Regarding transmission and reflection of EM-waves

[Kontilera]

Hello!
I´m trying to read Jacksons 'Classical Electrodynamics' and solving some problems. At the moment I´m stuck at problem 7.3. I started looking at suggested solutions (http://www-personal.umich.edu/~pran/jackson/P505/F07_hw11a.pdf) but I need some help I guess. Looking at how other people have done the boundary conditions at the first interface point (medium to air) with an electric field polarized perpendicular to the plane, they set up the condition of continuity as:
E_1 + E_2 = E_3 + E_4.
Where E_1 is the incident wave, E_2 the reflected, E_3 the transmitted, and E_4 the reflected wave of the second interface.

However the wave E_4 is never actually present at the first interface point! Why should we include this wave in the equation? It is reflected at the second interface point and then travels toward the medium again but at a distant point. Intuitional it seems to me that the boundary condition should be:
E_1 + E_2 = E_3.

Where does my logic fail?


Thanks in advance!

 

[Jano L.]

However the wave E_4 is never actually present at the first interface point!

Are you sure about this? Probably the assignment says that the incoming light wave if a plane wave, which means it has infinite extent in the directions perpendicular to the direction of propagation. Unless the wave is a beam narrower than the gap.

 

[Kontilera]

Haha of course! I don't know why I was so obsessed with thinking about the whole wave as a arrow... Now everything makes sense, thanks Jano! :-)

 

[Jano L.]

Great!

 

[sardar]

Hello! It's great that you're working on solving problems in Jackson's 'Classical Electrodynamics.' I can understand how you might be confused about the boundary conditions at the first interface point, so let me explain.

The equation E_1 + E_2 = E_3 + E_4 is actually the correct condition to use. This is because when an electromagnetic wave hits the first interface, it splits into two waves: one reflected wave (E_2) and one transmitted wave (E_3). The reflected wave (E_4) that is reflected at the second interface also contributes to the overall electric field at the first interface point.

Think of it this way: when the electromagnetic wave hits the first interface, some of it is reflected back and some of it passes through and continues into the second medium. The reflected wave (E_4) that is reflected at the second interface then travels back towards the first interface and adds to the electric field there. So, the equation E_1 + E_2 = E_3 + E_4 takes into account all the waves present at the first interface point.

I hope this helps clarify things for you. Keep up the good work with your problem solving!