# Explaining receiver antennas without using Reciprocity

I am trying to understand the physical behavior receiver antennas, but I am unsatisfied with all the explanations I have seen so far in my textbook (Stuzman & Thiele, Antenna Theory and Design) or online articles. Everyone seems to develop the theory of transmitting antennas thoroughly starting from the Maxwell equations, and then receiver antennas are just swept under the rug by saying something like "the receive pattern of the antenna must be the same as the transmit pattern because of the Reciprocity theorem".

I find this kind of explanation a little too high-level, and it does not provide sufficient detail. Note that I am not looking for a proof of the Reciprocity theorem. What I am interested is the following:

We can model the receiving antenna as a conductor with conductance $$\sigma$$ and some given shape (with surface normal $$\mathbf{\hat{n}})$$, which is exposed to an electromagnetic wave coming from a distant transmitter. As a good approximation we may assume the incoming wave to be a plane wave.

At the surface of the conductor, we will observe reflection and refraction of the incoming plane wave, and induction of surface currents and charges. The way this goes in detail, according to my understanding, is that the tangential electric field component that gets transmitted into the metal will induce a tangential current density $$\mathbf{J_t} = \sigma\mathbf{E_t}$$. The magnitude of the transmitted electric field decays exponentially, depending on the skin depth $$\delta$$, so the tangential current exists mainly very close to the surface. The normal component of the incoming electric field will induce surface charges (in reality it will be some volume charge density close to the boundary according to Poisson's equation and taking into account the exponential dependence of the skin-depth). Somehow the surface currents and surface charges will balance each other in order to remain consistent with charge conservation.

We can also look at what happens in terms of the incoming magnetic field. Suppose for simplicity that the incoming plane wave propagates in the $$-\mathbf{\hat{n}}$$ direction (i.e. directly normal onto the surface) with the $$\bf{E}$$ field pointing right and $$\mathbf{H}$$ pointing out of the page (let's say $$\mathbf{\hat{n}} = -\mathbf{\hat{z}}$$ and the fields are $$\mathbf{E} = E_x\mathbf{\hat{x}}$$ and $$\mathbf{H} = H_y\mathbf{\hat{y}}$$ with the wave moving downwards in the $$\mathbf{\hat{z}}$$ direction). As mentionned above, the $$\mathbf{E}$$ field will cause current density $$\mathbf{J} = \sigma\mathbf{E}$$ close to the surface in the $$\mathbf{\hat{x}}$$ direction. The magnetic field will be proportional to the curl of the electric field, and as the $$E$$-field is exponentially decaying into the metal, there will be a strong $$H$$-field at the surface.

But as the skin depth decreases (getting closer to an ideal conductor) the E-field becomes zero inside the conductor, and the ohmic current density also becomes $$0$$. The entire current is concentrated at the surface and exists as a surface current density $$\mathbf{K}$$, and the H-field becomes infinite at the surface due to the curl of E being infinite.

In addition, there will also be reflection of the incoming plane wave, causing a slight change in the steady-state EM-field around the antenna.

Ultimately, I would like to find a relationship between the incoming plane wave and the current or voltage induced at the terminals of the antenna, but in a way that treats all the reflections, boundary conditions, induced currents and charges on the conductor, more explicitly (i.e. from first principles). The way I see things, the parts of the antenna that are tangential to the electric field will have surface currents on them, moving toward parts of the antenna where the electric field is normal to the surface. At these points, the currents converge and produce surface charges. Ultimately, the terminals of the antenna will draw currents and produce a voltage into the reading circuit.

I must admit I am getting confused by how to analyze these effects systematically, and I find that the treatment using reciprocity glosses over these details to make a simpler discussion, at the expense of a lack of understanding of these subtleties.