Different treatment of E and B field in Brewster angle derivation

[Heirot]

Why does the Brewster angle phenomena happen only when the electric field vector lies in the plane of incidence (the magnetic field vector is perpendicular to that plane) but not when the magnetic vector field lies in the plane of incidence (now the electric field is perpendicular to that plane)?

 

[Antiphon]

It can, but the material has to have a dielectric constant of 1 and a magnetic permeability greater than 1.

When you sum the incident, transmitted and reflected tangential fields at the interface, there is no reflection needed to balance the electric and magnetic field ratios at the interface. The balance is accomplished by geometric projection in the air and a reduced electric field in the dielectric. In the other polarization, the electric vector has no geometric reduction with angle. This means field balance can only be achieved with the inclusion of a reflected wave at all angles.

The field balance I refer to is the wave impedance of plane waves. Reflection is governed by impedance mismatches at the interface. If the tangential fields can be balanced with only the incident and transmitted waves, the reflection goes away. This is what happens at the Brewster angle.