Worked example: An electromagnet has N = 500 turns on a soft-iron core (μr = 800), length l = 0.4 m, carrying I = 3 A. (a) Estimate B inside. (b) What current gives the same B with an air core? (c) Why is the air-core version impractical? Step 1: B = μ0μrNI/l = (4π × 10⁻⁷)(800)(500)(3)/0.4 = (4π × 10⁻⁷ × 1.2 × 10⁶)/0.4 = (1.508)/0.4 ≈ 3.77 T — in practice the iron saturates around 1.5–2 T, so the real field caps there; the linear formula overestimates past saturation. Step 2: Air core (μr = 1): need I′ = 800 × 3 = 2400 A for the same unsaturated B. Step 3: 2400 A needs massive conductors and megawatts of cooling — the iron core's μr is doing the work of a power station. This is the entire economic case for cores, and why saturation is the hard ceiling of electromagnet design.