Solution to Exercise 5.1-1: Derivation of Snellius Law

Show that you obtain Itr = IinIref  and Snellius law (sina/sinb = n) from energy and momentum conservation
     
Snellius law
Solution:
The intensity I of the beams is given by their power (energy /t) which is given by the nunber of photons/s in the beams: Ein, Ere, Etr. Everythin always per cm2 but that is not important for what follows.
Energy conservation demands
Ein   =   Ere + Etr
     
Etr  =     =   EinEre
     
Itr  =   IinIre
Looking at the x-component of the momentum p and considering that the wavelength in the material is l/n we have
|pz, in|   =   Iinkin · sina   =   Iin · 2p · sina
l
           
|pz, re|   =   Irekre · sina   =   Ire · 2p · sina
l
           
|pz, tr| Itrktr · sinb   =   Itr · 2p · sinb · n
l
Momentum conservation demands that pz, in + pz, trpz, re = 0, or
Iin sina + Itr sinb · nIre sina   =   0
Substituting Ire = IinItr leads straight ot
   
n   =   sina
sinb

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go to Exercise 5.1.1 Derivation of Snellius Law

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