In the Lorentz gauge the potentials **A** and Φ satisfy the inhomogeneous
wave equation.

**∇**^{2}Φ - (1/c^{2})∂^{2}Φ/∂t^{2} = -ρ/ε_{0}**,
∇**^{2}**A** - (1/c^{2})∂^{2}**A**/∂t^{2}
= -μ_{0}**j**.

Solutions are

Φ(**r**,t) = [1/(4πε_{0})]∫_{v'} dV' ρ(**r**',t_{r})/|**r**
- **r**'|,

**A**(**r**,t) = [μ_{0}/(4π)]∫_{v'} dV'
**j**(**r**',t_{r})/|**r** - **r**'|.

ρ(**r**',t_{r}) = ρ(t_{r}(**r**')) is evaluated at the
retarded time t_{r} = t - |**r** -
**r**'|/c.

For a point charge moving in an arbitrary this yields the
Lienard-Wiechert potentials

Φ(**r**,t) = [1/(4πε_{0})][q/((1 -
**β**∙**n**)|**r**
- **r**'|)]_{ret},

**A**(**r**,t) = [μ_{0}c/(4π)][q**β**/([1
- **β**∙**n**)|**r**
- **r**'|)]_{ret}.

Here **r**' is the position of the point charge at the retarded time, **n**|**r**
- **r**'| is the vector pointing from the position of the point charge
at the retarded time 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 also on the
acceleration.

We can find **E**(**r**,t) and
**B**(**r**,t) using
**E** = -∂**A**/∂t
-**∇**Φ, **B** =
**∇**×**A.**

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**.

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},

with e^{2} = q^{2}/(4πε_{0}). 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}).

An electric dipole radiates energy at a rate P_{rad} = <(d^{2}**p**/dt^{2})^{2}>/(6πε_{0}c^{3}).

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

A magnetic dipole radiates energy at a rate P_{rad} = <(d^{2}**m**/dt^{2})^{2}>/(6πε_{0}c^{5}).