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Spherical Harmonics Change of Coordinate System

  1. Jul 2, 2016 #1
    1. The problem statement, all variables and given/known data
    Let $$\vec H = ih_4^{(1)}(kr)\vec X_{40}(\theta,\phi)\cos(\omega t)$$
    where ##h## is Hankel function of the first kind and ##\vec X## the vector spherical harmonic.
    a) Find the electric field in the area without charges;
    b) Find both fields in a spherical coordinate system that is a rotation of 45 deg about the y axis.

    2. Relevant equations
    Maxwell's equations, addition theorem for spherical harmonics.

    3. The attempt at a solution
    For part a I used Maxwell's equations, namely
    $$\nabla \times H = -\epsilon_0\frac{\partial E}{\partial t} $$
    For part b I want to use the addition theorem, namely
    $$P_l(\cos \alpha) = \frac{4\pi}{2l+1}\sum_{m=-l}^lY^{*}_{lm}(\theta ',\phi ')Y_{lm}(\theta,\phi)$$
    using the specific transofrmation ##\theta = pi/4,\;\phi=0##, but I can't find a way to isolate ##Y_{lm}## in the old coordinate system and express it in the new because of the terms in the sum. Any directions? Thanks.
  2. jcsd
  3. Jul 7, 2016 #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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