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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).

## The 10 best physicists – no. 5 – James Clerk Maxwell

At number 5 in The Guardian’sten best physicists” is the Scottish theoretical physicist James Clerk Maxwell.

As the caption above says, James Clerk Maxwell is probably one of the least known of the physicists in this list, and yet one of the most important. It was through his theoretical work that a full understanding of electromagnetism became possible. His four equations, known as Maxwell’s equations, are as important to understanding electromagnetism as Newton’s laws of motion are to understanding mechanics. In some ways, Maxwell was to Faraday what Newton was to Galileo. Neither Galileo nor Faraday had the mathematical ability to write in mathematics the experimental results they obtained in mechanics and electricity and magnetism respectively. Newton wrote the mathematical equations to explain Galileo’s experiments in mechanics; and similarly Maxwell wrote the mathematical equations to explain the experiments Faraday had been doing on electricity and magnetism.

## Maxwell’s brief biography

James Clerk was born in 1831 in Edinburgh, Scotland. He was born into a wealthy family, and later added Maxwell to his name when he inherited an estate as part of his wider family’s wealth. His mother died when Maxwell was eight years old, and at ten years of age his father sent him to the prestigious Edinburgh Academy.

At the age of 16, in 1847, Maxwell left Edinburgh Academy and started attending classes at Edinburgh University. His ability in mathematics quickly became apparent. At the age of only 18 he contributed two papers to the Transactions of the Royal Society of Edinburgh, and in 1850, at the age of 19, he transferred from Edinburgh University to Cambridge. Initially he attended Peterhouse College, but transferred to the richer and better known Trinity College. He graduated from Trinity College Cambridge in 1854 with a degree in mathematics. He graduated second in his class.

In 1855 he was made a fellow of Trinity College, allowing him to get on with his research. But in November 1856 he left Cambridge, being appointed Professor of Natural Philosophy at the University of Aberdeen in Scotland. He was 25 years of age, and head of the Natural Philosophy department, some 15 years younger than any of his colleagues there.

Maxwell left Aberdeen in 1860 to become Professor of Natural Philosophy at Kings College, University of London. He resigned in 1865 to return to his estate in Scotland, and for the next 6 years he got on with his research, and due to being independently wealthy he didn’t need a formal paying position at a university. But in 1871 he was tempted back into university life, he was appointed the first Cavendish Professor of Physics at Cambridge University, and given the role of establishing the Cavendish Laboratory, which opened in 1875.

Maxwell died in 1879 of abdominal cancer, and is buried in Galloway, Scotland, near where he grew up.

## Maxwell’s scientific legacy

Maxwell is best known amongst physicists for two main areas of work, his work on electromagnetism and his work on the kinetic theory of gases. As I mentioned above, his work on electromagnetism can be likened to a certain extent to Newton’s work on mechanics. Newton was able to take the experimental results Galileo had found on the behaviour of moving bodies, and find the underlying laws of mechanics to explain them, and to express these in mathematical form. This is essentially what Maxwell did for the work of Faraday, he was able to find the laws which explained the observations Faraday had been making, and was a sufficiently skilled mathematician to write these laws in equation form.

Maxwell first presented his thoughts on electromagnetism in a paper in 1855, when he was 24 years old. In this he reduced all the known knowledge of electromagnetism as it existed in 1855 into a set of differential equations with 20 equations and 20 variables. In 1862 he showed that the speed of propagation of an electromagnetic field was approximately the same as the speed of light, something Maxwell thought was unlikely to be a coincidence. This was the first indication that light was a form of electromagnetism, but a form to which are eyes are senstive.

In 1873 Maxwell reduced the 20 differential equations of his 1855 paper into the four differential equations which are familiar to every undergraduate physics student. These equations, expressed in their differential form are
$\boxed{ \begin{array}{lcll} \nabla \cdot \vec{D} & = & \rho & (1) \\ & & & \\ \nabla \cdot \vec{B} & = & 0 & (2) \\ & & & \\ \nabla \times \vec{E} & = & - \frac{\partial \vec{B}}{\partial t} & (3) \\ & & & \\ \nabla \times \vec{H} & = & - \frac{\partial \vec{D}}{\partial t} + \vec{J} & (4) \end{array} }$

The symbol $\nabla$ is known as the vector differential operator, or del, and I have explained del in this blog.

The four equations can also be written in integral form, which many people find easier to understand. In integral form, the equations become

$\boxed{ \begin{array}{lcll} \iint_{\partial \Omega} \vec{D} \cdot d\vec{S}& = & Q_{f}(V) & (5) \\ & & & \\ \iint_{\partial \Omega} \vec{B} \cdot d\vec{S} & = & 0 & (6) \\ & & & \\ \oint_{\partial \Sigma} \vec{E} \cdot d\vec{\l} & = - & \iint_{\Sigma} \frac{\partial \vec{B} }{\partial t} \cdot d\vec{S} & (7) \\ & & & \\ \oint_{\partial \Sigma} \vec{H} \cdot d\vec{l} & = & I_{f} + \iint_{\Sigma} \frac{\partial \vec{D} }{\partial t} \cdot d\vec{S} & (8) \end{array} }$

## Kinetic theory of gases

Maxwell also did important work on one of the other main areas of physics research in the 19th Century – the kinetic theory of gases, a branch of thermodynamics. Maxwell and the Austrian physicist Ludwig Boltzmann independently came up with an expression which describes the distribution of speeds of the atoms or molecules in a gas. The speed depends on the temperature, the higher the temperature the faster the atoms or molecules move. But Maxwell and Boltzmann showed that not all the atoms will be moving at this speed; some will be moving faster and some will be moving slower. The distribution of speeds is known as the Maxwell-Boltzmann distribution.

## The theory of colour vision

Less well known to most physicists is the work Maxwell did on optics, and on colour vision. He was the first person to show how a colour image could be created from combining three images taken through red, green and blue filters. This lies at the foundation of practical colour photography, and is also how our colour television sets work.

I think it is true to say that most physicists would place Maxwell in their top four physicists. His contributions to physics are immense, creating the formal framework for our understanding of electromagnetism, without which the modern world would not be possible. That he is so unknown outside of the world of physics is strange, but whereas Newton had his $F=ma$ and Einstein his $E=mc^{2}$, Maxwell’s equations are far too complicated to have permeated into the consciousness of the public. But they are every bit as important as Newton and Einstein’s.