How Is Snell's Law Derived Using Boundary Conditions and Wave Equations?

[helpcometk]

Homework Statement

The electric field of an electromagnetic wave traveling in the dielectric can be written as
E = (E0) e(iωt−ikx)
where E0 is a constant vector and |k| = [ω(ε)^1/2]/c .

Consider an electromagnetic wave traveling in vacuum towards the dielectric at an angle θ
to the x-axis with its electric field polarised in the x, y plane. When this wave is incident
on the dielectric a scattered wave and a transmitted wave are produced.
By imposing the relevant boundary conditions on the electric field across the boundary, or
otherwise, derive Snell’s law.

Homework Equations

Snell's law : (sinθ1)(n1)=(sinθ2)(n2)
Where:
θ1 is the angle of incidence and θ2 is the angle of reflection,measured with respect to the normal.
n is the refractive index of the material
n1,2 = velocity of light in a vacuum / velocity of light in medium

ω=2πu=ku
where :
k is the wave number defined as:
λ=2π/k

ω is the angular frequency
u is the frequency

The Attempt at a Solution

 

[henry_m]

Hi,

What does the electric field of the incident wave look like? What about the reflected and transmitted waves? What are the boundary conditions?

 

[helpcometk]

So you actually say that the exponents of reflected and incident waves are equal with the exponent of the transmitted wave.which means xk1=xk2. ?

 

[henry_m]

I'm not quite sure what you mean. My point was that you should start by explicitly writing down what the electric field of an incoming wave as described in the problem would be, and write down what the boundary conditions are at the interface. You should then find that the forms of the reflected and transmitted waves are heavily constrained by the boundary conditions.