# Math Help - Electromagnetic wave equation

1. ## Electromagnetic wave equation

First of all I have to say that translating specific words from native language to english, is not easy. So I hope that you realize what is going on:

What did I do wrong ?

(Traveling waves from: Wave - Wikipedia, the free encyclopedia ).

2. OK. First, we should just state the form of the 'wave equations'. These have a lot of interesting physics behind them, but for now we'll just state them!

Say the light wave travelling in the $x$-direction, in free space. Then, say the wave is polarised, so the electric field is in the $y$-direction. Then the only non-zero components of the fields are $E_y$ and $B_z$:

$\bold{E} = (0, E_y, 0); \quad \bold{B} = (0, 0, B_z);$
$E_y = E_0 \sin (kx - \omega t);$
$B_z = B_0 \sin (kx - \omega t);$

$B_0 = E_0/c;$
$\omega = c k.$

Now, an explanation of the symbols I've used:
• $k$ is the wavenumber (or wavevector) of the wave. It's defined by $k = \frac{2\pi}{\lambda}$, where $\lambda$ is the wavelength.
• $\omega$ is the angular frequency of the wave. It has units of "radians per second", and relates to the 'true' frequency $\nu$ by $\omega = 2\pi \nu$.
• $c = 3 \times 10^8 \text{m~s}^{-1}$ is the speed of light.
In your question, you have been given the following information:

$E_0 = 1.5\times 10^6 \text{V~m}^{-1}$
$\lambda = 10.6\times 10^{-6} \text{m}$

and so you can calculate

$k = 2\pi / \lambda = 0.59\times 10^6 \text{m}^{-1}$.
$B_0 = E_0 / c = 0.005 T$ ( $T$ is a unit, the tesla.)

Put these into the expressions above, and ignoring the units, you get:

$E_y = (1.5 \times 10^6) \sin (kx - c k t)$
$= (1.5 \times 10^6) \sin [k(x - c t)]$
$= (1.5 \times 10^6) \sin [0.59\times 10^6 (x - 3\times 10^8~ t)]$;

$B_z = 0.005 \sin (kx - c k t)$
$= 0.005 \sin [0.59\times 10^6 (x - 3\times 10^8~ t)]$.

- - -

Your answers are written slightly diffierently: they have a minus sign, $(x - ct) \rightarrow (ct - x)$. This doesn't matter. However, I seem to have a different answer for $B_0$ (I don't know why!)

Also, the form used in Wikipedia has an "extra" phase factor $\phi$; I have made $\phi = 0$. You can do this just by choosing the origin of your $xyz$coordinates in the right place!

3. Originally Posted by Pterid
This doesn't matter. However, I seem to have a different answer for $B_0$ (I don't know why!)
B_0=E_0/c is in vacuum. Maybe for that reason.
Although you answer for B_0 is close to the original one.

4. Originally Posted by totalnewbie
B_0=E_0/c is in vacuum. Maybe for that reason.
Although you answer for B_0 is close to the original one.
Well, yes, but the $\omega(k)$relation is consistent with a vacuum - and if the problem were "set" in a medium, you would expect the question to tell you something about the medium...

If I were a more arrogant kind of person, I might suggest that the answers had a typo in!