Answer by hft for How to handle divergences in Poisson's equation in the...
Suppose an electron is moving through an electric field in some region...We know that the electron - as a charged particle - will experience a force from this field.Yes, but stop here for a second. The...
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Answer by basics for How to handle divergences in Poisson's equation in the...
Singularities are mathematical models of "small"-dimensional features. They provide information of what happens outside them.If your problem is to evaluate the force on a point particle, you could...
View ArticleAnswer by ACuriousMind for How to handle divergences in Poisson's equation in...
There is mathematically no problem with differentiating distributions like the $\delta$-distribution, you just need to use the notion of the distributional (or weak) derivative. The particular case of...
View ArticleAnswer by Puk for How to handle divergences in Poisson's equation in the...
Besides taking the electron to be a small ball of radius $r$ (and taking the limit $r\to 0$ if necessary after calculating the force), the other simple approach is to just not include the potential due...
View ArticleAnswer by Mauricio for How to handle divergences in Poisson's equation in the...
The mathematics is very clear, there is a singularity, the equation diverges at the particle position.However in physics, these kind of singularities are not a problem because they serve as effective...
View ArticleAnswer by JQK for How to handle divergences in Poisson's equation in the...
Maxwell's equations are valid everywhere except at charges. There's a long history of this problem the culminates in quantum electrodynamics (QED) that includes the self-energy of the electron. QED...
View ArticleAnswer by John Doty for How to handle divergences in Poisson's equation in...
Mostly it isn't resolved. The mathematical rabbit hole here is very deep, and poorly connected to real physics. The Euclidean point is not always a good model for an electron.
View ArticleHow to handle divergences in Poisson's equation in the presence of a point...
I'm looking for a (fairly) mathematically-rigorous resolution to the following). Suppose an electron is moving through an electric field in some region: $$\Omega \subset \mathbb{R}^d \ ,$$ where $d =...
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