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## Divergence of a vector field

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.

$\boxed{ \begin{array}{lcll} \nabla \cdot \vec{D} & = & \rho & (1) \\ & & & \\ \nabla \cdot \vec{B} & = & 0 & (2) \\ & & & \\ \nabla \times \vec{E} & = & - \frac{\partial \vec{B}}{\partial t} & (3) \\ & & & \\ \nabla \times \vec{H} & = & - \frac{\partial \vec{D}}{\partial t} + \vec{J} & (4) \end{array} }$

Here I have written equation (1), also known as Gauss’s equation, in terms of the electric displacement field $\vec{D}$. This is related to the electric field $\vec{E}$ in a vacuum via its definition, $\vec{D} = \epsilon_{0} \vec{E}$, where $\epsilon_{0}$ is the permittivity of free space [in the case of the displacement field when it is not in a vacuum, the equation becomes $\vec{D} = \epsilon_{0} \vec{E} + \vec{P}$ where $\vec{P}$ 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

$\nabla \cdot \vec{E} = \frac{ \rho }{ \epsilon{0} }$

In this previous blog I discussed the vector differential operator, often known as “del” and given the symbol $\nabla$ 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 $\vec{E} \text{ and } \vec{B}$ respectively.

## The divergence $\nabla \cdot$

The vector operator, $\nabla \cdot \vec{F}$ is also known as the divergence (or div) of the vector field. Mathematically, it is

$\nabla \cdot \vec{F} = \frac{\partial F_{x}}{\partial x} + \frac{\partial F_{y}}{\partial y} + \frac{\partial F_{z}}{\partial z}$

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.

A uniform vector field, each vector has the same direction and size.

A vector field with no divergence. This could be, for example, a vector field representing the flow of cars either side of a central reservation (middle row), with the length of the vectors showing the cars are going faster further from the central line.

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).

A vector field with divergence. This example shows positive divergence, or a “source” of flux.

This shows negative divergence, or a “sink” of flux.

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).

### 6 Responses

1. Am 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!)

• Hi 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.

2. on 27/05/2014 at 00:03 | Reply Estelle Asmodelle

Thanks Rhodri for this post, I particularly like the visual representation of the vector fields, nice post.

• Thank 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.

3. This 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.

• Thank you. There should be a search box in the top right of the blog page.