1The Bar Magnet

A bar magnet is the simplest, most familiar magnetic dipole — every magnet has two poles, north and south, that always come as a pair. Cut a bar magnet in half, and instead of isolating a single pole, you just get two smaller magnets, each with its own north and south. No isolated magnetic pole (monopole) has ever been observed.

A current-carrying loop produces a field pattern outside it that’s nearly identical to a bar magnet’s — which is why a bar magnet is often modelled as an equivalent solenoid, its magnetism arising from countless aligned atomic current loops rather than actual “magnetic charge.”

2Magnetic Field Lines

• Magnetic field lines always form closed loops — they emerge from the north pole, curve around, and re-enter at the south pole, continuing through the interior of the magnet back to the north pole. This is fundamentally different from electric field lines, which start and end on charges.
• Field lines never intersect, same reasoning as for electric fields.
• The density of field lines indicates field strength — closer together where B is stronger.

3Field Due to a Magnetic Dipole

A bar magnet’s magnetic moment m plays the same role the electric dipole moment p played in Chapter 1 — and, unsurprisingly, the field expressions look almost identical.

On the Axial Line (r ≫ size of magnet)
B_axial = (μ₀ / 4π) · 2m / r³
On the Equatorial Line
B_equatorial = (μ₀ / 4π) · m / r³
Exactly half the axial value at the same distance — the same ratio seen for the electric dipole in Chapter 1.

4Torque and Potential Energy in a Uniform Field

A bar magnet placed in an external uniform field feels a torque that tries to align it with the field, and has an orientation-dependent potential energy — both direct parallels to the electric dipole case.

Torque
τ = m × B, magnitude τ = mB sin θ
Potential Energy
U = − m · B = − mB cos θ
Minimum (most stable) when m is aligned with B; maximum when anti-aligned — this is why a compass needle settles pointing along the field.

5Gauss’s Law for Magnetism

Gauss’s Law for Magnetism
∮ B · dA = 0
The net magnetic flux through any closed surface is always exactly zero — a direct statement that magnetic monopoles don’t exist. Whatever field lines enter a closed surface must also leave it, since they never simply terminate.

6The Earth’s Magnetism

The Earth itself behaves approximately like a giant bar magnet tilted from its rotation axis — which is why a compass needle doesn’t quite point to true geographic north. Three quantities, called the elements of Earth’s magnetic field, describe it fully at any location:

• Declination: the angle between geographic north and the direction a compass points (magnetic north).
• Inclination (Dip): the angle the Earth’s field makes with the horizontal plane — the field isn’t horizontal everywhere; it dips into the ground at higher latitudes.
• Horizontal component: B_H = B cos(dip), the part of the field a horizontal compass actually responds to.

7Magnetization and Magnetic Intensity

When a material is placed in a magnetic field, its atomic magnetic moments respond, developing a net magnetic moment per unit volume called magnetization.

Magnetization
M = m_net / V
Total Field Inside a Magnetized Material
B = μ₀ (H + M)
H is the magnetic intensity (field due to free currents alone), M is the material’s own contribution.
Magnetic Susceptibility
χ = M / H
Tells you how strongly a material magnetizes in response to a given field — the sign and size of χ is exactly what distinguishes the three classes of magnetic materials below.

8Magnetic Properties of Materials

Diamagnetic Materials

Weakly repelled by a magnetic field; χ is small and negative, essentially independent of temperature. Examples: bismuth, copper, water. Every material has a diamagnetic response, but it’s usually too weak to notice unless nothing stronger is present.

Paramagnetic Materials

Weakly attracted by a magnetic field; χ is small and positive. Follows Curie’s Law:

Curie’s Law
χ = C / T
C is the Curie constant, T is absolute temperature — susceptibility falls as temperature rises, since thermal agitation disrupts the alignment of atomic moments. Examples: aluminium, platinum, oxygen.

Ferromagnetic Materials

Strongly attracted; χ is large and positive, and depends on field history (hysteresis) — this is what makes permanent magnets possible. Examples: iron, cobalt, nickel. Above a critical temperature called the Curie temperature, thermal motion overcomes the internal alignment entirely, and the material becomes merely paramagnetic.

Formula Summary

Axial FieldB = (μ₀/4π)(2m/r³)
Equatorial FieldB = (μ₀/4π)(m/r³)
Torqueτ = mB sinθ
Potential EnergyU = −mB cosθ
Gauss’s Law (Magnetism)∮B·dA = 0
Total FieldB = μ₀(H+M)
Susceptibilityχ = M/H
Curie’s Lawχ = C/T

Solved Examples

Example 1 · Axial Field of a Bar Magnet

A bar magnet has a magnetic moment of 5 A·m². Find the magnetic field at a point 20 cm from its centre, on the axial line.

Solution: B = (μ₀/4π)(2m/r³) = 10⁻⁷ × (2 × 5) / (0.2)³ = 10⁻⁷ × 10 / 0.008 = 1.25×10⁻⁴ T.

Example 2 · Earth’s Field Components

At a certain place, the horizontal component of the Earth’s magnetic field is 0.3 G and the angle of dip is 60°. Find the total field strength and the vertical component.

Solution: B_H = B cos(dip) → B = 0.3 / cos60° = 0.3 / 0.5 = 0.6 G.

Vertical component: B_V = B sin(dip) = 0.6 × sin60° ≈ 0.6 × 0.866 ≈ 0.52 G.

Quick Check

1. Magnetic field lines, unlike electric field lines, always:
Start and end on charges
✓ Form closed loops (correct)
Point radially outward
Never exist inside a magnet
2. A material with small, negative magnetic susceptibility is:
Ferromagnetic
Paramagnetic
✓ Diamagnetic (correct)
Non-magnetic
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