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So in my university course, we have studied the EM radiation infalling perpendicularly onto a dielectric boundary. I have now tried to go beyond and work out how the relationships look for oblique incidence, but I am running into trouble as some of my results are not consistent with eachother. I would be really happy if somebody could point out what I am doing wrong.

Throughout this analysis I assume $\mu = 1$.


Consider a dielectric boundary in the xy-plane, separating air at z>0 and a dielectric with relative permittivity $\epsilon$ at z<0. Take the incoming radiation to have angle $\theta$ to the surface normal and be polarised in the xz-plane, ie $$\vec{E_0} = (-E_0\cos \theta, 0, -E_0 \sin \theta) e^{-iwt} $$ and $$\vec{H_0} = (0,H_0,) e^{-iwt} $$

Now we have the continuity conditions for a dielectric boundary:

$D_{perpendicular}, B_{perpendicular} = $ cont.

$E_{parallel}, H_{paralell} = $ cont.

Hence inside the plasma (I believe) we should have:

$$\vec{E_p} = (-E_0\cos \theta, 0, \frac{-E_0}{\epsilon} \sin \theta) e^{-iwt} $$ and $$\vec{H_p} = (0,H_0,) e^{-iwt} $$

This gives the ratios of the fields as: $$ |\frac{E_p}{H_p}| = \frac{E_0}{H_0} \sqrt{(\cos\theta)^2+(\frac{\sin\theta}{\epsilon})^2} = Z_0\sqrt{(\cos\theta)^2+(\frac{\sin\theta}{\epsilon})^2}$$


However, we also know that $$|\frac{E}{H}| = \frac{Z_0}{n} $$ where $n = \sqrt{\epsilon}$

This is clearly not consistent with the above expression involving the angles, worryingly enough not even for perpendicular incidence $\theta = 0$.

Where do I go wrong in my line of thought?

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You obviously overlooked that you also have a reflected wave in air and that the total electric (magnetic) field in air at the boundary is the the sum of the electric (magnetic) field of the incident and reflected wave. You have to apply the mentioned boundary conditions to the total electric (magnetic) field in air at the boundary and the electric(magnetic) field of the transmitted wave in the dielectric.

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  • $\begingroup$ freecharly, wait a minute. The described influence of a dielectric does not play any role in slit experiments? I’ve asked questions like this: physics.stackexchange.com/questions/114384/… and physics.stackexchange.com/questions/158105/… $\endgroup$ Jan 14, 2018 at 18:03
  • $\begingroup$ @HolgerFiedler - The questions in your links relate to much more complicated situations than the simple reflection/transmission of an electromagnetic wave at a plane boundary. I presume that the materials of which the slit are made of have an influence in slit experiments. Just think of slowly changing the properties of the materials from opaque to transparent. $\endgroup$
    – freecharly
    Jan 14, 2018 at 18:40
  • $\begingroup$ So why not assume that the interaction of the boundary electrons of the slits edges is quantized in their interaction with photons or with an electron beam? $\endgroup$ Jan 14, 2018 at 19:02

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