Poynting vector

The Poynting vector is the power density of an electromagnetic wave. It is usually given the symbol s and is determined by the electric field E and magnetic field B at a point,

s=(EXB)/μ_o

where μ_o is the permeability of free space. The Poynting vector always points ('poynts'-yerricde) in the direction of propagation of the wave.

What follows is a derivation of the Poynting Theorem wherin the Poynting vector plays a part.

An electromagnetic field interacts with a particle of charge q travelling at a velocity v via the Lorentz force

F_Lorentz=q(vXB+E)=d/dt(mv)

Multiply this equation by v to get the energy relation. Notice that the magnetic field does not contribute to the particle's energy since v.(vXB) is zero. The particle's kinetic energy is augmented by the electric field via-

d/dt(1/2 mv²)= qvE

Multiply by the particle density n and introduce the current density J=nqv to obtain

dT/dt=J.E

where T is the kinetic energy of the ensemble of particles. Next use one of Maxwell's equations to express J in terms of the magnetic and electric fields.

J.E=E(curlB)/μ_o-ε_od/dt(E²/2)

where ε_o is the permittivity of free space. The final step before the Poynting vector makes an appearance is to use the vector identity

div.(EXB)=B(curl E)-E(curl B)

Implementing this identity one obtains

J.E= -div.(EXB/μ_o)-ε_od/dt(E²/2)-B(curl E)/μ_o

The last term in the equation above is actually the time derivative of the magnetic field energy density. This can be shown by using Faraday's law to substitute -dB/dt for the curl of E. The first term on the R.H.S contains the Poynting vector s.

J.E=-div.s-d/dt(ε_oE²/2 + (1/μ_o)B²/2)

Recognising that the electromagnetic field energy density U is given by

U= 1/2(ε_oE²+(1/μ_o)B²)

one arrives at the Poynting theorem for the case of an ensemble of free particles in an electromagnetic field in its most compact form.

-J.E=dU/dt + div.s

References
http://www.astro.warwick.ac.uk/warwick/chapter2/node8.html