Using Fresnel Formulae and Poynting vector problem

[tommowg]

Homework Statement

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

Homework Equations

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

given that:

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

[/B]
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!

Thanks,

Tom

 

[mark726]

Dear Tom,

Thank you for your post. I can understand your struggle with the fractions, but I assure you that with some practice and understanding of the concepts, you will be able to solve these problems easily.

To prove R+T=1, we need to use the conservation of energy principle, which states that energy cannot be created or destroyed, only transformed from one form to another. In this case, we are dealing with the energy of the incident, reflected, and transmitted waves at the interface between two dielectrics. By using the Poynting vector and the equations you have provided, we can show that the sum of the reflected and transmitted energy is equal to the incident energy, thus proving R+T=1.

For the second question, incident normally means that the wave is coming straight at the interface, perpendicular to it. In this case, the angle of incidence (θi) is 0 degrees, and the angle of refraction (θt) can be calculated using Snell's law. The equations you have provided for reflectivity and transmissivity can be used to find the amplitudes and intensities of the reflected and transmitted waves.

I hope this helps. If you have any further questions, please don't hesitate to ask.