Poynting Vector at an interface with no free current flowing

[Jon.G]

Homework Statement

How do the normal and tangential components of the Poynting vector in matter, S = E x H , behave at an interface between two simple media where no free current flows, but either free charge or polarization charge is present at the interface?

Homework Equations

E_1(para) - E_2(para)0

The Attempt at a Solution

Honestly, I'm having trouble starting this. Think I might just need a nudge in the right direction though.

I think I know what to do, but doing it is a different question :S
I think I should probably write E and H as the sum of their parallel and perpendicular components and then substitute this into the equation for the Poynting Vector (S), and then identify the parallel and perpendicular components of S.
(Surely it can't be as simple as E = E_para + E_perp and same for H)
Then I would somehow use the boundary conditions of the E and H vectors and apply this to S, but I'm not too sure how to do this.
The only boundary condition I can think of that I know is the one I wrote above.
I've also written that H is discontinuous by J_c but I do not know how to prove this (and the q states no free current flows)

This could be an insanely easy question I just feel like my mind is hitting a wall and I'm getting frustrated by how little I've got done :S

Thanks in advance for any help :)

 

[mark726]

Hello,

Thank you for your question. The behavior of the normal and tangential components of the Poynting vector at an interface between two simple media can be determined by using the boundary conditions for the electric and magnetic fields. These boundary conditions can be derived from Maxwell's equations and are as follows:

1. The tangential component of the electric field is continuous across the interface: E1t = E2t
2. The normal component of the magnetic field is continuous across the interface: H1n = H2n

Using these boundary conditions, we can write the electric and magnetic fields as the sum of their parallel and perpendicular components. This allows us to substitute these expressions into the Poynting vector equation. We then have:

S = (E1para + E1perp) x (H1para + H1perp)

Next, we can use the boundary conditions to simplify this expression. Since no free current flows at the interface, we know that the tangential components of both the electric and magnetic fields are equal on either side of the interface. This means that E1para = E2para and H1para = H2para. Therefore, we can rewrite the Poynting vector as:

S = (E1para + E1perp) x (H1para + H1perp) = (E2para + E1perp) x (H2para + H1perp)

Now, we can identify the parallel and perpendicular components of the Poynting vector. The parallel component, Spara, can be written as:

Spara = E2para x H2para

And the perpendicular component, Sperp, can be written as:

Sperp = E1perp x H1perp

Therefore, we can see that the normal component of the Poynting vector, Sperp, is discontinuous at the interface due to the presence of free charge or polarization charge. This is because the perpendicular components of the electric and magnetic fields are not equal on either side of the interface, leading to a discontinuity in the perpendicular component of the Poynting vector.

I hope this helps to nudge you in the right direction. If you have any further questions, please don't hesitate to ask. Good luck with your research!