**K.E. Schmidt
Department of Physics and Astronomy
Arizona State University
Tempe, AZ 85287
**

The potentials in free space with the boundary conditions that
the only sources of radiation are
and (i.e. no incoming
waves from infinity) in the Lorentz gauge are

(,t)
| = |
d^{ 3}r'dt'(t' - [t - c^{ - 1}| - |])
| |

(,t)
| = |
d^{ 3}r'dt'(t' - [t - c^{ - 1}| - |]) .
| (1) |

The charge and current densities of a point charge *q* at position
_{0}(*t*) are

(,t)
| = |
q( - _{0}(t))
| |

(,t)
| = |
q( - _{0}(t)) q(t)( - _{0}(t)) .
| (2) |

Plugging in the charge and current densities, the
integral
can be done immediately to give

(,t)
| = |
dt'(t' - [t - c^{ - 1}| - |])
| |

(,t)
| = |
qdt'(t' - [t - c^{ - 1}| - |]) .
| (3) |

The delta function integral is given in the usual way by changing variables
to its argument
in the region of the zeroes of its argument. For example
if *x*_{0} is the only
zero of *f* (*x*) in the range of integration, then

dx(f (x)) = .
| (4) |

(,t)
| = | ||

(,t)
| = | (5) |

t_{ret} = t - c^{ - 1}| - _{0}(t_{ret})| .
| (6) |

It simplifies the notation (although the first two hide some
of the dependencies)
to write

(t)
| = |
- _{0}(t)
| |

_{ret}
| = |
f (t_{ret})
| |

= | (7) |

(,t) = _{ret}
| |||

(,t) = _{ret}
| (8) |

= |
x
| ||

= | - - . | (9) |

Expanding the equation for the retarded time keeping constant gives the differential as

dt_{ret} = dt + [ ]_{ret}dt_{ret}
| (10) |

= _{ret}
| (11) |

dt_{ret} = - _{ret} + t_{ret}
| (12) |

= - _{ret}
| (13) |

The chain rule gives

_{ret}
| = |
_{ret}_{ret}
| |

_{ret}
| = |
_{ret} - _{ret}_{ret}
| (14) |

Applying this to the time derivative of the vector potential

= q_{ret}
| (15) |

= q_{ret}
| (16) |

= q_{ret} - _{ret}
| (17) |

x [( - ) x ] = ( - ) - (1 - )
| (18) |

= q_{ret} + _{ret}
| (19) |

The curl of the vector potential is calculated the same way,

x
| = |
- q_{ret} - _{ret}
| |

= |
- q_{ret} - _{ret}
| (20) |

= [ x ] _{ret}
| (21) |

dE
| = |
( x )dtd
| |

= |
dtd .
| (22) |

Generally we are not interested in the energy detected by an observer
in time interval *dt* far from our charge. Rather we are interested
in the amount of energy the particle loses to radiation in a particular
direction during a change in the particles time. That is, we want
*dE*/*dt*_{particle} not *dE*/*dt* , and since the particle is evaluated
at the retarded time, the power radiated into solid angle *d*
by the particle is then

= | . | (23) |

The cross product can be expanded to give

| x [( - ) x ]|^{2} = ()^{2}(1 + ) - ( )^{2} - 2()^{2} + 2 - ( )^{2} - ( [ x ])^{2}
| (24) |

Taking the *z* axis along , the integrations
are trivial and
the integrals needed to find the total power radiated are
of the form

dx
| = | ||

dx
| = | ||

dx
| = | (25) |

d
| = | 1 + ) | |

d
| = | ||

d
| = | ||

d
| = | . | (26) |

Adding the results together gives

P
| = | ||

= | |||

= | (27) |

Taking the nonrelativistic limit where 1 , the angular
distribution and the total power radiated are the Larmor results

= |
| x |^{2}
| ||

P
| = |
||^{2} .
| (28) |

It is considerably easier to derive these results from the
potentials by taking their nonrelativistic limit first and dropping
the velocity dependent terms in the derivatives.
Taking the curl will give the magnetic induction. Since derivatives
of *R* will give terms that go like 1/*r*^{ 2} etc., they will not contribute
to the radiation field. Derivatives of *R* with respect to *t*_{ret}
will give terms higher order in the velocity, and these can be dropped
in the nonrelativistic limit. The only term that survives is the derivative
of the velocity in the numerator of
with respect to *t*_{ret}
and the gradient of
*ct*_{ret} gives
- . The
radiation field for nonrelativistic motion is therefore

( ,t) - .
| (29) |

= |( ,t)|^{2} = | x |^{2}
| (30) |