Magnetism and Matter

A bar magnet always has two poles, a north (N) and a south (S) pole, that cannot be separated — cutting a magnet in half only produces two smaller magnets, each with its own N and S pole. The magnetic field lines of a bar magnet emerge from the N pole, curve through the space outside the magnet, enter the S pole, and continue inside the magnet from S back to N, forming continuous closed loops. Field lines never intersect, and they are more crowded where the field is stronger.

Ampere proposed that all magnetism ultimately arises from electric currents — tiny atomic current loops. This leads to the idea of the equivalent solenoid: a current-carrying solenoid of length 2l, radius a, N turns and current I produces, at points far away, a field with exactly the same pattern as a bar magnet of the same size. Both are treated as a magnetic dipole of moment m, where for the solenoid m = NIA (A = cross-sectional area) and, in the older pole model, m = q_m × 2l for a bar magnet (q_m = pole strength, 2l = separation between poles). This is the magnetic analogue of the electric dipole, with m playing the role p played there.

When the distance r from the magnet's centre is much larger than the magnet's own length (r ≫ l), the magnet can be treated as a point dipole, and its field has a simple closed form at two special points: on the axis (end-on) and on the equatorial line (broadside-on, the perpendicular bisector of the magnet). The axial field points along m; the equatorial field is exactly half the axial field in magnitude, at the same r, and points opposite to m.

Worked example. A short bar magnet has magnetic moment m = 0.40 A·m². Find the axial and equatorial fields at r = 0.50 m from its centre (r is much larger than the magnet's length, so the dipole formulas apply).

1. Axial field: B_axial = (μ₀/4π) × 2m/r³ = (1×10⁻⁷) × (2 × 0.40)/(0.50)³ = (1×10⁻⁷) × 0.80/0.125 = (1×10⁻⁷) × 6.4 = 6.4×10⁻⁷ T.
2. Equatorial field: B_eq = (μ₀/4π) × m/r³ = (1×10⁻⁷) × 0.40/0.125 = 3.2×10⁻⁷ T.
3. Check: B_eq = ½ B_axial as expected, and both fields are extremely small because r is large compared to atomic-scale currents but the magnet itself is weak (0.40 A·m² is typical of a small bar magnet).

When a bar magnet of moment m is placed in a uniform external magnetic field B, making an angle θ with it, the two poles experience equal and opposite forces (q_mB each) that do not cancel in effect because they act at different points — this produces a torque rather than a net force. The torque tends to rotate the magnet so as to align m parallel to B, exactly as an electric dipole experiences a torque in a uniform electric field.

Associated with this torque is a potential energy that depends on orientation. Taking the reference (zero of potential energy) at θ = 90°, the magnet has minimum (most negative) potential energy when m is parallel to B (θ = 0°, the stable equilibrium) and maximum potential energy when m is antiparallel to B (θ = 180°, the unstable equilibrium). Work must be done against the field to rotate the dipole away from its stable position.

Worked example. A bar magnet of moment m = 0.50 A·m² is held at θ = 60° to a uniform field B = 0.20 T. Find the torque and the potential energy in this position.

1. Torque: τ = mB sinθ = 0.50 × 0.20 × sin60° = 0.10 × 0.866 = 0.0866 N·m, directed so as to reduce θ toward zero.
2. Potential energy: U = −mB cosθ = −0.50 × 0.20 × cos60° = −0.10 × 0.5 = −0.050 J.
3. If instead the magnet were rotated all the way to θ = 0°, U would become −0.10 J (lower, more stable); at θ = 180°, U would be +0.10 J (highest, most unstable).

This description is mathematically identical (with m replacing p and B replacing E) to the electric dipole in a uniform electric field studied earlier — the same torque and energy expressions apply, which is why the bar magnet and the current loop / solenoid can be treated on the same footing as a magnetic dipole.

Just as Gauss's law in electrostatics relates the electric flux through a closed surface to the charge enclosed, there is a magnetic version of Gauss's law. It states that the net magnetic flux through any closed surface is always zero, no matter what shape the surface has or where it is placed relative to magnets or currents.

The reason is fundamental: unlike electric charges, which can exist singly (an isolated positive or negative charge), magnetic poles have never been found to exist in isolation — every north pole is inseparably paired with a south pole. As a result, for any closed surface, every magnetic field line that leaves the surface at some point (heading away from a region behaving like a N pole) must re-enter the surface somewhere else (heading into a region behaving like a S pole). The outward and inward contributions to the flux exactly cancel, giving zero net flux.

This is in direct contrast to the electrostatic case, ∮E·dA = q_enc/ε₀, where a nonzero enclosed charge gives nonzero flux — consistent with electric field lines actually starting or ending on charges. Magnetic field lines, having nowhere to start or end, must always close on themselves.

Worked example. A closed surface is drawn so that it encloses the N-pole end of a long bar magnet. Suppose magnetic flux of 7 × 10⁻⁴ Wb is calculated to be leaving through the curved part of the surface near the pole. What is the flux entering through the remaining (flat, far) part of the same closed surface?

1. By Gauss's law for magnetism, the total flux through the whole closed surface must be zero.
2. So flux entering through the rest of the surface = flux leaving = 7 × 10⁻⁴ Wb, i.e. exactly enough field lines re-enter the surface (looping back around through the magnet toward the S pole) to cancel what left near the N pole.

The Earth itself behaves approximately like a giant bar magnet (magnetic dipole) tilted at about 11.3° to its rotation axis, with its magnetic south pole located near the Earth's geographic north (which is why the north-seeking end of a compass needle points roughly toward geographic north). The three quantities that completely specify the Earth's magnetic field at a place are called the elements of the Earth's magnetic field.

Magnetic declination (D) is the angle between the geographic (true) north and the magnetic north as shown by a compass, i.e. the angle between the geographic and magnetic meridians at that place. Angle of dip / inclination (I) is the angle that the Earth's total magnetic field B makes with the horizontal plane at that place, measured using a dip needle free to rotate in the vertical plane. The field's horizontal component B_H is what an ordinary compass needle aligns with. These three are related through B_H = B cos I and the vertical component B_V = B sin I, so tan I = B_V/B_H. At the magnetic equator, I = 0° (field is fully horizontal); at the magnetic poles, I = 90° (field is fully vertical) and B_H = 0, so an ordinary compass becomes useless there.

Worked example. At a certain place, the horizontal component of the Earth's field is B_H = 0.35 G and the angle of dip is 60°. Find the total field B and its vertical component B_V.

1. B = B_H/cos I = 0.35/cos60° = 0.35/0.5 = 0.70 G (= 7.0 × 10⁻⁵ T, using 1 G = 10⁻⁴ T).
2. B_V = B sin I = 0.70 × sin60° = 0.70 × 0.866 = 0.606 G.
3. Check via tan I: B_V/B_H = 0.606/0.35 = 1.732 = tan60° ✓.

When a material is placed inside a magnetising field (such as inside a current-carrying solenoid), the atomic magnetic dipoles within it respond, and the material develops a net magnetic moment. Magnetisation (M) is defined as the net magnetic moment per unit volume of the material, M = m_net/V, and has SI unit A/m — the same unit as the applied field intensity, which makes them directly comparable.

The magnetic intensity (H) is the magnetising field produced purely by free (externally supplied) currents, independent of the material's response; for a long solenoid with n turns per unit length carrying current I, H = nI. Once a material is placed inside, the actual magnetic field B inside it has contributions from both the external H and the material's own induced magnetisation M, combined as B = μ₀(H + M). The ratio χ = M/H is the material's magnetic susceptibility, a dimensionless number that tells us how strongly the material magnetises in response to H: χ is small and negative for diamagnetic materials, small and positive for paramagnetic materials, and large and positive for ferromagnetic materials. The relative permeability μ_r = 1 + χ, and the material's absolute permeability μ = μ₀μ_r, so that B = μH.

Worked example. A long solenoid has n = 3000 turns/m and carries I = 2.0 A. Its core is filled with a material of relative permeability μ_r = 400. Find H, the susceptibility χ, the magnetisation M, and the resulting field B.

1. H = nI = 3000 × 2.0 = 6000 A/m.
2. χ = μ_r − 1 = 400 − 1 = 399.
3. M = χH = 399 × 6000 = 2.394 × 10⁶ A/m.
4. B = μ₀μ_rH = (4π×10⁻⁷) × 400 × 6000 = (1.2566×10⁻⁶) × 2.4×10⁶ ≈ 3.02 T.
5. Check with B = μ₀(H+M): μ₀(6000 + 2.394×10⁶) = (1.2566×10⁻⁶) × 2.4×10⁶ ≈ 3.02 T ✓ (same answer, as it must be).

Diamagnetic materials (e.g. bismuth, copper, water, gold) have atoms with no net magnetic moment; an external field induces a weak moment that opposes the applied field (Lenz's-law-like behaviour at the atomic level), so χ is small and negative (typically of order −10⁻⁵ to −10⁻⁶) and essentially independent of temperature. Diamagnetic materials are weakly repelled from strong-field regions and tend to move from stronger to weaker field.

Paramagnetic materials (e.g. aluminium, sodium, oxygen, platinum) have atoms with a net permanent magnetic moment due to unpaired electrons, but these moments are randomly oriented by thermal motion in the absence of a field. An external field partially aligns them, giving a small positive χ (typically 10⁻⁴ to 10⁻⁵) that decreases as temperature rises, since thermal agitation competes against alignment. This temperature dependence is captured by Curie's law: χ = C/T, where C is the Curie constant and T the absolute temperature.

Ferromagnetic materials (iron, cobalt, nickel and their alloys) have very strong interatomic ("exchange") interactions that align neighbouring atomic moments spontaneously within small regions called domains, even without an external field. In an unmagnetised sample the domains point in random directions and cancel out; an external field grows and aligns the domains, producing a very large positive χ (often 10² to 10ⁱ). Ferromagnetism disappears above a material-specific Curie temperature, above which thermal agitation destroys the domain alignment and the material becomes simply paramagnetic.

Worked example (Curie's law). A paramagnetic sample has susceptibility χ = 2.8 × 10⁻⁴ at T = 300 K. What is its susceptibility at 200 K?

1. Since χ ∝ 1/T, χ₂/χ₁ = T₁/T₂.
2. χ₂ = χ₁ × (T₁/T₂) = 2.8×10⁻⁴ × (300/200) = 2.8×10⁻⁴ × 1.5 = 4.2×10⁻⁴.
3. Susceptibility increases as temperature falls, exactly as Curie's law predicts (more alignment survives when thermal agitation is weaker).

This trio of behaviours directly governs material choice for practical magnets. A permanent magnet (e.g. steel, alnico) needs high retentivity (it keeps strong magnetisation after the external field is removed) and high coercivity (it strongly resists being demagnetised by stray fields or shocks) — steel's broad hysteresis loop delivers both. An electromagnet core (soft iron) instead needs high permeability so it magnetises strongly for a modest current, but low retentivity and low coercivity so it demagnetises almost completely the instant the current is switched off, and a narrow hysteresis loop so that little energy is wasted as heat on each magnetisation cycle.