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Electric and magnetic energy densities

Short answer

From a parallel-plate capacitor, u_E = U/volume = ½ ε₀ E². From a long solenoid, u_B = B²/(2μ₀). Both energy densities are proportional to the square of the respective field; in a medium, ε₀→ε, μ₀→μ.

Show that the energy per unit volume in electric field and magnetic field are proportional to the square of their fields.

Energy density u is the energy stored per unit volume.

Electric field

Take a parallel plate capacitor of plate area A and separation d, so the field is E = V/d, and C = ε₀A/d.

U = ½ C V² = ½ (ε₀A/d)(E d)² = ½ ε₀ E² (A d)
u_E = U/(A d) = ½ ε₀ E²

Magnetic field

Take a long solenoid, length ℓ, area A, n turns per metre. B = μ₀nI, L = μ₀n²Aℓ.

U = ½ L I² = ½ (μ₀ n² A ℓ) I²
I = B/(μ₀ n)  →  U = ½ μ₀ n² A ℓ · B²/(μ₀² n²) = B² A ℓ /(2μ₀)
u_B = U/(A ℓ) = B²/(2μ₀)

Result:

  • u_E = ½ ε₀E² ∝ E²
  • u_B = B²/(2μ₀) ∝ B²

Both are proportional to the square of the field. In a medium, replace ε₀ with ε and μ₀ with μ. In an EM wave E = cB, so u_E = u_B.

✓ Answer checked by Claude Opus 5.5 (high effort)

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