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I don't understand boundary conditions very well yet, it would seem. I'm sure this is very simple. In analyzing a situation where a monochromatic plane wave approaches an interface (with polarization perpendicular to the plane of incidence), why does the boundary condition:

$$\epsilon_1 E_1^{\perp}=\epsilon_2 E_2^{\perp}$$

hold true? The book I am following simply says that it is trivial and that $0=0$, but this doesn't make sense. If the electric component of the wave is given as:

$$\tilde{\vec{E}} = \tilde{E_0}e^{i(\vec{k}\cdot \vec{r} - wt)}\hat y$$

then of course we have a perpendicular component. Maybe this will help me understand - what exactly do we mean by $E^{\perp}$ and $E^{\parallel}$??

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The symbols $E_1^{\perp},E_2^{\perp}$ probably denote components of electric field perpendicular to the interface between the two different media. The equation

$$ \epsilon_1 E_1^{\perp}=\epsilon_2 E_2^{\perp} $$

follows from the fact that if there are no free charges near the interface (that is, if the dielectric media are not charged), flux of displacement field $\mathbf D$ through box centered at the interface plane is zero. This is explained in electrostatics.

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