Finding Fresnel coefficients from the interface conditions

[Cathr]

Homework Statement

We have an incident electric field, and there are two cases:
1) the field is polasised perpendicularly to the incidence plane (TE)
2) polarised in the plane (TM)

Here I must be able to correctly apply the limit conditions, to find the Fresnel formulas that give the amplitudes of the electric fields Er (reflected), Et (refracted) as a function of: Ei (incident field amplitude), ai (I meant so say alpha i, the angle of incidence) and ar (angle of refraction) and the constants that caracterise the environment for the two polarisations TE and TM.

Homework Equations

E2-E1=0 or D2-D1=sigma
B2-B1=0 H2-H1=j

The Attempt at a Solution

The relevant equations are the ones I found on the internet, to show what kind of equations I am looking for, but they are not adapted for this problem. I am not quite sure why the components of E that is parallel to the interface should be equal, and the ones perpendicular sshound be proportional, or maybe I'm wrong. Please help.

 

[ehild]

I am not quite sure why the components of E that is parallel to the interface should be equal, and the ones perpendicular sshound be proportional, or maybe I'm wrong. Please help.

The tangential components of both E and H are the same at both sides of an interface. It comes from Maxwell's equations curl E=-∂B/∂t and curl H =∂D/∂t, integrating them over an area that encloses the interface and applying Stokes' theorem.
You need to apply the boundary conditions for the tangential field components only. for the electric field E in case of TE waves, and for the magnetic field strength H in case of the TM waves.