There isn't any general equations though. But a static field is different from a wave. To be static means that the field is time-independent, which will never form an electromagnetic wave. jtbell is talking about the near-field that occurs around an electromagnetic wave source. What happens is that the near-field, the fields in the space surrounding the immediate volume of the source, traps energy in and around the electromagnetic source. I would not call it static since it is still a time-harmonic field, it is more or less static only in the sense of spatial propagation. The portions of the near-field that contribute fields in the direction of propagation do not propagate out, they are trapped and stored. The waves that do propagate out and leave the near-field will not have a component in the direction of propagation.
jtbell uses the term "radial" component because with a point source, or any source viewed from a large enough distance away such that it becomes point-like, the propagating waves in the far-field confine their electric and magnetic fields to the theta and phi directions in terms of spherical coordinates. The radial component would be in the direction of propagation.
If you look at the exact electric field equations for an idealized dipole antenna,
E_r=\frac{Z}{2\pi}\,I_0\,\delta l\left(\frac{1}{r^2}-i\,\frac{\lambda}{2\pi\,r^3} \right) e^{i(\omega t-k\,r)}\,\cos(\theta)
E_\theta=i\frac{Z}{2\lambda}\,I_0\,\delta l\left(\frac{1}{r}-i\,\frac{\lambda}{2\pi\,r^2}-\frac{\lambda}{4\pi^2\,r^3} \right) e^{i(\omega t-k\,r)}\,\sin(\theta)
You will notice that the radial component will drop off as 1/r^2 and 1/r^3 as opposed to the theta component which drops off as 1/r and more. When we move any appreciable distance away from the source, say where
kr >> 1
what happens is that when you look at the wave equations as a whole, the contribution from 1/r dominates in comparison to the contributions from 1/r^2 and 1/r^3. So, in the far-field, the theta component dominates dramatically. This is why we state that the near-field is trapped power, the radial component and higher order terms of the transverse components do not contribute to the fields at any distance away from the source.
The reason why we have this radial component as opposed to what I stated earlier is the presence of the source. If you solve for the wave equations with no sources, then my earlier statements hold. But a source can allow for components in the direction of propagation. At the same time though, since these components are non-propagating, can you still define a direction of propagation at all for them? In that sense, these are akin to the confined wave condition that I stated earlier in which you can get components in the direction of propagation.
EDIT:
jchodak2 said:
I find it very interesting that the magnitude of the force "felt" should be different for a static field as opposed to an oscillating field!
In response to this, the magnitude of the force acting upon a charge is independent of whether or not the applied field is static or time-harmonic. The Lorentz force,
\mathbf{F} = q(\mathbf{E} + \mathbf{v}\times\mathbf{B})
does not differentiate between the two. The only consequence with an oscillating field is that the force will oscillate along with it.
