# Maxwell's Eqns: EM Basis Functions Explored

• Dale
In summary, the various bases that are useful for Maxwell's equations depend on the specific problem at hand.

#### Dale

Mentor
Are there any sets of basis functions that are particularly useful for Maxwell's equations? I was thinking about Fourier just because it is the first basis I always think of, but I don't know that it would actually be a convenient basis. For example, I don't know that curl or divergence would be easy in the Fourier basis.

"Maxwell's Equations" are pretty open ended. There are bases that are useful for solving certain problems.
1. For scalar potential and Green function problems in magneto- and electro-statics, cylindrical and spherical harmonics come to mind as particularly useful. There are two different cylindrical harmonic bases as well, those with cosine/sines and ordinary Bessels, and those with cosh/sinh and modified Bessels. There are different versions of the spherical expansions, too. The alternate expansions sometimes have particular advantages, see
http://physics.princeton.edu/~mcdonald/examples/EM/cohl_apj_527_86_99.pdf" [Broken]
and
http://iopscience.iop.org/0022-3727/19/8/001?ejredirect=migration"
for two interesting applications.
2. Similar harmonics exist of course for spheroidal and all the other 8 or so coordinate systems that are separable under Laplace's equation.
3. The harmonic bases are useful for radiation problems as well (e.g., multipoles). For full EM wave problems that include polarization, the vector forms of the harmonic bases are used. Vector spherical harmonics are the best known of these.
4. Sines and cosines (your Fourier bases) are useful for problems in rectangular coordinates.

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Spin functions seem to be basis functions of the equations. I've heard it expressed that quaterions can be reduced to a pair of complex-valued 2 component spinors. And quaterions are the most reduced form of transformations for SR. Any combination of rotations or boosts, inclusive or exclusive, can be performed most simply with spinors and quaterions.

Since SR is a particular analysis of Maxwell's Equations regarding time and length (minus the conjectural things Einstein added) spinors and quaterions must be the most reduced form of transformations of variables with the Maxwell Equations.

Quaterions function in that way apparently because they encode symmetry properties which vectors do not. At least that is my understanding.

I happen to use the vector spherical harmonics recently. They are the basis functions to describe the electric and magnetic field in the spherical coordinate. However, I find the derivation of the vector spherical harmonics in Jackson's a little awkward. Bohren's book on absorption and scattering of light by small particles has a different derivation I find easier to chew.

Thanks for the replies so far! I looked into Fourier a bit more and I found that divergence and curl can be represented by reasonably straightforward operations on the Fourier domain, but then I realized that what is really needed is a basis where the inverse of the curl and divergence has a simple representation. Unfortunately, I got stuck there. I will look into the details of what you have all posted over the next day or so.

## 1. What are Maxwell's equations?

Maxwell's equations are a set of four fundamental equations in classical electromagnetism that describe the behavior of electric and magnetic fields, as well as their interactions with matter. They were first developed by James Clerk Maxwell in the 1860s and are essential for understanding the principles of electricity and magnetism.

## 2. What do Maxwell's equations describe?

Maxwell's equations describe the relationships between electric and magnetic fields, electric charges, and electric currents. They also describe how these fields and charges interact with each other and with matter. These equations play a crucial role in understanding the behavior of electromagnetic waves, such as light, and in the design of technologies such as radios, televisions, and cell phones.

## 3. What are the four Maxwell's equations?

The four Maxwell's equations are Gauss's law, which describes the relationship between electric fields and electric charges; Gauss's law for magnetism, which describes the relationship between magnetic fields and magnetic charges; Faraday's law of induction, which describes how changing magnetic fields can create electric fields; and Ampere's law, which relates magnetic fields to electric currents.

## 4. What is the importance of Maxwell's equations?

Maxwell's equations are considered to be one of the most significant achievements in the history of physics. They are essential for understanding and predicting the behavior of electromagnetic waves, such as light, and for developing technologies that rely on these waves. Maxwell's equations also played a crucial role in the development of Einstein's theory of relativity and the modern understanding of the nature of light.

## 5. How are Maxwell's equations used in science and technology?

Maxwell's equations are used in a wide range of fields, including telecommunications, electronics, and optics. They are essential for designing and optimizing technologies such as cell phones, satellite communication, and medical devices that use electromagnetic waves. They are also used in many scientific fields, including astrophysics, plasma physics, and geophysics, to understand and study the behavior of electromagnetic fields in various systems.

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