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Homework Help: Half wave antenna

  1. Mar 22, 2008 #1
    Vector Potential

    1. The problem statement, all variables and given/known data
    Consider two half wave antennas each ahving current

    [tex] I(z,t)=\hat{z} I_{0}\cos\omega t\sin k(\frac{d}{2}-|z|) [/tex]
    where [itex] k=\omega/c[/itex]
    Each antenna has length d and points in the z direction. Antenna 1 is at [itex] (\Delta/2,0,0)[/itex] and antenna two is at [itex](-\Delta/2,0,0)[/itex]

    a) Find the vector potential A
    b) Find the electric and magnetic field
    c) Find [tex] dP/d\Omega[/tex]
    d) Evalute [tex] dP/d\Omega [/tex] in the X Y plane when the antenna is seaparated by a distance lambda/2. Along what direction is the radiation preferentialy propagated?

    2. Relevant equations
    In CGS units so...
    [tex] \vec{A}(\vec{r},t)=\frac{1}{c}\int \frac{\vec{J}(\vec{r},t_{r})}{|\vec{r}-\vec{r'}|} d\tau [/tex]


    3. The attempt at a solution
    So we need the current as a function of z' and the retarded time

    [tex]I(z,t)=\hat{z} I_{0}\cos\omega t\sin k(\frac{d}{2}-|z|) [/tex]
    [tex]I(z,t)=\hat{z} I_{0}\cos\omega (t-\frac{\mathcal{R}}{c})\sin k(\frac{d}{2}-|z|) [/tex]

    where [tex]\mathcal{R}=\sqrt{z'^2+r^2-2z'r\cos\theta}[/tex]
    since we want the fields far away (radiation zone), expand
    [tex]\mathcal{R}\approx r\left(1-\frac{z'}{r}\cos\theta[/tex]
    so then
    [tex] \cos\omega (t-\frac{\mathcal{R}}{c})\approx\cos\omega\left(t-\frac{r}{c}\left(1-\frac{z'}{r}\cos\theta\right)\right)[/tex]
    [tex] \cos\omega (t-\frac{\mathcal{R}}{c})\approx\cos\omega t \cos\frac{r}{c}\left(1-\frac{z'}{r}\cos\theta\right)+\sin\omega t\sin\frac{r}{c}\left(1-\frac{z'}{r}\cos\theta\right)[/tex]

    So then A is calculated like this? make the approximation that [tex] \mathcal{R}\approx r [/tex]

    [tex] \vec{A} = \hat{z}\frac{I_{0}}{rc}\int \sin k\left(\frac{d}{2}-|z|\right)\left(\cos\omega (t-\frac{\mathcal{R}}{c})\left(\cos\omega t \cos\frac{r}{c}\left(1-\frac{z'}{r}\cos\theta\right)+\sin\omega t\sin\frac{r}{c}\left(1-\frac{z'}{r}\cos\theta\right)\right) dz'[/tex]

    Ok since there are two antennas how should the integration be performed...
    should i do for each antenna separately? That is integrate one of them from [itex]\Delta[/itex] to [itex]\Delta+\frac{d}{2}[/itex] and one of them from [itex]\Delta[/itex] to [itex]\Delta-\frac{d}{2}[/itex] ?? And then add the two results?

    Thanks for your help!!
     
    Last edited: Mar 22, 2008
  2. jcsd
  3. Mar 26, 2008 #2

    clem

    User Avatar
    Science Advisor

    Since it is a radiation problem, I think you are allowed to assume r>>r' and
    kr>>1, which greatly simplifies the problem.
     
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