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 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).