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For an ideal permanent magnet whose demagnetization curve is the straight line B = Br + µ₀H (with a sufficiently large HcJ) and whose remanent flux density is Br = 1.40 T, which value is closest to the maximum energy product (BH)max? Take µ₀ = 4π×10⁻⁷ H/m.

  1. AAbout 390 kJ/m³
  2. BAbout 780 kJ/m³
  3. CAbout 1560 kJ/m³
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Answer: A. About 390 kJ/m³

On the straight line in the second quadrant, the magnitude of the product of B and H, |BH| = (Br + µ₀H)(−H), is greatest at H = −Br/(2µ₀), B = Br/2, giving (BH)max = Br²/(4µ₀). Substituting values, 1.40²/(4 × 4π×10⁻⁷) = 1.96/(5.03×10⁻⁶) ≈ 3.9×10⁵ J/m³, about 390 kJ/m³ (about 49 MGOe). About 1560 kJ/m³ multiplies the two ends of the line, Br and Br/µ₀, and about 780 kJ/m³ divides by 2 only once. The theoretical upper limit estimated by the same formula from the saturation magnetization of Nd₂Fe₁₄B (about 1.6 T) is about 510 kJ/m³, and high-performance commercial neodymium magnets reach about 80% of it (about 400 to 440 kJ/m³).

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