Magnetic energy density per unit volume is given by which expression?

Magnetic energy density per unit volume is given by which expression?

• A. u_B = B^2 /(2 μ0)
• B. u_B = μ0 B^2 /2
• C. u_B = B /(2 μ0)
• D. u_B = B^2 /(μ0)

Answer: (A)

Magnetic energy per unit volume in a vacuum grows with the square of the magnetic flux density and scales with 1/μ0. The correct expression is u_B = B^2/(2 μ0). This comes from the relation u = ∫ H · dB; in vacuum H = B/μ0, so u = ∫ (B/μ0) dB = B^2/(2 μ0). The energy density must be proportional to B^2 and inversely proportional to μ0, and the 1/2 factor is the exact coefficient that arises from the integral. The other forms either miss the 1/2, place μ0 in the numerator, or use the wrong dependence on B, leading to incorrect units or magnitudes.

Magnetic energy per unit volume in a vacuum grows with the square of the magnetic flux density and scales with 1/μ0. The correct expression is u_B = B^2/(2 μ0). This comes from the relation u = ∫ H · dB; in vacuum H = B/μ0, so u = ∫ (B/μ0) dB = B^2/(2 μ0). The energy density must be proportional to B^2 and inversely proportional to μ0, and the 1/2 factor is the exact coefficient that arises from the integral. The other forms either miss the 1/2, place μ0 in the numerator, or use the wrong dependence on B, leading to incorrect units or magnitudes.