physics question

[Flanders]

A physics question:

Can anybody explain the second equation in this: http://en.wikipedia.org/wiki/Surface_plasmon_polariton#Dispersion_relation

It is supposedly a relation for the continuity on the interface between two semi-infinite solids. But I can’t figure out where it comes from. I’m not that familiar with Maxwells equations and I know some of you are interested in physics

[demagogue]

I don't know the answer, but it sounds like a cool effect I've never read about before.

Light rippling across the surface of a metal, huh?

[Flanders]

They are EM-waves that travel along the surface yes. In the direction normal to the surface they decay exponentially and they don't transfer any energy unless another object is placed in this evanescent field.

[Obsttorte]

You have that the divergence of E is zero. Calculate that and you get the second equation. Nonsense, sorry.

 

It holds that the function is steady at the boundaries between the two phases and that maxwell holds (divergence of E is zero). If you use both you get the boundary conditions shown.

[Flanders]

Divergence of E is zero? Then where does the epsilon come into play? I know that the displacement field has epsilon and E in it.

[Sotha]

That's why we have colorful windows in medieval churches, by the way.

[Flanders]

I still don't get it

[Obsttorte]

I'm just assuming that they are using the vacuum equations: http://en.wikipedia.org/wiki/Maxwell%27s_equations#Vacuum_equations.2C_electromagnetic_waves_and_speed_of_light

[AluminumHaste]

Head=asploded

[Briareos H]

My physics are very rusty but if I had to prove that equation I'd definitely use Gauss's law (which is basically one of the Maxwell's equations -- only not in a vacuum) which states that div E = density of charge / epsilon. But if you use its integral form flux = charge / permittivity you can deduce the continuity of the tangential component of E at an interface. I remember being asked to prove it once, if I had to do it again I'd integrate on an infinitesimally small surface at the interface using the E wave in its exponential form: the k_x components get thrown away, the density of charge eliminated on both sides of the equation and the wave numbers come out of the exponential upon integration. Something like that.

Edited May 20, 2014 by Briareos H

[Diego]

My physics are very rusty but if I had to prove that equation I'd definitely use Gauss's law (which is basically one of the Maxwell's equations -- only not in a vacuum) which states that div E = density of charge / epsilon. But if you use its integral form flux = charge / permittivity you can deduce the continuity of the tangential component of E at an interface. I remember being asked to prove it once, if I had to do it again I'd integrate on an infinitesimally small surface at the interface using the E wave in its exponential form: the k_x components get thrown away, the density of charge eliminated on both sides of the equation and the wave numbers come out of the exponential upon integration. Something like that.

 

That's it. Get Maxwell's equations on integral form and choose the volume and contour that simplifies the integrals.