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Spherical Harmonics Change of Coordinate System
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[QUOTE="Yoni V, post: 5511624, member: 545683"] [h2]Homework Statement [/h2] 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. [h2]Homework Equations[/h2] Maxwell's equations, addition theorem for spherical harmonics. [h2]The Attempt at a Solution[/h2] 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. [/QUOTE]
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