Quite a while ago I posted a blog about the statue in Edinburgh to the Scottish mathematical physicist James Clerk Maxwell. On this statue are his famous equations, Maxwell’s equations, which describe electromagnetism. I also mentioned these equations in my blog about James Clerk Maxwell in my list of the “ten best physicists” (here and here).

Let me remind you of the four equations we now know as Maxwell’s equations. They succinctly describe electromagnetism, and are the equivalent in electromagnetic theory to Newton’s laws of motion and his three equations of motion in classical mechanics.

Here I have written equation (1), also known as Gauss’s equation, in terms of the electric displacement field . This is related to the electric field in a vacuum via its definition, , where is the permittivity of free space [in the case of the displacement field when it is not in a vacuum, the equation becomes where is the density of the induced and permanent electric dipole moment, also known as the polarisation density]. So we can also write Maxwell’s first equation as

In this previous blog I discussed the vector differential operator, often known as “del” and given the symbol in mathematics. Today I thought I would discuss a key concepts in vector field theory, namely the *“divergence”* of a vector field. This crops up a lot in physics and engineering, and going back to Maxwell’s equations, the first two are the divergence of the electric and magnetic fields respectively.

## The divergence

The vector operator, is also known as the *divergence* (or *div*) of the vector field. Mathematically, it is

The resulting quantity is a *scalar*, as the so called *dot* product produces a scalar (see my blog on the scalar product of two vectors here).

What the divergence of a vector field is actually measuring is how much of the vector field flows outwards from a given point. This is best illustrated by a few diagrams. First I will show a uniform vector field, then secondly a vector field with *no divergence*, even though it is not a uniform vector field.

In these last two diagrams I show two examples of vector fields with divergence. The first shows positive divergence (a source), the second shows negative divergence (a sink).

In a future blog I will explain how the divergence of an electric field is related to the charges enclosed within a surface surrounding the charges (Gauss’s law or the first of Maxwell’s equations).

on 26/05/2014 at 16:56 |Clive AlabsterAm I not right in thinking that Maxwell originally wrote his equations describing EM propagation using quarterions and that it was Oliver Heaviside who put them into the vector form we are perhaps more familiar with today? Clive Alabaster (Hi there, old chum!)

on 26/05/2014 at 17:03 |RhEvansHi Clive, lovely to hear from you!!

Yes, you are absolutely correct. The 4 “Maxwell Equations” we now know and love were written in that form my Heaviside, in Maxwell’s papers the same information was written in about 20-odd equations.

on 27/05/2014 at 00:03 |Estelle AsmodelleThanks Rhodri for this post, I particularly like the visual representation of the vector fields, nice post.

on 27/05/2014 at 02:40 |RhEvansThank you Estelle. I always found when I learnt about divergence and curl of a vector field that visual representations of what they meant were lacking in the lectures.

on 14/01/2015 at 20:27 |john cThis was very useful to me, thanks. I started a project to learn what I could about Maxwell’s equations (over a year ago) and am making intermittent progress. Please tell me how I can find other significant material on your blog ( see a long list of general subjects). I need to get some idea of the relevent Calculus also, if it’s available on your logs.

on 14/01/2015 at 22:25 |RhEvansThank you. There should be a search box in the top right of the blog page.