Show that an oscillating electric monopole does not radiate

  • #1

Homework Statement


Given
[tex] \rho(r',t)=\frac{Q}{4\pi (r')^2} \delta(r'-(R_0-\Delta{R}\cos(wt))) [/tex]
[tex] J(r',t)=\frac{wQ\cos(wt)}{4\pi r'} [/tex],
which is an oscillating sphere with uniform charge distribution,
find the vector potential, then the magnetic and electric fields.
Hint: Integrate in the following order: [tex] \phi , \theta , r' [/tex]

Homework Equations


[tex] A(r',t)=\frac{\mu_0}{4\pi}\int\frac{J(r',t_R)}{|r-r'|}d^3r' [/tex]
Also, due to symmetry, we are told to find it at a point z along the z axis.
Since the point is to show there is no radiation, we must be in the radiation zone, so z>>r'

The Attempt at a Solution


I am not that great at latex, so I do not want to retype my whole solution, but I did get A being proportional to [tex] \frac{1}{z} [/tex]. However, I expected it to be proportional to [tex] \frac{1}{z^2} [/tex]. Which, if either, would be correct? Also, wouldn't the current density need a delta function? The reason I am wondering is my units do not work if I do not include one, which is why I suspect I am not getting A to be proportional to [tex] 1/z^2 [/tex].
 

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