# Does an oscillating electrical monopole radiate?

For example, the displacement R(r,t) of the single charge q is given by

$$R = R_0\cos(\omega t)$$

In the view of multipole expansion, what is the monopole moment and the dipole moment for this configuration? Why the monopole does not contribute to the final EM field?

-
An electric monopole is just a charge, isn't it? Yes, oscillating charges radiate. Any accelerating charge radiates. –  Colin K Sep 28 '12 at 4:01
Its important to note that the directivity of this oscillation is the worst. You will have a field, but you won't be able to do much with only one. –  Mikhail Sep 28 '12 at 7:57
Your electric monopole has a dipole moment $d(t)=q\cdot R(t)$. –  Vladimir Kalitvianski Sep 28 '12 at 13:24
Yes, there seems to be a dipole moment, but just hard to understand in that there is no actual opposite charge. –  Tao Sheng Sep 28 '12 at 20:48
FWIW, here's an applet that aids in picturing the field of a moving point charge: cco.caltech.edu/~phys1/java/phys1/MovingCharge/… –  Alfred Centauri Sep 28 '12 at 21:41

I believe that to get radiation you need a "jerk," that is a non-zero third time-derivative of the displacement. Hence, this will radiate.

-
Accelerating charges radiate. You don't need jerk. –  Colin K Sep 28 '12 at 4:02
This doesn't deserve a downvote +1--- the time derivative of the acceleration is the back-reaction in Dirac's point limit, and this led to the question of whether an eternally accelerating charge radiates, with back and forth going on for 50 years, until the question petered out with no completely agreed upon answer. I am not sure the question is well defined in classical mechanics, but it is definitely true that radiation reaction goes as the jerk. –  Ron Maimon Sep 28 '12 at 7:32
@ColinK, "Does a uniformly accelerating charge radiate?" mathpages.com/home/kmath528/kmath528.htm –  Alfred Centauri Sep 28 '12 at 21:40

An electrical monopole is just charged particle. Any charged particle that accelerates will definitely radiate an electromagnetic wave. A sinusoidal oscillation you gave as an example will produce radiation at the frequency of the sinusoidal acceleration. But any acceleration of any kind will produce some kind of electromagnetic wave.

-
Not any acceleration will result in radiation. please see the Nonradiation condition <en.wikipedia.org/wiki/Nonradiation_condition>; for more details. This one will radiate. But I need find a rigid derivation to convince myself. Thanks anyway. –  Tao Sheng Sep 28 '12 at 18:37