# Multipole Expansions

This lecture is based on HRW, Sections 23:5
• Scientists of all kinds often try to approximate a complicated phenomenon by adding together a few simple pieces.
• Fourier series
• Bessel functions
• Primary Component Analysis
• Spherical harmonics
Another of these methods, multipole expansion, is especially suitable for electric fields of common molecules.
• A monopole is a point charge. Its electric field falls off as 1/r^2.
• A dipole is a pair of opposite charges, separated by a small distance. Its electric field falls off as 1/r^3 at large distances
• A quadrupole is a set of 4 charges, 2 positive and 2 negative, arranged in a symmetric cross. Its electric field falls off as 1/r^4 at large distances

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An example of using the Fourier series to approximate a sawtooth wave.

A complicated function -- the sawtooth wave -- which we want to approximate with simple sinusoidal functions.

The first Fourier term: a single sinusoid.

How well does it fit the complicated function? Not very.

The second Fourier term.

Add together the first and second Fourier sinusoids -- how well do they fit the sawtooth?

The third Fourier term.

Add together the first, second and third Fourier sinusoids -- how well do they fit the sawtooth? Pretty well.

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