Differential Form of Gauss's Law

4 hours ago 1

Before you read this, I suggest you read posts 25.9 and 25.11

In 1865, the Scottish physicist Maxwell published a series of differential equations that described the properties of electromagnetic fields. But he had published several previous papers on the subject while he was developing his ideas. Maxwell derived over 20 equations to express his final conclusions. But, using vector notation, they can be expressed as the four independent equations that we now call Maxwell’s equations. These vector equations were published in 1884 by the English electrical engineer, Oliver Heaviside (1850-1925). Heaviside is remarkable for never having attended university – his parents were too poor to support him as a student. He also gives his name to the Heaviside layer (sometimes called the Kennelly-Heaviside layer), a region of the upper atmosphere that reflects radio waves. (What’s the connection between the Heaviside layer and the introduction to post 23.5?)

The first two of the four Maxwell’s equations are Gauss’s law for an electric field and a magnetic field. But they are not expressed in the same way as in post 25.11. Instead, they are expressed as partial differential equations, known as the differential forms of Gauss’s law. They are shown (highlighted in yellow and green) at the beginning of this post.

To derive these forms, we need to use Gauss’s divergence theorem (also called the divergence theorem). In post 25.9, I defined the flux of a vector field, F, by

Here u is a unit vector perpendicular to an elemental area δA. The integration is over the whole surface S that encloses a volume of space. I am now going to define the elemental area as a vector δA whose direction is specified by the direction of u; F.u is the dot product of F and u. Now I can define the flux of F as

According to Gauss’s divergence theorem (see appendix), we can then write

Here the vector operator ∇ is defined by

(see post 20.34) and the definite integral on the right-hand side is over the whole enclosed volume, V.

Electric field

According to post 25.11, the electric flux is given by

where q is the total charge in a defined volume of space. From equation 2,

when the definite integral is evaluated over the whole defined volume. From equations 3 and 4

which is Gauss’s law for electric fields in differential form and Maxwell’s first equation. This result can also be written as

(see post 20.34).

Magnetic field

According to post 25.11, the magnetic flux, in a defined volume of space, is given by

From equation 2,

When the definite integral is evaluated over the whole defined volume of space. So, from equations 5 and 6,

which is Gauss’s law for magnetic fields in differential form and Maxwell’s second equation. It can also be written

Appendix – Gauss’s divergence theorem

The purpose of this appendix is to explain the idea of the divergence theorem but not to give a rigorous proof.

The divergence theorem states that

We can write

where F_x, F_y and F_z are the x, y and z components of F.

Now let’s consider the field in an infinitessimally small cube in space whose sides define the x, y and z axes of an orthogonal Cartesian coordinate system. We’ll start by thinking about the x component of the field. The change in F_x in moving from one face of the cube to the opposite face, in the x-direction (a distance δx), is given by

Now δy and δz are the faces of the cube perpendicular to the x-direction so that

Similarly

Now let’s think of a surface made by putting lots of these cubes together. At a point where the resultant field is F, the flux is given by

where δA is an elemental area surrounding the point whose direction (perpendicular to the surface) is the direction of F.

Now let’s suppose that our surface encloses a volume. Then the total flux at a point is given by

where, in each case, δA is perpendicular to the component of F. Since equation 7 was derived for a cube, δA is the same whatever its orientation. From equations 8 and 10