Spherical Harmonics Change of Coordinate System

[Yoni V]

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.

 

[mark726]

For part a, using Maxwell's equations is the correct approach. You can use the equation you mentioned, $$\nabla \times H = -\epsilon_0\frac{\partial E}{\partial t}$$ to solve for the electric field, since the magnetic field is already given in the form of the Hankel function and vector spherical harmonic.

For part b, you are on the right track by using the addition theorem for spherical harmonics. To express the spherical harmonics in the new coordinate system, you can use the transformation matrix for a rotation of 45 degrees about the y axis. This matrix will allow you to express the spherical harmonics in terms of the new coordinate system. Once you have done this, you can substitute the new expressions for the spherical harmonics into the addition theorem and solve for the electric and magnetic fields in the new coordinate system.

I hope this helps. Good luck with your calculations!