Maxwell’s Equations; Simple Explanations and Mathematical Formulations


I obtained my Ph.D. degree in the field of Computational Electromagnetics (CEM). It is the science of numerical techniques that allow for fast and efficient computations of Maxwell’s equations.

In this post, my aim is to give a brief mathematical explanations of these 4 equations as well as simple explanations. If you are trying to understand these equations, the more you read about them, the more insight you will get. So I decided to share my own view in the hope that it helps you build a better understanding.

In section 2.1.1 of my Ph.D. dissertation (University of Manitoba 2016), I wrote:

In 1873, James Clerk Maxwell published A Treaties on Electricity and Magnetism, his complete presentation of the laws of electromagnetics (EM) which is regarded as one of the greatest contributions in the 19th century and the most impactful discovery in the history of humanity (according to Einstein). In it, he mathematically formulated/ corrected the EM laws that were discovered by Gauss, Faraday and Ampére through experiments. What is quite interesting (at least to me), is that for about a decade there was not a general understanding of his work as Maxwell made no effort to condense or simplify his work, aiming to be comprehensive rather than understandable. In the key chapter, entitled “General Equations of the Electromagnetic Field”, he wrote:

These may be regarded as the principal relations among the quantities we have been considering, some could be combined, but our object is not to obtain compactness in the mathematical formulae.

It was Oliver Heaviside who effectively discovered Maxwell’s equations from about the thousand pages of Maxwell’s treaties and presented them to the research community in 1883 similar to the form we know them today as four compact and understandable equations. As explained in the thesis, under certain conditions, Maxwell’s equations are as follows:

If you have any questions about the mathematical formulation, please let me know. You can post your questions in the comments section and I will be glad to help. The following 10 minutes video, gives a simple explanation of the equations.

This video is obtained from CrashCourse – YouTube.