Direction of polarization for monochromatic wave?

[magnesium12]

Homework Statement

Write down the (real) electric and magnetic fields for a monochromatic plane wave of amplitude E_o, 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.

Homework Equations

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

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!

 

[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 $\hat{k}$.

 

[magnesium12]

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 $\hat{k}$.

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?

 

[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

$n = \sqrt(a^2 + b^2)) = 1$ Therefore: $a^2= 1-b^2$

.
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 $kn\cos \theta$.