This Wikipedia entry tells me that the Thevenin equivalent circuit for an arbitrary receiving antenna on which an electric field $E_b$ is incident is a voltage source $V_a$ in series with an impedance $R_a + j X_a$ where (I have re-arranged the terms a bit to frame my question…)
$$
V_a = E_b\;
\frac{\cos {\psi}}{\sqrt{\pi Z_0}} \;
\left( \lambda \sqrt{R_a G_a} \right)
$$
… given that $G_a$ is the directive gain of the antenna in the directive gain of the antenna in the direction of incidence, and $\psi$ is the angle by which the electric field is 'misaligned' with the antenna.
The article does mention that this is derived from reciprocity, from which I assume that there should be some reasoning beginning with the Rayleigh-Carson theorem:
$$
\iiint_{V_1} \vec{J}_1 \cdot \vec{E}_2 \;dV
=
\iiint_{V_2} \vec{J}_2 \cdot \vec{E}_1 \;dV
$$
I am trying to understand how I can apply this, and in fact how I can approach an arbitrary antenna structure in general (I do understand how a dipole and a loop can be analyzed)
Unfortunately the article itself doesn't point out any sources where this relationship is derived, so I was wondering if anyone could point me to any textbook or paper where this derivation may be found?
My motivation is something like this — the relation mentioned in the Wikipedia article is actually for a sinusoidal input — and the frequency determines $\lambda$, $R_a$, and $G_a$ in the expression (and $X_a$ in the equivalent circuit). I am trying to understand if any insight can be obtained about the equivalent voltage source $V\left(t\right)$ given an arbitrary $E\left(t\right)$ — maybe, for example, as a differential or integral equation? The $X_a$ can be replaced by frequency independent $C_a$ and $L_a$ in series — and for the voltage source I would integrate over $\lambda$ — but I don't know how to deal with $R_a\left(\lambda\right)$ (which is ideally only the radiation resistance) and $G_a\left(\lambda\right)$ for arbitrary antenna geometries. So I was hoping that the derivation would offer me some clues…
Update
OK, so it seems that I went on the wrong track here — it is actually quite easy. I am answering my own question below.
Best Answer
A slight variant on your fine answer...
A reference is Ramo et al, Fields and Waves in Communication Electronics, chapter 12.
First, reciprocity: $Z_{21}=Z_{12}$ tells you that (assuming a conjugate-matched load):
$$ g_{dt} A_{er} = g_{dr} A_{et}$$
For both transmitting (subscript t) and receiving (r) antennas, $g_d$ is the antenna directional gain.
$A_{er}$ is the effective area of the receiving antenna, defined as the ratio of useful power removed from the receiving antenna $W_r$ to average power density $P_{av}$ in the incoming radiation.
Thus the ratio $g_d/A_e$ is the same for both transmitting and receiving antennas.
For large aperture antennas, it can be shown that the maximum possible gain satisfies: $$ \frac{(g_d)_{max}}{A_e} = \frac{4 \pi}{\lambda^2} $$
For other geometries, $A_e$ is defined to give the same result. For example, for a Hertzian dipole, with a maximum directivity of 1.5: $$ (A_e)_{max} = \frac{\lambda^2}{4 \pi} (g_d)_{max} = \frac{3}{8 \pi} \lambda^2 $$
Anyway, for the problem at hand, as you deduced, the useful power removed from the receiving antenna is:
$$ W_r = P_{av} A_{er} \text{, with the power density } P_{av} = \frac{E_b^2}{2 Z_o} , Z_o=377 \text{ ohms} $$
(Here, electric field and voltage are sinusoids measured as peak values.)
With a conjugate-matched load with real part $R_L$, equating load power dissipated with power delivered gives for the receiving antenna's Thevenin equivalent source voltage $V_a$:
$$\frac{(V_a/2)^2}{2 R_L} = \frac{E_b^2}{2 Z_o} A_{er} $$
$$ V_a = 2 \sqrt{A_{er}} \sqrt{\frac{R_L}{Z_o}} \, E_b $$
Substituting for $A_{er}$ from the reciprocity relation, the maximum voltage $V_{a,max}$ is:
$$ V_{a,max} = \sqrt{\frac{(g_{dr})_{max}}{\pi }} \sqrt{\frac{R_L}{Z_o}} \,\, \lambda E_b $$
I'm cautious about the $\cos \psi$ factor because beam patterns differ for different antennas.