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,

where μo 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

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
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.
where εo 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.(EXBo)-εod/dt(E2/2)-B(curl E)/μo
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(εoE2/2 + (1/μo)B2/2)
Recognising that the electromagnetic field energy density U is given by
U= 1/2(εoE2+(1/μo)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


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