I EM Power transmitted from one region to another (normal incidence)

[Fernandopozasaura]

High!
I have a EM plane wave hitting normally a surface dividing universe in media 1 and 2, both without losses.
So we have incident, reflected and transmitted waves.

It's a simple exercise in which you are given the basic data about two media and wave incident amplitude H in medium 1.
I get Ei, Er, Hr, Et and Ht in first medium. So far so good.
I went further and performed calculation for mean poyinting vector for incident, reflected and transmitted waves.

Evertything fine becouse I get that Stransmitted + Sreflected = Sincident so I conclude there is no errors in my calculations.
I then, went on and calculated the mean poyinting vector in medium 1 using fields E and H in medium 1 as the sum of both fields in first medium (E1 = Ei+Er, and H1 = Hi+Hr)
And what I get is that S in medium 1 is the same that S in medium 2 (almost, I assume some discrepance due to computations)

At a first glance, I thougth that this result didn't make sense, but trying to find a reason I concluded that as long as media are ideal there is no losses and no power is lost in any place.
So, is my conclussion correct?

Thank you very much for your opininos, and sorry for my english

 

[vanhees71]

Indeed, if there are no losses you should get energy and momentum conservation, and that seems to be what you get :-).

 

[Fernandopozasaura]

Great!, I think I needed any kind of support.
What made me unsure was the fact that, somehow, in medium 1 should be more power (S) because, there, you have two powers and only one in medium 2. I think there is something counterintuitive. At least to my brain.
Thanks a lot for your help.

 

[tech99]

In medium 1 you have forward and reflected power, and the difference should equal power in medium 2.

 

[Fernandopozasaura]

Yes, that's what I get. More precisely S_incident = S_reflected + S_transmited.
Further calculation was to get S_medium1 (which is not S_incident neither S_reflected) starting from total E (Ei + Er) and total H (Hi + Hr) in that medium 1.

And then S_medium2 from E and H in medium 2. And I get the same value, and that's what I couldn't justify. My intuitive understanding was that they should be different, but I looks like I have to assume this fact. (as far as someone send a different reasoning)

 

[vanhees71]

No, that's the correct reasoning. $\vec{S} =\vec{E} \times \vec{B}/\mu_0$ (SI) everywhere, i.e., if you have two pieces (as the incident and reflected in medium 1) you have to superimpose these two pieces first to get the total em. field in this region and then calculate $\vec{S}$ from this total field.

The reason is that the local (microscopic) energy-conservation law in electromagnetic reads
$$\partial_t u +\vec{\nabla} \cdot \vec{S}=-\vec{j} \cdot \vec{E}, \qquad(*)$$
where
$$u=\frac{\epsilon_0}{2} \vec{E}^2 + \frac{1}{2 \mu_0} \vec{B}^2$$
is the energy density of the em. field. On the right-hand side of (*) is the power loss due to work done on all (microscopic charge-current densities), i.e., the power on the electric charge due to the Lorentz force density
$$\vec{f}=\rho (\vec{E}+\vec{v} \times \vec{B}),$$
where $\rho \vec{v}=\vec{j}$.

 

[Fernandopozasaura]

Ok, so, if I have understood, the power expressed by $$\vec{S}$$ is a characteristic of the whole universe, in this case: nothing in it but the materials and the frontier and whatever source generating this fields, and the existence of this (the border) doesn't affect this quantity.
And that's why the power calculation in both media starting from different fields are the same.
Mmm, I admit I need to spend more time thinking of it... or learning more details about the physics involved.
Thanks!