Magnetism & Matter

Magnetic dipole moment of a bar magnet: product of pole strength qₘ and magnetic length 2l (distance between poles, slightly less than geometric length). Direction: from S-pole to N-pole. SI unit: A·m².

Field at a point on the axis of a bar magnet at distance r from centre. Direction is along the dipole moment (N→S outside). The approximation holds when r ≫ l. Axial field is twice the equatorial field at the same distance.

Field at a point on the equatorial line (perpendicular bisector) at distance r from centre. Direction is antiparallel to the dipole moment. Magnitude is half the axial field at the same distance.

Field magnitude at a point making angle θ with the dipole axis at distance r (for r ≫ l). α is the angle between B and the radial direction. At θ = 0° (axial): B = 2M·μ₀/4πr³; at θ = 90° (equatorial): B = M·μ₀/4πr³.

Torque experienced by a magnetic dipole in a uniform field B. θ is the angle between m and B. Torque is zero at θ = 0° (stable equilibrium) and θ = 180° (unstable equilibrium). Maximum at θ = 90°.

PE of a magnetic dipole in external field B. Zero reference at θ = 90°. Minimum (stable) at θ = 0°: U = −mB. Maximum (unstable) at θ = 180°: U = +mB. Analogous to electric dipole in electric field.

Work done by external agent in rotating a magnetic dipole from angle θ₁ to θ₂ against the field. To bring from equilibrium (θ = 0°) to angle θ: W = mB(1 − cosθ). This equals the gain in potential energy.

A bar magnet or compass needle oscillates with this period about its equilibrium in a uniform field B. I is the moment of inertia of the magnet about the oscillation axis. Used in magnetometers to measure B or m.

Net magnetic flux through any closed surface is always zero. Consequence: magnetic monopoles do not exist. Every field line that enters a closed surface must exit it. Contrasts with Gauss's law for electricity where net flux = Q_enc/ε₀.

Net magnetic dipole moment per unit volume of a material. SI unit: A m⁻¹. Measures how strongly a material is magnetised in response to an external field. Zero for unmagnetised materials.

Total magnetic field B in a medium is the sum of applied field component μ₀H and the field due to magnetisation μ₀M. H is the magnetic intensity (source field), SI unit A m⁻¹. In vacuum: B = μ₀H since M = 0.

Dimensionless ratio of magnetisation to applied magnetic intensity. χₘ < 0 for diamagnetics (repelled, e.g. bismuth), 0 < χₘ ≪ 1 for paramagnetics (weakly attracted), χₘ ≫ 1 for ferromagnetics (strongly attracted, e.g. iron).

Relative permeability μr relates permeability of medium to that of free space. For diamagnetics: μr < 1. For paramagnetics: μr slightly > 1. For ferromagnetics: μr ≫ 1 (can be 10³–10⁵). B = μH is the constitutive relation.

For paramagnetic materials, susceptibility is inversely proportional to absolute temperature T. C is the Curie constant (material-specific). Physical reason: thermal agitation opposes alignment of dipoles with the field. At higher T, randomisation wins.

Above the Curie temperature Tc, a ferromagnetic becomes paramagnetic and obeys the Curie-Weiss law. Susceptibility diverges as T → Tc⁺, signalling the ferromagnetic transition. Below Tc, the material has spontaneous magnetisation with domains.

Retentivity Br: value of B when H is reduced to zero (permanent magnetism). Coercivity Hc: reverse H needed to demagnetise the material. Area of hysteresis loop = energy dissipated per cycle per unit volume. Hard magnets: high Br, high Hc. Soft magnets: high Br, low Hc.

Earth's total field B resolved into horizontal component BH and vertical component BV. δ is the angle of dip (inclination). At magnetic equator: δ = 0°, BV = 0. At magnetic poles: δ = 90°, BH = 0. BH drives compass needles.

Angle of dip δ is the angle made by Earth's total magnetic field with the horizontal plane at a given location. Measured by a dip circle. Increases from 0° at magnetic equator to 90° at magnetic poles.

Apparent dip δ' observed when a dip circle is set in a vertical plane making angle φ with the magnetic meridian. Always δ' ≥ δ (apparent dip ≥ true dip). True dip is observed only when the dip circle is in the magnetic meridian (φ = 0°).

Declination α is the horizontal angle between true geographic north and the direction of Earth's horizontal magnetic field (magnetic north). Varies by location and changes slowly over time. Essential for navigation correction. Positive east, negative west of true north.