Question 1

Dimensions: - S = emf per unit temperature = V/K. Voltage V = W/q = [M L² T⁻²]/[I T] = [M L² T⁻³ I⁻¹]. So S = [M L² T⁻³ I⁻¹ K⁻¹]. - σ (electrical conductivity): resistance R = V/I = [M L² T⁻³ I⁻²]; resistivity ρ = R·A/ℓ = [M L³ T⁻³ I⁻²]; σ = 1/ρ = [M⁻¹ L⁻³ T³ I²]. - κ (thermal conductivity): from P = κ A ΔT/ℓ → κ = P·ℓ/(A·ΔT) = [M L² T⁻³]·[L]/([L²][K]) = [M L T⁻³ K⁻¹]. Now Z = S²σ/κ: - S² = [M² L⁴ T⁻⁶ I⁻² K⁻²] - S²σ = [M¹ L¹ T⁻³ I⁰ K⁻²] - Divide by κ: [M⁰ L⁰ T⁰ I⁰ K⁻¹] So Z has dimensions [M⁰L⁰T⁰I⁰K⁻¹] (this is the thermoelectric figure of merit, with units of 1/K).

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