to \[\vec M\].

Check: B_a = 2B_e. Hence, the answer is consistent.