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.
Coulomb's Law experiment with torsion balance with a mirror on the balance to amplify rotations by uclaphysics (2010)
Source. 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.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".
Existence and uniqueness of solutions to Maxwell's equations by 
Ciro Santilli 37 Updated 2025-07-16
Ciro Santilli 37 Updated 2025-07-16In 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.
Hall effect experimental diagram
. Source. The Hall effect refers to the produced voltage , AKA on this setup.An intuitive video is:
The key formula for it is:where:
- : current on x direction, which we can control by changing the voltage
- : strength of transversal magnetic field applied
- : charge carrier density, a property of the material used
- : height of the plate
- : 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:
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.
A different and more elegant way to express Maxwell's equations by using the:instead of the:
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.
Pinned article: Introduction to the OurBigBook Project
Welcome to the OurBigBook Project! Our goal is to create the perfect publishing platform for STEM subjects, and get university-level students to write the best free STEM tutorials ever.
Everyone is welcome to create an account and play with the site: ourbigbook.com/go/register. We belive that students themselves can write amazing tutorials, but teachers are welcome too. You can write about anything you want, it doesn't have to be STEM or even educational. Silly test content is very welcome and you won't be penalized in any way. Just keep it legal!
Intro to OurBigBook
. Source. We have two killer features:
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This feature makes it possible for readers to find better explanations of any topic created by other writers. And it allows writers to create an explanation in a place that readers might actually find it.
Figure 1. Screenshot of the "Derivative" topic page. View it live at: ourbigbook.com/go/topic/derivativeVideo 2. OurBigBook Web topics demo. Source. - local editing: you can store all your personal knowledge base content locally in a plaintext markup format that can be edited locally and published either:This way you can be sure that even if OurBigBook.com were to go down one day (which we have no plans to do as it is quite cheap to host!), your content will still be perfectly readable as a static site.
- to OurBigBook.com to get awesome multi-user features like topics and likes
- as HTML files to a static website, which you can host yourself for free on many external providers like GitHub Pages, and remain in full control
Figure 3. Visual Studio Code extension installation.
Figure 4. Visual Studio Code extension tree navigation.
Figure 5. Web editor. You can also edit articles on the Web editor without installing anything locally.Video 3. Edit locally and publish demo. Source. This shows editing OurBigBook Markup and publishing it using the Visual Studio Code extension.Video 4. OurBigBook Visual Studio Code extension editing and navigation demo. Source. 
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