Applying Maxwell's equations to plane waves

[cksoon11]

Homework Statement

For a harmonic uniform plane wave propagating in a simple medium, both [tex]\vec{E}[/tex] and [tex]\vec{H}[/tex] vary in accordance with the factor exp(-i [tex]\vec{k}[/tex].[tex]\vec{R}[/tex])

Show that the four Maxwell’s equations
for a uniform plane wave in a source-free region reduce to the following:

[tex]\vec{k}[/tex] [tex]\times[/tex][tex]\vec{E}[/tex]= [tex]\omega\mu[/tex][tex]\vec{H}[/tex]

[tex]\vec{k}[/tex] [tex]\times[/tex][tex]\vec{H}[/tex] = [tex]\omega\epsilon[/tex][tex]\vec{E}[/tex]

[tex]\vec{k}[/tex] [tex]\bullet[/tex] [tex]\vec{E}[/tex] = 0

[tex]\vec{k}[/tex] [tex]\bullet[/tex] [tex]\vec{H}[/tex] = 0

Apparently "Vector k and Vector R are the the general forms of wave number and position vector(or direction of propagation)"

Question Source(no.4) :http://www.lib.yuntech.edu.tw/exam_new/96/de.pdf"

Homework Equations

You supposed to use Maxwell's Equations for a plane wave

[tex]\nabla[/tex] x [tex]\vec{E}[/tex] = -i[tex]\omega\mu[/tex][tex]\vec{H}[/tex][tex]\nabla[/tex] x [tex]\vec{H}[/tex] = i[tex]\omega\mu[/tex][tex]\vec{E}[/tex]

[tex]\nabla . [/tex] [tex]\vec{E}[/tex] = 0

[tex]\nabla . [/tex][tex]\vec{H}[/tex] = 0

The Attempt at a Solution

First off,I am confused as to how k can be a vector when it is the wave number(a scalar).
From what I can tell,the wave is propagating in the radial direction in spherical coordinates.
I then assumed the electric and magnetic fields to be orthogonal in the [tex]\theta[/tex]and [tex]\phi[/tex] direction.

But just simply substituting the phasor form of the plane wave into Maxwell's equations:

E[tex]_{o}[/tex] exp((-i [tex]\vec{k}[/tex].[tex]\vec{R}[/tex]) into Maxwell equations doesn't seem to yield the desired results because I don't understand how they obtained the cross-products and dot products of [tex]\vec{k}[/tex]with [tex]\vec{E}[/tex] and
[tex]\vec{H}[/tex].

I just can't seem to grasp the concept of a vector as my wave number.Could someone please phrase the question in more concise terms?What am I misunderstanding here?

 

[gabbagabbahey]

The vector [itex]\textbf{k}[/itex] is called the "wave vector". It's magnitude is the wavenumber and it points in the direction of propagation.

Just substitute [tex]\textbf{E}=\textbf{E}_0e^{i(\textfb{k}\cdot\textbf{r}-\omega t)}[/itex] and [tex]\textbf{H}=\textbf{H}_0e^{i(\textfb{k}\cdot\textbf{r}-\omega t)}[/itex] into Maxwell's equations and take the derivatives...what do you get?