# Direction of polarization for monochromatic wave?

1. Dec 9, 2015

### magnesium12

1. The problem statement, all variables and given/known data
Write down the (real) electric and magnetic fields for a monochromatic plane wave of amplitude Eo, frequency w, and phase angle zero traveling in the direction from the origin to the point (1,1,1) with polarization parallel to the xz plane.

I understand how to write the equations, I just don't understand how to get the correct direction for the electric and magnetic fields.

2. Relevant equations
$E(z,t) = E_o\cos(\hat k \cdot \hat r - \omega t) \hat n$
$B(z,t) = \frac{E_o}{c}\cos(\hat k \cdot \hat r - \omega t) ( \hat k x \hat n)$
$k = -\frac{\omega}{c}$
$\hat n \cdot \hat k = 0$

3. The attempt at a solution
This is what I did:

$\hat n = \hat x + \hat z$
$\hat k = \frac{\omega}{c} (\hat x + \hat y + \hat z)$

So I thought that was all I was supposed to do to find the direction, but the solutions manual says these are the actual directions of n and k:

$\hat n =\frac { \hat x - \hat z}{\sqrt{2}}$
$\hat k = \frac{\omega}{c} \frac{(\hat x + \hat y + \hat z)}{\sqrt{3}}$

So where did those factors of sqrt(2) and sqrt(3) come from?
I appreciate any help!

2. Dec 9, 2015

### blue_leaf77

First, the question asks you to find the unit vector, so the magnitude of the vector which is supposed to be the answer should be unity. Second, you only know that $\hat{n}$ only has components along $\hat{x}$ and $\hat{z}$ but you are not given the length of each component, these are what you should calculate subject to the condition that the length of $\hat{n}$ is unity and that this vector is perpendicular to $\hat{k}$.

3. Dec 9, 2015

### magnesium12

I don't think I understand.
So I would do
$n = \sqrt(a^2 + b^2)) = 1$ Therefore: $a^2= 1-b^2$
$k = \sqrt(c^2 + d^2 + e^2) = 1$
And then use this somehow:
$\hat n \cdot \hat k = nkcos\theta = 0$
$nkcos\theta = \sqrt((1-b^2) + b^2)\sqrt(c^2 + d^2 + e^2)cos\theta$
But since n = 1 and k =1, wouldn't that just leave me with nothing again?

4. Dec 9, 2015

### blue_leaf77

If $\mathbf{k}$ is denoted such that it has components $c$, $d$, and $e$ then they must be known already since the problem tells you that $\mathbf{k}$ goes from the origin to the point (1,1,1). What you don't know yet are just $a$ and $b$, i.e. two unknowns. You have figured out one equation relating these unknowns, which is
.
The other equation you need is the orthogonality condition between $\mathbf{k}$ and $\hat{n}$. To do this, it will be easier with component-by-component multiplication instead of the one like $kn\cos \theta$.