**Kevin Schmidt
Department of Physics and Astronomy
Arizona State University
Tempe, AZ
**

If a localized current is given as a function of space and time, the vector potential in the Lorentz gauge is

(,t) = d^{ 3}r' .
| (1) |

The retardation is often simpler if the current is Fourier transformed in time,

_{}() = dt(,t)e^{ it}
| (2) |

(,t) = _{}()e^{ - it} .
| (3) |

Since the current is real, the complex conjugate of the last
equation shows that
_{}() = ^{*}_{- }() .
The current can be written in terms of positive frequencies alone as

(,t) = Re _{}()e^{ - it}
| (4) |

Plugging this form in, the vector potential is

(,t) = Re e^{ - it}.
| (5) |

To extract the radiation fields, the distance from the source to
observation point is expanded assuming the source is localized and
the observation point goes to
*r* . The vector
potential is

(,t) = Re d^{ 3}r'_{}()e^{ - i }
| (6) |

Notice that the current integral is just the ``on energy shell'' space-time Fourier transform of the current. That is defining the Fourier transform as

(,) = d^{ 3}rdt(,t)e^{ - i + it}
| (7) |

(,t) = Re
| (8) |

There are two cases of interest. The first is where the current (or at least the radiating part) is localized in time. In that case the total radiated energy in a small frequency range is finite. This of course is the only physical situation. However, in many cases, the motion is periodic for a very long time, and it is the energy radiated per cycle or equivalently the power radiated that is desired. If the motion continues indefinitely, the total energy radiated is infinite, so the mathematics needs to be slightly modified as shown in the spectral resolution notes.

The magnetic field in the radiation zone is given by taking the
curl of
, and to give a *r*^{ - 1} dependence, the gradient
must operate on the exponent. Similarly, the electric field is
given for a harmonic time dependence as

x _{} = i_{} .
| (9) |

(,t)
| = |
Re
| |

(,t)
| = |
Re
| (10) |

The
`x` ( `x` ) is the
transform of the transverse component of the current.
The Poynting vector is
`x` , and with
direction along
and magnitude given by
*c*||^{2}/4 .

As shown in the spectral resolution notes,
if the time integral of the product of two functions of time

f (t)
| = |
Re ()
| |

g(t)
| = |
Re ()
| (11) |

= Re ()() .
| (12) |

= 2 | (13) |

Again applying results of the spectral resolution notes
to the case where the motion is periodic with period
*T* = , the power radiated per solid angle
at the frequency *n* is

= 2 | (14) |

() = dtd^{ 3}rJ(,t)e^{ - i + int} .
| (15) |

For reference, the spectral resolution of signals localized in time is
given in Jackson section 14.5. The relationship between
*dP*_{n}/*d* and
*d*^{ 2}*I*/*d**d* is given a physical meaning
on Jackson page 681 around Eq. 14.92. Essentially, if you imagine that
you have a signal that repeats with fundamental frequency ,
but in each period is localized to time much smaller than the period,
then the intensity of radiation for one period would be given by
*d*^{ 2}*I*/*d**d* where the current integration would be over
just one period. Since the motion repeats, the power
radiated in a frequency *n* is the energy radiated at that
frequency
= *n*
divided by the period
2/ to get the power, and
multiplied by to convert from *d* to *dn* . The overall
factor is
/2 which is the difference in the prefactors
of the two expressions above.

If the current is localized in a region or a set regions, each which has
spatial extent much smaller than a wavelength,
= 2*c*/ , the origin of integration
can be translated to the approximate center of a region, and
the exponential expanded in a power series. The terms in the power series
are roughly powers of the extent of the region of nonzero current divided
by the wavelength. The exponential becomes

e^{ - i } = 1 - i + ...
| (16) |

I^{ (0)}_{k}
| = |
d^{ 3}r'J_{ k}()
| |

I^{ (1)}_{jk}
| = |
d^{ 3}r'x'_{j}J_{ k}() .
| (17) |

(x_{k}_{})
| = |
x_{k} _{} + J_{ k}
| |

= |
ix_{k} + J_{ k}
| ||

(x_{j}x_{k}_{})
| = |
x_{j}x_{k} _{} + [x_{j}J_{ k} + x_{k}J_{ j}]
| |

= |
ix_{j}x_{k} + [x_{j}J_{ k} + x_{k}J_{ j}] .
| (18) |

I^{ (1)}_{jk} =
| (19) |

I^{ (0)}_{k}
| = |
- id^{ 3}r'x'_{k}() = - id_{ k}
| |

I^{ (1)}_{jk}
| = | . | (20) |

d^{ 3}r'[x'_{j}J_{ k}() - x'_{j}J_{ k}()] = 2cm_{} .
| (21) |

Noting that in the field equation, one component is dotted with , while the other is crossed with , a constant diagonal element can be added to it without changing the result. That is

r_{j}r_{i}_{} = 0 .
| (22) |

d^{ 3}r'x'_{j}x'_{k}() d^{ 3}r'_{}() = Q_{ jk}
| (23) |

= |
- i_{} + i x _{} - _{} + ...
| (24) |

_{n}
| = |
- in_{n} + in x _{n} - _{n} + ...
| (25) |

As a check against Jackson's results, look at the case where only the
diagonal quadrupole terms contribute with
*Q*_{33} = *Q*_{0} , and
*Q*_{11} = *Q*_{22} = - *Q*_{0}/2 each oscillating as
cos(*t*) , so that
only the *n* = 1 term survives, and the time integration gives

_{1} = .
| (26) |

_{1}
| = | - | (27) |

= | 2 | ||

= | |||

= | (28) |