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Radiation fields calculus

  1. Oct 14, 2016 #1
    Hello.Looking at Jackson's ch 9 on radiation, I am trying to calculate the fields E and B from the potentials in the far field but it is very confusing. Given now the approximation for he vector potential

    [tex] \textbf{A}_{\omega}(x) = -ik \frac{e^{ikr}}{r} \textbf{P}_{\omega} [/tex]

    with [tex]\textbf{P}_{\omega} = \int{d^{3}x^{\prime} \textbf{x}^{\prime} \rho_{\omega} (\textbf{x}^{\prime})} [/tex] the electric dipole moment. The fields are the calculated from

    [tex]\textbf{B}_{\omega}(x) = \nabla\times\textbf{A}_{\omega}(x) [/tex] and [tex]\textbf{E}_{\omega}(x)= \frac{i}{k}\nabla\times\textbf{B}_{\omega}(x)[/tex]
    So first I tried to computed the B fiels and I get to an expression like

    [tex]\textbf{B}_{\omega}(x) = \Big[k^2 \frac{e^{ikr}}{r} + ik\frac{e^{ikr}}{r^2}\Big](\hat{n}\times\textbf{P}_{\omega} ) + \Big[-ik\frac{e^{ikr}}{r} (\nabla\times\textbf{P}_{\omega})\Big][/tex]

    and it seems that [tex]\nabla\times\textbf{P}_{\omega} =0 [/tex] but I am not sute why exactly. I think maybe that the dipole is calculated at the prime coordinates and so the curl is with respect to the not prime coordinates so maybe that's why. Also I am confused on how to proceed with the calculations for the E field


    [tex]\textbf{E}_{\omega}(x)= \frac{i}{k} \Big[\big(k^3\frac{e^{ikr}}{r} - k^2\frac{e^ikr}{r^2}\big)[\hat{n}\times(\hat{n}\times\textbf{P}_{\omega})] + k^2\frac{e^{ikr}}{r} \nabla\times(\hat{n}\times\textbf{P}_{\omega})\Big][/tex]

    I believe the second term must vanish as i suspect from the result but don't why again. Also the dipole moment which direction has?

    I tried to investigate the second term using the vector identity relevant here so

    [tex] \nabla\times(\hat{n}\times\textbf{P}_{\omega}) =\hat{n}(\nabla\cdot\textbf{P}_{\omega}) - \textbf{P}_{\omega}(\nabla\cdot\hat{n}) + (\textbf{P}_{\omega}\cdot\nabla)\hat{n}- (\hat{n}\cdot\nabla)\textbf{P}_{\omega} [/tex]

    Is this thing zero and why?
     
  2. jcsd
  3. Oct 14, 2016 #2

    Charles Link

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    Homework Helper

    All the vector operations on ## P_{\omega} ## are zero because ## P_{\omega} ## is just a constant. The vector operations on ## \hat{n} ## I think need a little consideration, but I think ## \hat{n}=1\hat{a_r} ## in spherical coordinates. You can google the operations of curl and divergence in spherical coordinates. I think the curl vanishes, but not the divergence. The ## P_{\omega}\cdot \nabla \hat{n} ## term also needs to be carefully evaluated. I think this last term is zero because the "1" in front of ## \hat{a_r} ## is a constant. ## \\ ## Just an additional comment: I think the first equation for ## A_{\omega} ## comes from the polarization current formula ## J_p=\dot{P} ## along with ## A(x)=\int {\frac{J(x')}{c|x-x'|} } \, d^3x' ##. (My formula for ## A ## may be a steady state formula, so I may need to do some further reading on the subject). ## \ ## In the ## x' ## integral, the sinusoidal dipole is assumed to be localized near the origin.
     
    Last edited: Oct 14, 2016
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