Is there a differential equation that can represent a circuit with an arbitrary voltage source connected acrorss an antenna? An RLC circuit with a voltage source can be characterized by the differential equation:
$$
LC\;\ddot{I}\left(t\right) + RC\; \dot{I}\left(t\right) + I\left(t\right)-C \;\dot{V}\left(t\right) = 0
$$
... but this will be true only as long as radiation is ignored.
My question is, what will this equation become in the general case, when the radiation cannot be ignored and also has to be accounted for? I ask this because the radiation resistance on any antenna depends on the frequency of the (sinusoidal) current passing through it -- so I suppose it cannot simply add it on to the $R$ term.
I expect $\dot{I}\left( t\right)$ to be involved -- as the far field $\vec{\mathrm{E}}$ depends on $\dot{\vec{\mathrm{J}}}$ which must derive from $\dot{I}$ somehow. But I don't seem to be able to go beyond that.
Looking for pointers -- are there any textbooks that address this part? So far my Googling hasn't turned up anything, so I guess I don't know what I should be searching for or maybe this is something that has fallen through the cracks between circuit theory textbooks and antenna theory textbooks (most electromagnetism texts seem to jump to $\cos{\omega t}$ or $e^{j\omega t}$ and stay there).
Another thing -- would the L & C values that can be computed from the antenna geometry be sufficient to address the reactance in the high frequency case, or is there something else to it?
I'm looking for an understanding within the bounds of classical electrodynamics (that is, leaving quantum mechanics out).
Thanks...
 A: I'd like to build on Lionel Brits's answer and the electronic book reference you cite Sophocles J. Orfanidis, "Electromagnetic Waves and Antennas", Chapter 22.  
The structure of the relationship is more readily understood if you think of the Laplace or Fourier transforms of all your impedances. A relationship between $I(t)$ and $V(t)$ that is given by a constant co-efficient linear differential equation always gives rise to an impedance $Z(i\omega)$ or $Z(s)$ ($\omega$ = Fourier transform frequency variable, $s$ = Laplace transform complex frequency) which is a rational function of $i\omega$ or $s$. 
Now an antenna's impedance is in general made up of reactions from the electromagnetic field which contain pure delays. This is owing to the finite speed of light. Unless the antenna is perfectly matched everywhere, the impedance is made up of components arising from delayed reflexions from different parts of its structure. One cannot match the antenna everywhere at all frequencies. So there will always be terms of the form $\exp(-s\,\tau)$ in the numerator and denominator of the impedance $Z(s)$ (here $s$ is the complex frequency), where $\tau$ is a there-and-back delay for a reflexion. When expanded out, $\exp(-s\,\tau)$ corresponds to the infinite differential operator:
$$1 - \tau {\rm d}_t + \frac{\tau^2}{2!} {\rm d}_t^2 - \frac{\tau^3}{3!} {\rm d}_t^3 + \cdots$$
so you'll never exactly represent such a thing by a finite differential equation.
That's the theory. In practice, depending on the important timescales, you could likely get a good approximation using only a finite number of terms. How many is hard to answer: this would need to be answered by someone with experience in numerically modeling antennas. But, by dint of the $\exp(-s\,\tau)$ terms, there is no theoretical limit to the order of derivatives that will arise in your expressions.
A: Suppose you have a dipole radiator. Then you can look up or calculate it's radiation resistance
$R_\mathrm{rad} = \frac{2 \pi}{3} Z_{0} \left( \frac{\ell}{\lambda}\right)^{2}$
, which might be off by a factor of 4.
I would say that you can use the superposition principle to just treat the problem in frequency space. There is also reactance to think about.
