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

where

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

(SI units), (Gaussian units).

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

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

The radiation field of an oscillating electric dipole with **p** =
**p**_{0}cos(ωt)
is

**E**_{R}(**r**,t) = -(1/(4πε_{0}c^{2}r''))(d^{2}**p**_{⊥}(t - r''/c)/dt^{2}),

and the average total power radiated is <P> =
ω^{4}p_{0}^{2}/(12πε_{0}c^{3}).

The potentials of a point charge moving in an arbitrary way are

(SI units) | (Gaussian Units) | |

Here
is the vector pointing
from the point charge to the observer at **r**. The
potentials of a point charge depend only on the position and the velocity at the
retarded time. The fields **E** and **B** depend on the
acceleration.