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**=(**E**X**B**)/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(**v**X**B**+**E**)=d/dt(m**v**)

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**.(

**v**X

**B**) is zero. The particle's kinetic energy is augmented by the electric field via-

d/dt(1/2 mv^{2})= q**vE**

Multiply by the particle density n and introduce the

current density **J**=nq

**v** 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**(curl**B**)/μ_{o}-ε_{o}d/dt(E^{2}/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 identitydiv.(**E**X**B**)=**B**(curl** E**)-**E**(curl** B**)

Implementing this identity one obtains

**J.E**= -div.(**E**X**B**/μ_{o})-ε_{o}d/dt(E^{2}/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 -d

**B**/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_{o}E^{2}/2 + (1/μ_{o})B^{2}/2)

Recognising that the electromagnetic field energy density

*U* is given by

U= 1/2(ε_{o}E^{2}+(1/μ_{o})B^{2})

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