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# Electromagnetism

As of the 20th century, this can be described well as "the phenomena described by Maxwell's equations".
Back through its history however, that was not at all clear. This highlights how big of an achievement Maxwell's equations are.

## Maxwell's equations (1861)

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Unified all previous electro-magnetism theories into one equation.
Explains the propagation of light as a wave, and matches the previously known relationship between the speed of light and electromagnetic constants.
The equations are a limit case of the more complete quantum electrodynamics, and unlike that more general theory account for the quantization of photon.
The equations are a system of partial differential equation.
The system consists of 6 unknown functions that map 4 variables: time t and the x, y and z positions in space, to a real number:
• , , : directions of the electric field
• , , : directions of the magnetic field
and two known input functions:
• : density of charges in space
• : current vector in space. This represents the strength of moving charges in space.
Due to the conservation of charge however, those input functions have the following restriction:
Equation 1. Charge conservation.
Also consider the following cases:
• if a spherical charge is moving, then this of course means that is changing with time, and at the same time that a current exists
• in an ideal infinite cylindrical wire however, we can have constant in the wire, but there can still be a current because those charges are moving
Such infinite cylindrical wire is of course an ideal case, but one which is a good approximation to the huge number of electrons that travel in a actual wire.
The goal of finding and is that those fields allow us to determine the force that gets applied to a charge via the Equation 6. "Lorentz force", and then to find the force we just need to integrate over the entire body.
Finally, now that we have defined all terms involved in the Maxwell equations, let's see the equations:
Equation 2. Gauss' law.
Equation 3. Gauss's law for magnetism.
Equation 5. Ampere's circuital law.
You should also review the intuitive interpretation of divergence and curl.

### Lorentz force

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Equation 6. Lorentz force.
A little suspicious that it bears the name of Lorentz, who is famous for special relativity, isn't it? See: Maxwell's equations require special relativity.

### Explicit scalar form of the Maxwell's equations

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For numerical algorithms and to get a more low level understanding of the equations, we can expand all terms to the simpler and more explicit form:

#### Overdetermination of Maxwell's equations

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As seen from explicit scalar form of the Maxwell's equations, this expands to 8 equations, so the question arises if the system is over-determined because it only has 6 functions to be determined.
As explained on the Wikipedia page however, this is not the case, because if the first two equations hold for the initial condition, then the othe six equations imply that they also hold for all time, so they can be essentially omitted.
It is also worth noting that the first two equations don't involve time derivatives. Therefore, they can be seen as spacial constraints.
TODO: the electric field and magnetic field can be expressed in terms of the electric potential and magnetic vector potential. So then we only need 4 variables?

### Coulomb's law

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Static case of Maxwell's law for electricity only.
The "static" part is important: if this law were true for moving charges, we would be able to transmit information instantly at infinite distances. This is basically where the idea of field comes in.

### Solutions of Maxwell's equations

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#### Maxwell's equations with pointlike particles

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In the standard formulation of Maxwell's equations, the electric current is a convient but magic input.
Would it be possible to use Maxwell's equations to solve a system of pointlike particles such as electrons instead?
The following suggest no, or only for certain subcases less general than Maxwell's equations:
This is the type of thing where the probability aspect of quantum mechanics seems it could "help".

### Maxwell's equations in 2D

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TODO it would be awesome if we could de-generalize the equations in 2D and do a JavaScript demo of it!
Not sure it is possible though because the curl appears in the equations:

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### Electric field ()

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#### Electric charge

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##### Electric current
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In the context of Maxwell's equations, it is vector field that is one of the inputs of the equation.
Section "Maxwell's equations with pointlike particles" asks if the theory would work for pointlike particles in order to predict the evolution of this field as part of the equations themselves rather than as an external element.

#### Electric potential ()

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##### Volt (V)
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###### Electronvolt
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After the 2019 redefinition of the SI base units it is by definition exactly Joules.

### Hall effect

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The voltage changes perpendicular to the current when magnetic field is applied.
An intuitive video is:
The key formula for it is:
where:
Applications:
• the direction of the effect proves that electric currents in common electrical conductors are made up of negative charged particles
• measure magnetic fields, TODO vs other methods
Other more precise non-classical versions:

#### Hall resistance (, )

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In some contexts, we want to observe what happens for a given fixed magnetic field strength on a specific plate (thus and are also fixed).
In those cases, it can be useful to talk about the "Hall resistance" defined as:
So note that it is not a "regular resistance", it just has the same dimensions, and is more usefully understood as a proportionality constant for the voltage given an input current:
This notion can be useful because everything else being equal, if we increase the current , then also increases proportionally, making this a way to talk about the voltage in a current independent manner.
And this is particularly the case for the quantum Hall effect, where is constant for wide ranges of applied magnetic field and TODO presumably the height can be made to a single molecular layer with chemical vapor deposition of the like, and if therefore fixed.

### Electromagnetic four-potential

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A different and more elegant way to express Maxwell's equations by using the:

#### Lorentz gauge condition

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There are several choices of electromagnetic four-potential that lead to the same physics.
E.g. thinking about the electric potential alone, you could set the zero anywhere, and everything would remain be the same.
The Lorentz gauge is just one such choice. It is however a very popular one, because it is also manifestly Lorentz invariant.
##### Coulomb gauge
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Alternative to the Lorentz gauge, but less used in general as it is not as nice for relativity invariance.

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Implementations: