# Show that an oscillating electric monopole does not radiate

## Homework Statement

Given
$$\rho(r',t)=\frac{Q}{4\pi (r')^2} \delta(r'-(R_0-\Delta{R}\cos(wt)))$$
$$J(r',t)=\frac{wQ\cos(wt)}{4\pi r'}$$,
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: $$\phi , \theta , r'$$

## Homework Equations

$$A(r',t)=\frac{\mu_0}{4\pi}\int\frac{J(r',t_R)}{|r-r'|}d^3r'$$
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 $$\frac{1}{z}$$. However, I expected it to be proportional to $$\frac{1}{z^2}$$. 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 $$1/z^2$$.