Big picture
A bar magnet behaves exactly like a current-carrying solenoid: both are magnetic dipoles, with a north and a south pole that can't be separated. This chapter treats magnets with the same tools used for electric dipoles: field on the axis and the equator, torque in a field, and potential energy.
It then asks how materials respond to a magnetic field. Some are weakly repelled (diamagnetic), some weakly attracted (paramagnetic), and some strongly attracted and able to become permanent magnets (ferromagnetic).
Learning goals
Explain the bar magnet as an equivalent solenoid and state the properties of magnetic field lines.
Calculate the magnetic field of a short bar magnet on its axial and equatorial lines.
Calculate the torque on, and potential energy of, a magnetic dipole in a uniform field, and the period of oscillation.
State Gauss's law for magnetism.
Define magnetisation, magnetic intensity, susceptibility and permeability.
Compare diamagnetic, paramagnetic and ferromagnetic materials and state Curie's law.
Core ideas
1. The bar magnet
A freely suspended magnet aligns roughly north–south.
Like poles repel, unlike poles attract.
Magnetic poles always exist in pairs: cutting a magnet gives two smaller magnets, each with a north and south pole. There are no magnetic monopoles.
Bar magnet as an equivalent solenoid. The field lines of a bar magnet and a current-carrying solenoid are identical. A magnet's magnetism comes from tiny circulating currents (electrons' motion) inside it, so a bar magnet is equivalent to a solenoid.
Properties of magnetic field lines
They form continuous closed loops: outside the magnet from N to S, and inside from S to N. (Electric field lines, by contrast, start and end on charges.)
The tangent at any point gives the direction of the field.
The closer the lines, the stronger the field.
Field lines never intersect, because the field has only one direction at a point.
2. Magnetic dipole moment and the field of a short bar magnet
The magnetic dipole moment m points from S to N, with SI unit A m² (or J/T). For a current loop, m = NIA.
For a short bar magnet at distance r (r much larger than the magnet's length):
Position | Field | Direction |
|---|---|---|
Axial line | B = (μ₀/4π) × 2m/r³ | Along m |
Equatorial line | B = (μ₀/4π) × m/r³ | Opposite to m |
At the same distance, the axial field is twice the equatorial field. Both fall off as 1/r³.
Example. A short bar magnet has m = 0.48 J/T. Field at 10 cm (0.1 m):
Axial: B = 10^-7 × 2 × 0.48 / (0.1)³ = 9.6 × 10^-5 T (along m)
Equatorial: B = 10^-7 × 0.48 / (0.1)³ = 4.8 × 10^-5 T (opposite to m)3. Dipole in a uniform magnetic field
Torque: τ = m × B, with magnitude τ = mB sin θ, where θ is the angle between m and B. The torque tries to align m with B.
Potential energy: U = −m · B = −mB cos θ.
Angle θ | Potential energy | Equilibrium |
|---|---|---|
0° (m along B) | −mB (minimum) | Stable |
90° | 0 | – |
180° (m opposite to B) | +mB (maximum) | Unstable |
Work done in rotating the dipole from θ₁ to θ₂ = U₂ − U₁ = mB (cos θ₁ − cos θ₂).
Example. A magnet with m = 0.32 J/T is placed at 30° to a uniform field of 0.15 T.
Torque = mB sin 30° = 0.32 × 0.15 × 0.5 = 0.024 N m
Work to turn it from 0° to 90° = mB (cos 0° - cos 90°) = 0.32 × 0.15 = 0.048 JOscillation of a magnet. A magnet suspended in a uniform field and slightly displaced oscillates with period
T = 2π √(I/mB)
where I is its moment of inertia. A stronger field or a larger dipole moment gives faster oscillations.
4. Gauss's law for magnetism
The net magnetic flux through any closed surface is zero: ∮ B · dA = 0.
Every field line that enters a closed surface also leaves it, because magnetic field lines are closed loops and there are no isolated poles (monopoles). Compare Gauss's law for electricity, where the flux equals the enclosed charge divided by ε₀.
5. Magnetisation, magnetic intensity and susceptibility
Quantity | Definition | Unit |
|---|---|---|
Magnetisation (M) | Net magnetic moment per unit volume of the material: M = m_net/V | A/m |
Magnetic intensity (H) | The magnetising field due to external currents: for a solenoid H = nI | A/m |
Magnetic field in the material | B = μ₀(H + M) | T |
Magnetic susceptibility (χ) | M = χH; measures how easily a material is magnetised | No unit |
Relative permeability (μr) | μr = 1 + χ; B = μ₀μr H = μH | No unit |
6. Magnetic materials
Property | Diamagnetic | Paramagnetic | Ferromagnetic |
|---|---|---|---|
Behaviour in an external field | Weakly repelled; moves from stronger to weaker field | Weakly attracted; moves from weaker to stronger field | Strongly attracted |
Susceptibility χ | Small and negative (−1 ≤ χ < 0) | Small and positive | Large and positive (hundreds to thousands) |
Relative permeability μr | Slightly less than 1 | Slightly more than 1 | Much greater than 1 |
Field lines | Pushed out of the material | Slightly concentrated inside | Strongly concentrated inside |
Effect of temperature | Nearly independent | χ ∝ 1/T (Curie's law) | Becomes paramagnetic above the Curie temperature |
Cause | No permanent atomic moments; induced moments oppose the field | Permanent atomic moments, randomly oriented, partly aligned by the field | Atomic moments aligned in domains |
Examples | Bismuth, copper, lead, silicon, water, nitrogen, sodium chloride | Aluminium, sodium, calcium, oxygen, copper chloride | Iron, cobalt, nickel, gadolinium, alloys like alnico |
Superconductors are perfect diamagnets (χ = −1): they expel the magnetic field completely (the Meissner effect).
Curie's law (for paramagnetic materials): χ = C/T, where C is the Curie constant and T is the absolute temperature. Heating a paramagnet reduces its susceptibility, because thermal motion disturbs the alignment of atomic moments.
Example. A paramagnetic salt has χ = 2.8 × 10⁻⁴ at 300 K. At 150 K (temperature halved), χ doubles to 5.6 × 10⁻⁴.
Domains. A ferromagnet is made of tiny regions called domains, in each of which all atomic moments are aligned. In an unmagnetised sample the domains point randomly. In an external field, domains aligned with the field grow and others rotate, so the sample becomes strongly magnetised.
Permanent magnets and electromagnets. Materials for permanent magnets need high retentivity (keep magnetism when the field is removed) and high coercivity (resist demagnetisation), such as steel and alnico. Materials for electromagnets and transformer cores need high permeability and low coercivity, such as soft iron, so they magnetise and demagnetise easily.
A reliable problem-solving routine
For the field of a short magnet, decide axial (2m) or equatorial (m), then use (μ₀/4π) = 10⁻⁷ T m/A and r in metres.
For torque, use mB sin θ; for energy, −mB cos θ; for work, the difference in energy.
For susceptibility questions, use M = χH, μr = 1 + χ, and Curie's law χ ∝ 1/T.
To classify a material, check the sign and size of χ.
Connections that matter
Moving charges and magnetism gives the field of a current loop and solenoid, the basis of the equivalence.
Electric charges and fields gives the analogous electric dipole results.
Electromagnetic induction and AC uses soft iron cores in transformers.
Common mistakes
Saying field lines start at N and end at S. They form closed loops, continuing inside the magnet.
Forgetting the factor 2 on the axial line.
Calling θ = 180° stable. It is unstable equilibrium with maximum energy.
Giving diamagnetic materials positive susceptibility.
Applying Curie's law to diamagnets. It applies to paramagnets.
Revision checklist
I can explain the bar magnet–solenoid equivalence and field line properties.
I can calculate axial and equatorial fields of a short magnet.
I can calculate torque, potential energy, work and period of oscillation.
I can state Gauss's law for magnetism.
I can define M, H, χ and μr and use their relations.
I can compare the three types of magnetic materials and use Curie's law.