# Applying Maxwell's equations to plane waves

1. Nov 15, 2009

### cksoon11

1. The problem statement, all variables and given/known data

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

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

$$\vec{k}$$ $$\times$$$$\vec{E}$$= $$\omega\mu$$$$\vec{H}$$

$$\vec{k}$$ $$\times$$$$\vec{H}$$ = $$\omega\epsilon$$$$\vec{E}$$

$$\vec{k}$$ $$\bullet$$ $$\vec{E}$$ = 0

$$\vec{k}$$ $$\bullet$$ $$\vec{H}$$ = 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"

2. Relevant equations

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

$$\nabla$$ x $$\vec{E}$$ = -i$$\omega\mu$$$$\vec{H}$$

$$\nabla$$ x $$\vec{H}$$ = i$$\omega\mu$$$$\vec{E}$$

$$\nabla .$$ $$\vec{E}$$ = 0

$$\nabla .$$$$\vec{H}$$ = 0

3. 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 $$\theta$$and $$\phi$$ direction.

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

E$$_{o}$$ exp((-i $$\vec{k}$$.$$\vec{R}$$) 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 $$\vec{k}$$with $$\vec{E}$$ and
$$\vec{H}$$.

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?

Last edited by a moderator: Apr 24, 2017
2. Nov 16, 2009

### gabbagabbahey

The vector $\textbf{k}$ 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?