Magnetism & Matter— Notes, Formulas & Revision

A bar magnet is a permanent magnet with two opposite POLES at its ends — by convention North (N) and South (S).

Magnetic field lines emerge from N, curve around, and enter S externally. Inside the magnet they go S → N (closed loops, no monopoles).

Magnetic dipole moment: m = NIA (for a current loop equivalent) or m = qm × 2L for a magnetic-charge bar magnet.

Field along the AXIAL line (on the magnet's axis, distance r ≫ L): B_axial = (μ₀/4π)·(2m/r³).

Field along the EQUATORIAL line (perpendicular bisector, distance r ≫ L): B_equatorial = (μ₀/4π)·(m/r³). Half the axial field.

Like poles repel, unlike attract — force ∝ 1/r⁴ for short bar magnets along axial line.

Cutting a bar magnet in half produces TWO smaller bar magnets, each with its own N and S — NO isolated monopoles.

Earth itself acts as a giant bar magnet (approximately) — used in compass navigation.

Axial point: lies on the line through the center of the dipole, along its axis (extending the N-S line).

Equatorial point: lies on the perpendicular bisector of the dipole at the center.

Axial field is PARALLEL to the dipole moment; equatorial field is ANTIPARALLEL.

B_axial = (μ₀/4π)·(2m·r)/(r² − L²)². For r ≫ L: B_axial → (μ₀/4π)·(2m/r³).

B_equatorial = (μ₀/4π)·m/(r² + L²)^(3/2). For r ≫ L: B_equatorial → (μ₀/4π)·(m/r³).

Ratio at same r: B_axial / B_equatorial = 2 (for short dipoles).

Direction of axial field: from S to N of the source dipole (same as m).

Direction of equatorial field: opposite to m (from N to S externally).

Earth's magnetic field is approximately that of a bar magnet inside the Earth, tilted ~11° from the rotation axis.

Magnetic North pole is in the geographic SOUTH (Antarctic) — that's why a compass needle's N pole points to geographic N.

Three elements describe the field at any point: (i) declination D (angle between true N and magnetic N), (ii) dip angle / inclination I (angle between B and horizontal), (iii) horizontal component B_H.

Total field: B = B_H/cos I. Vertical component: B_V = B_H tan I.

Magnitude of Earth's B at the surface: ~25 to 65 μT. Strongest at poles (~65 μT), weakest near equator (~25 μT).

Earth's field has REVERSED polarity many times over geological history (last ~780,000 years ago) — recorded in basaltic rocks.

Origin: convection currents in the molten iron-nickel OUTER CORE — geodynamo theory.

Magnetosphere shields Earth from solar wind; auroras occur near magnetic poles where charged particles spiral down field lines.

Magnetization M is the magnetic moment per unit volume of a material: M = m/V. Units: A/m.

When a material is placed in an external field H, it develops a magnetization that adds to the total field: B = μ₀(H + M).

For LINEAR materials: M = χ·H, where χ is the magnetic susceptibility (dimensionless).

Permeability: μ = μ₀(1 + χ) = μ₀·μ_r, where μ_r is the relative permeability.

Three classes: diamagnetic (χ < 0, very small, μ_r ≈ 1), paramagnetic (χ > 0, small, μ_r ≈ 1), ferromagnetic (χ ≫ 1, very large, nonlinear).

H is determined by free currents alone; B includes contributions from magnetization.

In free space M = 0 ⇒ B = μ₀H.

Magnetization saturates in ferromagnets at high H — all spins aligned, can't increase M further.

Three classes of magnetic materials, distinguished by χ (susceptibility):

DIAMAGNETIC (χ < 0, small): Bi, Cu, Ag, Au, water, most non-metals. Repelled weakly by magnetic fields. Property of ALL matter; usually masked by para- or ferro-magnetism.

PARAMAGNETIC (χ > 0, small ~10⁻³): Al, Pt, Mn, Cr, O₂. Attracted weakly. Follows Curie's law: χ = C/T.

FERROMAGNETIC (χ huge, ~10³-10⁵): Fe, Co, Ni, Gd, and many alloys. Spontaneous magnetization persists without external field. Exists below Curie temperature T_C.

ANTIFERROMAGNETIC: adjacent spins anti-align (Cr, MnO). Net M ≈ 0 below Néel temperature. Special class.

FERRIMAGNETIC (ferrites): Fe₃O₄, MnFe₂O₄. Sublattice spins anti-align but unequal ⇒ net M ≠ 0. Used in transformers due to high resistivity (low eddy currents).

Soft magnets: easy to magnetize/demagnetize (electromagnets, transformer cores).

Hard magnets: retain magnetization (permanent magnets: alnico, neodymium NdFeB).

Hysteresis: the B vs H curve of a ferromagnet does NOT retrace itself when H is reversed. It forms a CLOSED LOOP.

Starting from unmagnetised material (B = 0, H = 0): as H increases, B follows the INITIAL MAGNETIZATION CURVE up to saturation B_s.

Reducing H to zero: B does NOT return to zero. It retains REMANENT MAGNETIZATION B_r (residual magnetism).

Reversing H: B decreases through zero at COERCIVE FIELD H_c (the negative H required to demagnetize).

Further reverse H: saturates at −B_s. Then reversing again traces out a closed loop.

Area enclosed by the loop = energy dissipated per unit volume per cycle (heat). Source of transformer core losses.

Soft magnets (Si-iron): narrow loop, small area ⇒ small hysteresis loss. Used in transformers.

Hard magnets (alnico, NdFeB): wide loop ⇒ large H_c ⇒ retain magnetization strongly. Used as permanent magnets.