References: Griffiths, David J. (2007), Introduction to Electrodynamics, 3rd Edition; Pearson Education – Chapter 11, Post 11.
In analyzing radiation from an arbitrary configuration of charge, we made the assumption that the maximum dimension of the source is much smaller than the observation distance, so that we can retain only first order terms in , the variable that is integrated over the source. In some cases, the first order contribution is zero and in that case, we need to look at the next order. This leads to electric quadrupole (and magnetic dipole, but we’ll leave that for now) radiation. A simple model that illustrates this is as follows.
Suppose we have two oscillating electric dipoles situated on the axis, with at and at . The dipole oscillate exactly out of phase, so that the dipole moment of the upper dipole is always the negative of the dipole moment of the lower one. We can work out the fields of this setup by using the same approximations we used in deriving the ordinary oscillating dipole. First, we need to define a few terms. (I’d draw a diagram, but that’s a painful process, so bear with me.)
Let the observation point make an angle with the axis, and let the vector from to be and the vector from to be . The vectors make angles with the axis.
The potential formulas for a dipole at the origin are
However, here, the dipoles are not at the origin so we need to adapt these formulas. For we must use and so we have
From the law of cosines we have
and from the geometry of the setup
Now assuming we have
to first order in .
Plugging these into 3 we get
Under our approximation of we can drop the middle term to get
For we can do the same calculation to get (note the opposite sign of since the dipole is opposite to the top one)
Putting this together, we get
The total potential is
For the vector potential, we get
With the potentials, we can calculate the fields. To simplify the notation, we’ll use the shorthand
and so on.
Using the approximation we can drop all but one term to get
For the magnetic field, we get
Again, using the approximation we drop the second term to get
The Poynting vector is
The intensity is the time average of :
and the power is the integral of this over a sphere of radius :