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,
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
) 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
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.
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/μo)-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
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