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Radiation E field of a rotor, can't get the result

  1. Jul 24, 2015 #1

    fluidistic

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    Gold Member

    1. The problem statement, all variables and given/known data
    Two equal and opposite charges are attached to the ends of a rod of length s. The rod rotates counterclockwise at a speed ##\omega=ck##. The electric dipole moment of the system at ##t=0## is ##\vec p_0=qs\hat x##.
    Show that the electric field in the radiation zone is ##\vec E_{\text{rad}}(r,\theta, \phi ,t)=\frac{k^2p_0}{4\pi\varepsilon_0}(\cos \theta \hat \theta +i\hat \phi )\frac{e^{i(kr-\omega t + \phi)}}{r}##.
    (It's a part of a problem in Zangwill's book).

    2. Relevant equations
    [tex]\vec E _{\text{rad}}=\frac{\mu_0}{4\pi} \frac{\hat r (\hat r \cdot \ddot {\vec p} _{\text{ret}} - \ddot {\vec p}_{\text{ret}})}{r}[/tex]

    3. The attempt at a solution
    I tried the brute force approach by using the formula I just wrote above, in the relevant equations.
    I got that ##\vec p (t)=p_0 [\cos (\omega t)\hat x + \sin (\omega t) \hat y]##.
    So that [tex]\ddot {\vec p}_\text{ret}=p_0 \omega ^2 \{ -\cos \left [ \omega \left ( t-\frac{r}{c} \right ) \right ] \hat x -\sin \left [ \omega \left ( t-\frac{r}{c} \right ) \right ] \hat y \}[/tex]

    Converting the Cartesian unit vectors to spherical unit vectors: ##\hat x =\sin \theta \cos \phi \hat r + \cos \theta \cos \phi \hat \phi -\sin \phi \hat \phi##. Plug and chug that into the formula for the E field... and I got a non vanishing ##\hat r## component which shouldn't happen.
    I reached [tex]\frac{\mu_0 p_0 \omega ^2}{4\pi r}\{ -\sin \left [ \omega \left ( t-\frac{r}{c} \right ) \right ] \sin \theta \cos \phi +\cos \left [ \omega \left ( t-\frac{r}{c} \right ) \right ] \sin \theta \sin \phi +\cos \left [ \omega \left ( t-\frac{r}{c} \right ) \right ] \sin \theta \cos \phi +\sin \left [ \omega \left ( t-\frac{r}{c} \right ) \right ] \sin \theta \sin \phi \} \hat r[/tex]. I see I can factor out the sine of theta but I don't see how it's going to help me.
    I don't seem to reach the desired result.
    Any comment is welcome.


    Edit: Nevermind I see my error. I miscalculated ##\hat r \cdot \ddot {\vec p}_{\text{ret}}##. In fact I did ##\hat r \cdot \dot {\vec p}_{\text{ret}}##. I now get 0 for the ##\hat r## component as it should. Problem solved!
     
    Last edited: Jul 24, 2015
  2. jcsd
  3. Jul 29, 2015 #2
    Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
     
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