Electromagnetism

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:

$E_{x} (t, x, y, z)$ , $E_{y} (t, x, y, z)$ , $E_{z} (t, x, y, z)$ : directions of the electric field $E : R^{4} \to R^{3}$
$B_{x} (t, x, y, z)$ , $B_{y} (t, x, y, z)$ , $B_{z} (t, x, y, z)$ : directions of the magnetic field $B : R^{4} \to R^{3}$

and two known input functions:

$ρ : R^{3} t o R$ : density of charges in space
$J : R^{3} \to R^{3}$ : 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:

\frac{\partial ρ}{\partial t} + \nabla \cdot J = 0

Equation 1.

Charge conservation

Also consider the following cases:

if a spherical charge is moving, then this of course means that $ρ$ is changing with time, and at the same time that a current exists
in an ideal infinite cylindrical wire however, we can have constant $ρ$ 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

E

and

B

is that those fields allow us to determine the force that gets applied to a charge via the Equation "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:

d i v E = \frac{ρ}{ε _{0}}

(2)

Equation 2.

Gauss' law

d i v B = 0

(3)

Equation 3.

Gauss's law for magnetism

\nabla \times E = - \frac{\partial B}{\partial t}

(4)

Equation 4.

Faraday's law

\nabla \times B = μ_{0} (J + ε_{0} \frac{\partial E}{\partial t})

(5)

Equation 5.

Ampere's circuital law

You should also review the intuitive interpretation of divergence and curl.

Lorentz force

force_density = ρ E + J \times B

Equation 1.

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.

Ampère's force law

Force between two wires carrying an electric current.

Caused by the Lorentz force.

Explicit scalar form of the Maxwell's equations

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:

\frac{\partial E _{x}}{\partial x} + \frac{\partial E _{y}}{\partial x} + \frac{\partial E _{z}}{\partial x} = \frac{ρ}{ε _{0}} \frac{\partial B _{x}}{\partial x} + \frac{\partial B _{y}}{\partial x} + \frac{\partial B _{z}}{\partial x} = 0 \frac{\partial E _{z}}{\partial y} - \frac{\partial E _{y}}{\partial z} = - \frac{\partial B _{x}}{\partial t} \frac{\partial E _{x}}{\partial z} - \frac{\partial E _{z}}{\partial x} = - \frac{\partial B _{y}}{\partial t} \frac{\partial E _{y}}{\partial x} - \frac{\partial E _{x}}{\partial y} = - \frac{\partial B _{z}}{\partial t} \frac{\partial B _{z}}{\partial y} - \frac{\partial B _{y}}{\partial z} = μ_{0} (J_{x} + ε_{0} \frac{\partial E _{x}}{\partial t}) \frac{\partial B _{x}}{\partial z} - \frac{\partial B _{z}}{\partial x} = μ_{0} (J_{y} + ε_{0} \frac{\partial E _{y}}{\partial t}) \frac{\partial B _{y}}{\partial x} - \frac{\partial B _{x}}{\partial y} = μ_{0} (J_{z} + ε_{0} \frac{\partial E _{z}}{\partial t})

Equation 1

Overdetermination of Maxwell's equations

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?

Bibliography:

physics.stackexchange.com/questions/20071/do-maxwells-equations-overdetermine-the-electric-and-magnetic-fields

Coulomb's law

Static case of Maxwell's law for electricity only.

Is implied by Gauss' law if Maxwell's equations: physics.stackexchange.com/questions/44418/are-the-maxwells-equations-enough-to-derive-the-law-of-coulomb

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.

Video 1.

Coulomb's Law experiment with torsion balance with a mirror on the balance to amplify rotations by uclaphysics (2010)

Source.

Solutions of Maxwell's equations

Video 1.

Understanding Electromagnetic Radiation! by Learn Engineering (2019)

Source. Shows animations of a dipole antenna which illustrates well how radiation is emitted from moving charges and travels at the speed of light.

Maxwell's equations with pointlike particles

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

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:

Existence and uniqueness of solutions to Maxwell's equations

TODO: I'm surprised that the Wiki page barely talks about it, and there are few Google hits too! A sample one: www.researchgate.net/publication/228928756_On_the_existence_and_uniqueness_of_Maxwell's_equations_in_bounded_domains_with_application_to_magnetotellurics

Electric field ( $E$ )

Electric charge

Electric charge measure unit

Coulomb (C)

Charge conservation

Electric current

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.

Measured in amperes in the International System of Units.

Electric potential ( $ϕ$ )

Volt (V)

Voltmeter

Nanovoltmeter

Voltage

Electronvolt

After the 2019 redefinition of the SI base units it is by definition exactly

1.6021766341 0^{- 19}

Joules.

Hall effect

The voltage changes perpendicular to the current when magnetic field is applied.

Figure 1.
Hall effect experimental diagram
. Source. The Hall effect refers to the produced voltage $V_{H}$ , AKA $V_{y}$ on this setup.

An intuitive video is:

Video 1. Source.

The key formula for it is:

V_{H} = \frac{I _{x} B _{z}}{n t e}

where:

$I_{x}$ : current on x direction, which we can control by changing the voltage $V_{x}$
$B_{z}$ : strength of transversal magnetic field applied
$n$ : charge carrier density, a property of the material used
$t$ : height of the plate
$e$ : electron charge

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:

quantum Hall effect

Bibliography:

hyperphysics.phy-astr.gsu.edu/hbase/magnetic/Hall.html

Hall resistance ( $R_{x y}$ , $R_{H}$ )

In some contexts, we want to observe what happens for a given fixed magnetic field strength on a specific plate (thus

t

and

n

are also fixed).

In those cases, it can be useful to talk about the "Hall resistance" defined as:

R_{x y} = \frac{V _{y}}{I _{x}}

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

I_{x}

current:

V_{y} = R_{x y} I_{x}

(2)

This notion can be useful because everything else being equal, if we increase the current

I_{x}

, then

V_{y}

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

R_{x y}

is constant for wide ranges of applied magnetic field and TODO presumably the height

t

can be made to a single molecular layer with chemical vapor deposition of the like, and if therefore fixed.

Charge carrier density ( $n$ )

Hall effect sensor

Electromagnetic four-potential

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

instead of the:

Electromagnetic tensor ( $F^{μ ν}$ )

Four-current

Lorentz gauge condition