Poynting vector

S = E \times B / μ_{0}

is electromagnetic energy flux.

Poynting theorem

For electromagnetic energy density

u = \frac{1}{2} (ϵ_{0} E^{2} + \frac{B ^{2}}{μ _{0}}),

\partial_{t} u + \nabla \cdot S = - J \cdot E .

The field energy lost from a volume becomes outward electromagnetic flux or work on charges.

In vacuum the electromagnetic energy density is

u = \frac{ε _{0} E ^{2}}{2} + \frac{B ^{2}}{2 μ _{0}},

and the field energy in a region is

U_{EM} = \int u d V

E_{B} = \int B^{2} / (2 μ_{0}) d V

B = k \times E / c

, so

S = ϵ_{0} c E^{2} k .

The electric and magnetic energy densities are equal, and the time averages satisfy

⟨ S ⟩ = c ⟨ u ⟩ k

For symmetric time-independent constitutive tensors,

u = \frac{1}{2} (ε_{ij} E_{i} E_{j} + μ_{ij} H_{i} H_{j}), S = E \times H,

and the macroscopic Maxwell equations give

\partial_{t} u + \nabla \cdot S = - E \cdot J

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The Poynting vector is a vector that represents the directional energy flux (the rate of energy transfer per unit area) of an electromagnetic field.