Results 1 to 4 of 4

Thread: Electromagnetic wave equation

  1. #1
    Member
    Joined
    Jul 2005
    Posts
    187

    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 ).
    Attached Thumbnails Attached Thumbnails Electromagnetic wave equation-412.jpg  
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Junior Member
    Joined
    Jun 2007
    From
    Cambridge, UK
    Posts
    41
    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 $\displaystyle x$-direction, in free space. Then, say the wave is polarised, so the electric field is in the $\displaystyle y$-direction. Then the only non-zero components of the fields are $\displaystyle E_y$ and $\displaystyle B_z$:

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

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

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

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

    and so you can calculate

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

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

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

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


    - - -

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

    Also, the form used in Wikipedia has an "extra" phase factor $\displaystyle \phi$; I have made $\displaystyle \phi = 0$. You can do this just by choosing the origin of your $\displaystyle xyz$coordinates in the right place!
    Last edited by Pterid; Jun 8th 2007 at 05:10 PM. Reason: cleanup
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Jul 2005
    Posts
    187
    Quote Originally Posted by Pterid View Post
    This doesn't matter. However, I seem to have a different answer for $\displaystyle B_0$ (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.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Junior Member
    Joined
    Jun 2007
    From
    Cambridge, UK
    Posts
    41
    Quote Originally Posted by totalnewbie View Post
    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 $\displaystyle \omega(k)$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!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Wave equation
    Posted in the Differential Equations Forum
    Replies: 0
    Last Post: Dec 31st 2010, 09:33 AM
  2. Electromagnetic wave
    Posted in the Calculus Forum
    Replies: 0
    Last Post: Apr 12th 2010, 01:21 PM
  3. Inhomogeneous electromagnetic wave equation
    Posted in the Differential Equations Forum
    Replies: 0
    Last Post: Apr 12th 2010, 11:07 AM
  4. Partial differential equation-wave equation(2)
    Posted in the Differential Equations Forum
    Replies: 4
    Last Post: Sep 6th 2009, 08:54 AM
  5. Partial differential equation-wave equation - dimensional analysis
    Posted in the Differential Equations Forum
    Replies: 3
    Last Post: Aug 28th 2009, 11:39 AM

Search Tags


/mathhelpforum @mathhelpforum