Do incident and reflected wave normal components have opposite signs?

[_Andreas]

Homework Statement

An electromagnetic wave is incident upon a planar interface at an oblique angle $$\theta_i$$, where it is reflected. For the wave vector components parallel to the interface, we have $$k_{xi}=k_{xr}$$. Thus, $$\theta_i=\theta_r$$. The wave numbers for the incident and reflected waves are equal. Find the relation between the wave vector components normal to the interface for the incident and reflected waves.

Homework Equations

$$\cos\theta_i=\cos\theta_r$$

See attached picture.

The Attempt at a Solution

Thus, from the picture, $$\frac{k_{zr}}{k}=\frac{k_{zi}}{k}\Longrightarrow k_{zr}=k_{zi}$$.

To me this seems to imply that both normal components point in the same direction, in addition to being of the same magnitude. But shouldn't the normal components have opposite signs, since the incident and reflected waves travel in opposite normal directions?

 

[tiny-tim]

Hi _Andreas!

(have a theta: θ and try using the X₂ tag just above the Reply box )

Thus, from the picture, $$\frac{k_{zr}}{k}=\frac{k_{zi}}{k}$$

I don't understand how you get k_zr/k = k_zi/k

 

[_Andreas]

Hi!

Hi _Andreas!

(have a theta: θ and try using the X₂ tag just above the Reply box )I don't understand how you get k_zr/k = k_zi/k

It follows from cosθ_i=k_zi/k and cosθ_r=k_zr/k, since θ_i=θ_r and k=|k_i| = |k_r|.

 

[tiny-tim]

Perhaps I'm misunderstanding your picture, but isn't cosθ_i = -k_zi/k ?

 

[_Andreas]

Perhaps I'm misunderstanding your picture, but isn't cosθ_i = -k_zi/k ?

Uhm... can you explain how you get this result?

 

[tiny-tim]

k_zi points right?

 

[_Andreas]

k_zi points right?

Sure, but that's in the positive z direction.

 

[tiny-tim]

ah, then isn't cosθ_r = -k_zr/k ?

 

[_Andreas]

ah, then isn't cosθ_r = -k_zr/k ?

I guess so, and there's my problem. I tend to think of k_zr without the sign as the z component of k_r.