Which expression correctly gives the magnetic energy density in vacuum?

Which expression correctly gives the magnetic energy density in vacuum?

• A. u_B = B^2 /(2 μ0)
• B. u_B = μ0 B^2
• C. u_B = ε0 E^2
• D. u_B = B^2 /(2 ε0)

Answer: (A)

Magnetic energy density is the energy stored per unit volume in the magnetic field, and it follows u_B = (1/2) B · H. In vacuum, the magnetic field and auxiliary field relate by B = μ0 H, so H = B/μ0. Substituting gives u_B = (1/2) B (B/μ0) = B^2/(2 μ0). This is the correct expression for vacuum. The electric energy density is u_E = (1/2) ε0 E^2, and the total energy density is the sum u = u_E + u_B. The other options either omit the 1/2 factor, represent electric energy density, or mix constants in the wrong way.

Magnetic energy density is the energy stored per unit volume in the magnetic field, and it follows u_B = (1/2) B · H. In vacuum, the magnetic field and auxiliary field relate by B = μ0 H, so H = B/μ0. Substituting gives u_B = (1/2) B (B/μ0) = B^2/(2 μ0). This is the correct expression for vacuum.

The electric energy density is u_E = (1/2) ε0 E^2, and the total energy density is the sum u = u_E + u_B. The other options either omit the 1/2 factor, represent electric energy density, or mix constants in the wrong way.