Poynting vector, S = E×B/μ0

Definition

The Poynting vector S is the rate of flow of electromagnetic energy per unit area, in the direction of propagation of the wave. Its unit is W/m². It is given by S = E × H = (E × B)/μ₀.

Proof

We use energy conservation with Maxwell's equations (in vacuum, in a region with current density J):

∇×E = −∂B/∂t and ∇×B = μ₀J + μ₀ε₀ ∂E/∂t.

1. Use the vector identity ∇·(E×B) = B·(∇×E) − E·(∇×B).

2. Substitute: ∇·(E×B) = −B·∂B/∂t − E·[μ₀J + μ₀ε₀ ∂E/∂t].

3. Divide by μ₀:

   ∇·(E×B)/μ₀ = −(1/μ₀) B·∂B/∂t − ε₀ E·∂E/∂t − E·J

4. Note B·∂B/∂t = ½ ∂(B²)/∂t and E·∂E/∂t = ½ ∂(E²)/∂t. So

   ∇·[(E×B)/μ₀] = −∂/∂t [ε₀E²/2 + B²/(2μ₀)] − E·J

5. The bracket is the energy density u = u_E + u_B. E·J is the power given to the charges per unit volume. Define S = (E × B)/μ₀:

   ∇·S = −∂u/∂t − E·J

6. Integrate over a volume and use the divergence theorem: ∮S·dA = −d/dt ∫u dV − ∫E·J dV.

The left side is the energy flowing out through the surface per second. The right side is the decrease in stored field energy minus the work done on the charges per second. So S·dA is the power crossing dA, and S = (E×B)/μ₀ is the energy flux, pointing along E × B (the direction of travel).