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## Homework Statement

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.

## Homework Equations

Maxwell's equations, addition theorem for spherical harmonics.

## 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.