# Week 1: Learning the Theory

March 1, 2024

Hello my fellow human beings. Welcome back to my blog, where I will be tracking my progress on my EM field simulation project. This week, I worked on reviewing some theoretical knowledge on the behavior of electromagnetic fields. This knowledge will be necessary in the future when I develop, test, and debug the simulation algorithm.

For the first two days, I reviewed the basics of electrostatics. This subfield of electrodynamics aims to describe the behavior of electric fields when all the charges are at rest (hence electro*statics*). The main building blocks of electrostatics are Gauss’ Law and Coulomb’s Law. Both laws relate the behavior of the electric field to the distribution of the electric charge producing the field. Together, the two laws predict that the electrostatic field is both spherically symmetrical and, like the Newtonian gravitational field, obeys the inverse square law.

Gauss’ Law is Lorentz invariant, which means that the law remains the same when shifting relativistically from one inertial frame to another. Coulomb’s Law, however, is not Lorentz invariant. This means that electrostatic theory fails for moving charges. As we take into account systems of charges moving relativistically, the situation becomes more complicated. According to Maxwell’s equations, charges moving relativistically produce both electric and magnetic fields. Moreover, the force experienced by a charge moving through an EM field is given by the Lorentz force law, which essentially expresses that the total force can be divided into a magnetic and an electric force.

For the next two days of the project, I reviewed the basics of magnetostatics, which investigates the behavior of magnetic fields produced by steadily moving charges. The main building blocks of magnetostatics are Ampere’s Law and the Biot-Savart Law, which are essentially analogous to Gauss’ Law and Coulomb’s Law for electrostatics. Together, these laws state that magnetic fields arise from moving charges and that there are no magnetic monopoles.

Of course, both magnetostatics and electrostatics are both subsets of the full theory of classical electrodynamics in which certain assumptions are made to simplify the physical analysis. However, situations in both areas are easy to solve by hand and offer a more intuitive understanding of electrodynamics. This makes electrostatic and magnetostatic systems optimal for constructing simple tests for my algorithm.

In my next week, I will review the other two Maxwell’s equations (in addition to the ones we already have reviewed, which are Gauss’ and Ampere’s Laws). Moreover, I will begin finding and comparing algorithms for solving Maxwell’s equations. Goodbye, my fellow human beings.

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