Making sense of Maxwell’s Equations

Ralf Gerlich


I’m currently learning about modelling of electromagnetic systems, and in that process, I’m trying to wrap my head around Maxwell’s Equations. Maxwell’s Equations can be used to determine electric and magnetic fields, which are important for machines using them for work, such as electrical motors.

There are some aspects to those that are similar to fluid dynamics, with electrical fields having their source in electrical charges, while magnetic fields are free of sources. Much of that is expressed using curl and divergence. These are differential operators used to characterise vector fields.

I came across a pretty good video by 3Blue1Brown of Khan Academy fame introducing the concepts of curl and divergence, and also showing the connection to fluid and electrodynamics. The other video, which goes over Maxwell’s Equations and explains them pretty well is by DrPhysicsA.

Be careful, though: At or around minute 18:13, DrPhysicsA explains how Maxwell’s Second Law (or Gauss’ Law for Magnetism) breaks down when you place your control surface such that only one pole of the magnet is inside and the other is outside. According to DrPhysicsA, that would lead to non-zero flow through the control surface, violating Gauss’ Law for Magnetism.

But actually, that is not the case. One has to consider that the field also flows inside the magnet. Namely, there is flow between the north- and south-pole inside the magnet in opposite direction to that outside the magnet. Thus, the flow inside the magnet and the flow outside that magnet through the control surface still cancel out. Gauss’ Law for Magnetism therefore very elegantly states that magnetic poles always come in pairs.