I'm having a problem using the fresnel formulae for these two problems below:
1. Using Fresnel formulae and expression for the Poynting vector show that at the interface between two dielectrics in the absence of absorption R+T=1, where R – is reflectivity and T – transmissivity
2. Using Fresnel formulae, find expressions for the amplitudes and intensities of the reflected and transmitted plane electromagnetic waves incident normally at an interface between two materials with refractive indices n1 and n2
I know reflectivity is given by R = j(r)/j(i) = ∣R^2∣/∣A^2∣
and transmissivity is given by T = j(t)/j(i) = (n2/n1)*(cosθt/cosθi)* ∣T^2∣/∣A^2∣
and the Poynting vector is: S = EH = 1/(v*μ0*E^2) = n*c*ϵ0*E^2
J(i) = S(i)cosθi = n1*c*ϵ0*∣A^2∣*cosθi
J(r) = S(r)cosθi = n1*c*ϵ0*∣R^2∣*cosθi
and J(t) = S(t)cosθi = n2*c*ϵ0*∣T^2∣*cosθi
The Attempt at a Solution
In theory, I can prove that R+T=1 but I am struggling with the fractions as they have always been my worst enemy throughout university. If someone can help me out with this please I would appreciate it. My main problem is working this using the 'conservation of energy', which I don't understand where the conservation of energy comes into place for this question, and why we need it to prove R+T=1.
For the second question I am struggling to understand it at all, when it talks about incident normally to the plane. We haven't covered waves being incident parallel or perpendicular between two materials at all.
Any help and insight will be greatly appreciated!