(a) Discuss the origin of hysteresis in ferromagnetic materials on the basis of domain theory. Explain how domain wall motion and…

(a) Domain theory of hysteresis and energy loss Domains: in a ferromagnet the quantum-mechanical exchange interaction aligns neighbouring atomic moments, so the material splits into Weiss domains, each spontaneously magnetised to saturation. In an unmagnetised sample the domain moments point in different easy directions and the net magnetisation is zero; this arrangement lowers the magnetostatic (stray-field) energy. Neighbouring domains are separated by Bloch walls in which the spin direction turns gradually; the wall width is a balance between exchange energy (favours a wide wall) and magnetocrystalline anisotropy energy (favours a narrow wall).

Magnetisation process: when a field H is applied,

• at low fields, walls bow and shift reversibly, so domains aligned with H grow;
• at intermediate fields, walls make large irreversible jumps;
• at high fields, domain moments rotate towards H until saturation.

Pinning and irreversibility: real crystals contain inclusions, voids, dislocations, grain boundaries and strain fields. A wall passing through such a defect has lower energy, so it sticks there (pinning). The field must rise until the wall breaks free; it then jumps suddenly to the next pinning site. These jumps are the Barkhausen effect. Because walls do not return along the same path when H is reduced, M lags behind H: at H = 0 some magnetisation remains (remanence Bᵣ), and a reverse field (coercivity H_c) is needed to remove it. This path-dependence traced over a full cycle is the hysteresis loop.

Energy loss: in each sudden jump the energy released is not recovered but is dissipated as heat through local eddy currents and spin-wave (lattice) damping. The work done by the source per cycle per unit volume is W = ∮ H dB = area of the B–H loop, and this appears as heat in the material (hysteresis loss, P = Wf per unit volume for frequency f).

• Soft materials (Si-steel, ferrites): few pinning sites, small H_c, narrow loop, low loss (transformer cores).
• Hard materials (alnico, NdFeB): strong pinning, large H_c, wide loop (permanent magnets). Above the Curie temperature the exchange ordering is destroyed, domains disappear and there is no hysteresis.

(b) Isothermal and adiabatic expansion of blackbody radiation Photon gas: u = aT⁴ with a = 4σ/c = 4 × 5.672 × 10⁻⁸/3.0 × 10⁸ = 7.57 × 10⁻¹⁶ J m⁻³ K⁻⁴ Pressure P = u/3 = aT⁴/3; internal energy U = aT⁴V; entropy S = (4/3)aT³V. At T = 2000 K: aT⁴ = 7.57 × 10⁻¹⁶ × 1.6 × 10¹³ = 1.21 × 10⁻² J m⁻³; ΔV = 10³ cm³ = 10⁻³ m³.

(i) Heat transferred (isothermal, reversible): Q = TΔS = (4/3)aT⁴ΔV = (4/3)(1.21 × 10⁻²)(10⁻³) = 1.61 × 10⁻⁵ J (absorbed)

(ii) Work done: at constant T the pressure is constant, so W = PΔV = (1/3)aT⁴ΔV = 4.04 × 10⁻⁶ J Check: ΔU = aT⁴ΔV = 1.21 × 10⁻⁵ J = Q − W; so Q = 4W.

(iii) Reversible adiabatic expansion: S = constant ⇒ VT³ = constant (PV⁴/3 = constant). V₁ = 10 cm³, V₂ = 10 + 10³ = 1010 cm³: T₂ = T₁(V₁/V₂)¹/3 = 2000 × (10/1010)¹/3 = 2000 × 0.2148 ≈ 429 K ΔT = 429 − 2000 ≈ −1.57 × 10³ K (the radiation cools by about 1571 K). If the expansion is taken to a final volume of 10³ cm³, T₂ ≈ 431 K and ΔT ≈ −1.57 × 10³ K as well.

(c) Motional emf in the moving wire Wire perpendicular to the X–Y plane ⇒ along z: L = 2k̂ m. Motional emf: ε = ∫(V × B)·dL = (V × B)·L for uniform V and B. V × B = | î ĵ k̂ ; 2 3 1 ; 1 2 0 | = î(3·0 − 1·2) − ĵ(2·0 − 1·1) + k̂(2·2 − 3·1) = −2î + ĵ + k̂ (V/m) ε = (−2î + ĵ + k̂)·(2k̂) = 2 V Only the component of V × B along the wire contributes; the magnitude of the potential difference is 2 V, with the end towards +z at the higher potential for the given directions.