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Amit Kumar Basistha

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- Homework Statement
- Recently I came across the following problem:

Suppose ##\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1## is an ellipse with surface charge density ##\sigma=\sigma_0\sin(\theta)\cos(\phi)## where ##\theta## is the angle with the ##z-## axis and ##\phi## is with the ##x-## axis. Find the potential and multipole moments at a point far away from the ellipse.

- Relevant Equations
- Maxwell's Equations

Spherical Harmonics

My initial idea was to first parametrize the ellipse as ##(a\sin(\theta')\cos(\phi'),b\sin(\theta')\sin(\phi'),c\cos(\theta'))## and then calculate ##\theta,\phi## in terms of these coordinates. I then did the coordinate transform ##x\to\frac{x}{a},y\to\frac{y}{b},z\to\frac{z}{c}## to convert it to the sphere case where you can find the potential and multipole moments using spherical harmonics. But the whole calculation is messy because you have to find the fundamental vector product and all those stuff for the coordinate change and the expression for the angles in terms of these coordinates.

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