Magnetism and Matter

In this chapter: field lines of a bar magnet, the bar magnet as an equivalent solenoid, a magnetic dipole in a uniform field, the electrostatic analogy, Gauss's law for magnetism, magnetisation, magnetic intensity H, susceptibility and permeability, diamagnetism, paramagnetism, ferromagnetism, Curie temperature and hysteresis.

The bar magnet

A freely suspended bar magnet settles roughly along the geographic north-south line. Its two poles cannot be isolated: cut a magnet in two and each piece is a complete magnet with its own north and south pole. Magnetic field lines of a bar magnet have these properties:

• They form continuous closed loops: outside the magnet they run from N to S, and inside from S to N. Electric field lines of a dipole, by contrast, begin on +q and end on −q.
• The tangent at any point gives the direction of B, and the density of lines shows its magnitude.
• Two field lines never cross.

Bar magnet as an equivalent solenoid

The field of a bar magnet and that of a current-carrying solenoid of the same size look almost identical, and the axial field of a finite solenoid at large distance has exactly the form of a dipole field. So a bar magnet can be treated as a solenoid with magnetic moment m = NIA, directed from S to N inside the magnet. For a short magnet (or any magnetic dipole) at distance r:

Dipole in a uniform magnetic field

The energy is minimum (−mB) when m is along B, which is stable equilibrium, and maximum (+mB) when it is opposite, which is unstable. This is how a compass needle aligns itself.

Worked example: A short bar magnet of moment 0.32 J T⁻¹ is placed in a uniform field of 0.15 T. Find the torque when its axis is at 30° to the field, and its potential energy in the stable and unstable positions.
Solution: τ = mB sin 30° = 0.32 × 0.15 × 0.5 = 0.024 N m. Stable (θ = 0°): U = −mB = −0.048 J. Unstable (θ = 180°): U = +0.048 J.

The electrostatic analogy

Results for the magnetic dipole can be read off from those for the electric dipole by the replacements below.

Magnetism and Gauss's law

The net magnetic flux through any closed surface is zero, because every field line that enters a closed surface also leaves it. In electrostatics the flux equals q/ε₀; the magnetic version is zero because isolated magnetic poles (monopoles) are not known to exist. The simplest magnetic element is a dipole or a current loop.

Magnetisation and magnetic intensity

Electrons in atoms have magnetic moments from their orbital motion and their spin. In a material, these add up to a net moment per unit volume, the magnetisation:

Place a material inside a solenoid with n turns per unit length carrying current I. The field due to the current alone is B₀ = μ₀nI. We define the magnetic intensity H = B₀/μ₀, so H = nI depends only on the external current. The total field is

χ is the magnetic susceptibility, a dimensionless number that measures how a material responds to a field. μ_r is the relative permeability and μ = μ₀μ_r the permeability.

Worked example: A solenoid with 1000 turns per metre carries 2 A and has a core of relative permeability 400. Find H, B and M in the core.
Solution: H = nI = 1000 × 2 = 2 × 10³ A m⁻¹. B = μ_rμ₀H = 400 × 4π × 10⁻⁷ × 2 × 10³ ≈ 1.0 T. M = (μ_r − 1)H = 399 × 2 × 10³ ≈ 8 × 10⁵ A m⁻¹. Notice that M is far larger than H; almost all of B comes from the core.

Magnetic properties of materials

(In the table, ε stands for a small positive number.)

Diamagnetism

Atoms of a diamagnetic substance have no net magnetic moment. An external field changes the orbital motion of the electrons so that a small moment is induced opposite to the field (a Lenz's law effect). Field lines are pushed out of the material. Diamagnetism is present in all substances but is usually masked by the other effects. A superconductor cooled below its critical temperature is a perfect diamagnet: χ = −1, μ_r = 0, and field lines are completely expelled. This is the Meissner effect.

Paramagnetism

Each atom has a permanent magnetic moment, but thermal motion keeps the moments randomly oriented. An external field aligns them partly, giving a small magnetisation along the field. Lower temperature and stronger field increase the alignment. Curie's law:

Ferromagnetism

In a ferromagnet the atomic moments of neighbouring atoms line up spontaneously over small regions called domains, each about 1 mm in size and containing around 10¹¹ atoms. Without a field the domains point in random directions and cancel. An applied field makes favourably oriented domains grow and others rotate, so the material becomes strongly magnetised.

Above the Curie temperature T_c, a ferromagnet becomes paramagnetic, with χ = C/(T − T_c) for T > T_c. NCERT lists T_c as 1043 K for iron, 1394 K for cobalt, 631 K for nickel and 317 K for gadolinium. Materials whose magnetisation persists after the field is removed are called hard ferromagnets (alnico, used for permanent magnets); those that lose it easily are soft ferromagnets (soft iron).

Hysteresis

Hysteresis, and the choice of materials for permanent magnets and electromagnets, belonged to a section that the rationalised NCERT book dropped. They are kept here because older papers and some state exams still use them, so check your current syllabus before spending long on this part.

If H is increased from zero, B rises along the initial curve to saturation. When H is brought back to zero, B does not return to zero; the value left over is the retentivity (remanence). A reverse field, the coercivity, is needed to make B zero. Taking H through a full cycle traces the hysteresis loop, and the area enclosed is the energy dissipated per unit volume per cycle.

• Permanent magnets need high retentivity and high coercivity, which means a broad loop (steel, alnico, cobalt steel, ticonal).
• Electromagnet cores need high permeability and low retentivity, so that the magnetism disappears when the current is switched off (soft iron).
• Transformer cores are taken through many cycles, so they need a narrow loop (low hysteresis loss), high permeability and high resistivity.

Common mistakes: (1) Saying field lines of a magnet go from N to S inside it; inside they go from S to N. (2) Mixing up the signs: diamagnetic χ is negative, paramagnetic χ is small and positive. (3) Using Curie's law χ ∝ 1/T for a ferromagnet below T_c. (4) Choosing soft iron for a permanent magnet; soft iron has low retentivity and coercivity. (5) Forgetting that H depends only on the free current, while B includes the material's contribution.

JEE and NEET focus

• Torque, potential energy and work done in rotating a magnetic dipole.
• Relations among B, H, M, χ and μ_r, with numericals on a solenoid with a core.
• Properties and examples of dia-, para- and ferromagnetic substances.
• Curie's law and the Curie temperature.
• Reading retentivity and coercivity from a hysteresis loop and choosing materials for magnets and cores (outside the rationalised NCERT book, but still seen in older papers).

Practice questions