**Radiation produced by moving charges
**Assume an observer is located at the origin.

The electric field produced by a point charge q which moves in an arbitrary way at the location of the observer is

**E**(t) = -(q/(4πε_{0}))[(**r**'/r'^{3}) + (r'/c)(d(**r'**/r'^{3})/dt)
+ (1/c^{2})(d^{2}(**r**'/r')dt^{2})].

Here **r**' is the position of the charge at the retarded time (t -
r'/c); **r**' points from the observer to the charge. [Note **
r**'/r'
is the unit vector.}

**E** = **E**_{1} + **E**_{2} + **E**_{3}.

**E**_{1} = **E**_{c}(t - r'/c) = retarded Coulomb
field. **E**_{2} = (r'/c)(d**E**_{1}/dt).

**E**(t) = **E**_{1}(t - r'/c) + (r'/c)(d**E**_{1}(t
- r'/c)/dt) + ...

The retardation is removed to first order. For the **near field** it is a
better approximation to use the instantaneous Coulomb field than to use the
retarded Coulomb field.

**E**_{3} is the radiation field. For a point charge moving non-relativistically
we have

E_{3} = -(q/(4πε_{0}c^{2}r'))**a**_{⊥}(t - r'/c).

If the observer is not located at the origin but at position **r** then the
**radiation field** **E**(**r**,t) of a point charge moving non-__relativistically__
is

**E**(**r**,t) = -(4πε_{0})^{-1}[(q/(c^{2}r''))**a**_{⊥}(t
- r''/c)

where

**r**'' = **r** - **r**'(t - |**r **- **r**'|/c),

i.e. the vector from the charge to the observer at the retarded time t -|**r
**-** r**'|/c, and **r**' is the position of the charge at the retarded
time.

For the radiation field we have **B** = **r''**/(r''c) ×** E **
(SI units), **B** = **r''**/r'' ×** E ** (Gaussian units).

The energy flux associated with the fields of a point charge is calculated
from the Poynting vector **S**.

The **total power radiated** by a point charge moving non-relativistically is

P =∮_{A}** S∙**d**A** = ⅔e^{2}a^{2}/c^{3},

in SI and Gaussian units, with e^{2} = q^{2}/(4πε_{0})
in SI units. This is the **Larmor formula**.

An electric dipole radiates energy at a rate P

For an oscillating dipole the average total power radiated is <P> = ω

A magnetic dipole radiates energy at a rate P