Maxwell's Equations

Released: May 9, 2026 8:00 AM CST

Here is the twenty-fourth episode of Quantum Foam, Maxwell's Equations. It is probably a good idea to go over a background on who came up with certain areas of science. James Clerk Maxwell derived equations using hydrodynamics, but there were others he used to build upon. We pay special attention to André-Marie Ampère, who the ampere is named after. And there was also Michael Faraday, who contributed vastly to the study of Electrochemistry and Electromagnetism. He didn't have any formal training. This is generally required for coming up with equations of Physics and Chemistry. Albert Einstein says that he has stood on the shoulders of James Clerk Maxwell. Isaac Newton's mathematical formalism was important here. We are doing some primers on the mathematics that are needed in order to understand Maxwell's Equations. We can show 2D and 3D mathematical grids. We describe unit vectors and other vectors. I-hat, j-hat, and k-hat components making up the XYZ components of vector A. This stuff can get dry but we need to make sure that we don't mess up magnitudes and vector specifics describing mathematical operations between vectors. In a 2D grid, you split up the X and Y components that are modeled after the X-axis and the Y-axis. This is done to describe different operations. We use the dot product and the cross product. The cross product is not exactly the same thing as multiplication. It can give you specifically the X component of a vector. The direction and size of a vector that is superimposed on another vector is apparent in its components. The cross product gives you the orthogonal vector component with a magnitude. The cross product between 2 vectors IS another vector. An example is firing a charged particle into a field. We are using sines and cosines. Other operations use the arcsine and arccosine. For example, the arcsine of x equals 0 is derived and given by the relationship sine of theta is equal to x. You may need a pen and paper to follow along with some of this. Relationship between vectors is paramount in the understanding of Maxwell's Equations.

We are interested in how this formalism relates to vector fields. A curl operator describes the infinite rotation of a vector field. Rotation at a point is what we are talking about. We theoretical physicists of the twentieth and twenty-first century are adding curl mathematics to Newton's Calculus today using computer analysis. There is the divergence operator and then the curl law. We can look at time-dependent surfaces to describe various integrals over surfaces or other structures. Charged density is key as well. Part of these equations are the same as Gauss' Law Of Magnetism. The law of inductions. This is work done by Faraday and Maxwell. We are looking at surfaces of integration. A way of adding up little pieces of reality. We are dealing with energy densities and flux. Maxwell is the father of Mechanical Engineering. He is one of the most well-known scientists since Sir Isaac Newton. I am standing on the shoulders of Albert Einstein and Stephen Hawking. Isaac Newton was the first to describe gravity and set the stage for developing further physics. Michael Faraday and James Clerk Maxwell formalized Electromagnetism with the current mathematics, which is otherwise known as The Calculus. Maxwell was from Edinburgh, Scotland and died at a young age of 48 in 1879. This also happens to be the year when Albert Einstein was born. Maxwell went to Cambridge after Edinburgh. His seminal work was in 1873 and was called Treatise On Electricity And Magnetism. This covered the partial differential equations demonstrating Electromagnetism. The mathematical flux between the commutability of electricity and magnetism was now one and the same. The induced forces by magnetism were of a new science. There are important equations that contribute to the standardization of the fundamental forces and these are included. This laid the groundwork for General Relativity. The equations have to do with a magnetic flux going through a material. There is also the Lorentz Force, which is the combination of electric and magnetic force on a point charge. Maxwell was also basically the father of Electrical Engineering. Understanding the energy densities of 2 different opposing fields was now possible. We need to know what the displacement current is in dealing with these equations. We need to know this for the entire equation. We are figuring out energy densities across a wave. This is an attempt to put everything into a nice, small formula. The formalisms outlined here are sure to be included when describing The Theory Of Everything. The permeability of the electric and magnetic fields can be defined using Maxwell's Equations.