State and prove Poynting's theorem

Statement

The net power flowing out of a closed surface equals the rate of decrease of the electromagnetic energy stored in the enclosed volume, minus the ohmic power dissipated inside it. The Poynting vector P = E × H (W/m²) gives the power flowing per unit area:

∮ (E × H)·dS = −d/dt ∫ (½εE² + ½μH²) dv − ∫ σE² dv

Proof

Start with the vector identity

∇·(E × H) = H·(∇×E) − E·(∇×H)

Substitute Maxwell's equations ∇×E = −∂B/∂t = −μ ∂H/∂t and ∇×H = J + ∂D/∂t = σE + ε ∂E/∂t:

∇·(E × H) = −μ H·∂H/∂t − σ E² − ε E·∂E/∂t

Since H·∂H/∂t = ½ ∂(H²)/∂t and E·∂E/∂t = ½ ∂(E²)/∂t:

∇·(E × H) = −∂/∂t (½μH² + ½εE²) − σE²

Integrate over a volume v bounded by the closed surface S and use the divergence theorem, ∫∇·(E×H) dv = ∮(E×H)·dS:

∮_S (E × H)·dS = −d/dt ∫_v (½εE² + ½μH²) dv − ∫_v σE² dv

Meaning of the terms

• Left side: total power leaving the volume through S.
• ½εE² + ½μH²: electric and magnetic energy densities (J/m³). The first term on the right is the decrease of stored energy per second.
• σE² = J·E: power dissipated as heat per unit volume (Joule loss).

If a source inside the volume also supplies power, an extra term −∫ E·J_source dv is added on the right. For time-harmonic fields the average power density is P_avg = ½ Re(E_s × H_s*) W/m².