How to determine the direction of the E-field of an EM wave

[Luk]

Homework Statement

Consider a wave vector which hits a plain boundary between water and air. The wave vector hits the boundary with an angle α1 measured from the vertical axis. The magnetic field amplitude has a y-component only. Also, notice: The z-axis is the horizontal axis, the x-axis is the vertical axis and the y-axis is the axis that goes into the paper. The question is, in what direction does the electrical field amplitude point?
The wave vector has components $$ \vec{k} = \begin{array}{c}\sin(\alpha_1)\\0\\-\cos(\alpha_1)\end{array} $$

Homework Equations

Well I know that the magnetic and the electrical field are both perpendicular to each other and to the wave vector. I also know that $$\vec{H} = 1/\eta * \vec{e_k} \times \vec{E} $$

The Attempt at a Solution

I was thinking that, maybe, this equation holds: $$ \vec{E} = \eta * \vec{e_k} \times \vec{H} $$
But I'm not sure about the correct order within the cross product. So it could also be this: $$ \vec{E} = \eta * \vec{H} \times \vec{e_k} $$
I simply don't know.

 

[kuruman]

The order of the cross product is such that $\vec{E}\times\vec{H}$ points in the direction of $\hat{e}_k$.

 

[Luk]

right. Well since H points in the positive y-direction, is the cross product then $$ \begin{array}{c}-\sin(\alpha_e)\\0\\-\cos(\alpha_e)\end{array} $$ ?

 

[kuruman]

... is the cross product then ...

Which cross product is this? Do you mean the electric field? It is best to make a drawing showing the three principal axes, the k-vector and the H-vector. Add to this drawing the E-vector in such a way that $\vec{E}\times\vec{H}$ points in the direction of $\hat{e}_k$. Then look at the drawing and find the components of the E-vector.

 

[Luk]

Ok. Well, here is my gemoetric solution and how I choose the cross product. I hope it is readable:
.. wait. How can I upload a photo ?

allright, should be uploaded. You find the images at the end of this page

 

[Luk]

 

[Luk]

So, what's confusing me is: This answer appears to be correct. But as you can see on the last page, the cross product of E x H does NOT yield a vector in k-direction, but in negative k-direction. So, WTF ?

 

[kuruman]

It seems you have applied the right hand rule incorrectly. If you use this definition of the rhr and your sketch, you should have index finger along E, middle finger along H then the thumb along k is opposite to what you have drawn. Now the directions of H and k are given, so you need to redraw your E-vector.