Gauss's Law,
the Divergence Theorem,
and the Electric Field

Theory

The Divergence Theorem

The Divergence Theorem relates the divergence of a function within some region to the values of that function on the boundary of that region. Let be some volume of space, and let be its surface. If is the function in question, then

or

where we have introduced the "Del-dot" symbol for the divergence operator.

Gauss's Law

Gauss's law for the electric field says that the electric flux through any closed surface is proportional to the amount of electric charge contained within that surface. Again let be some volume of space, and let be its surface. Then

where is the total charge contained in . We can write this in terms of the charge density as

so that Gauss's law becomes

where is a fundamental constant. You can look at a not-entirely rigorous proof.

Gauss's Law in Differential Form

The formulas for the Divergence Theorem and Gauss's Law have some similarities, which suggest the following development of Gauss's law into a differential form. The Divergence Theorem tells us that

(we have replaced F with E), so that

Now, this is true for any region , which is only possible if the integrands are equal:

which is Gauss's Law in differential form. Notice that while the integral form was concerned with the behavior of the electric field and the charge density over some spread-out region, the differential form is about their behavior at a point.

Exercise

Exploring a Field

In this exercise you will explore the electric field of a (not necessarily uniformly) charged cylinder. The cylinder is much bigger than the applet screen: (not to scale). Because of this, you cannot explore the region outside the cylinder. (Or if you can, you shouldn't.)

Use the following tools to explore the field:

1. Watch the field arrow as it grows and shrinks.
2. Shift-control-alt-left click (or S-C-middle) to drop a field arrow.
3. Shift-right to change the field arrow into a divergence meter. See how its edges bow outward or inward.
4. Shift-alt-left click (or S-middle) to drop a divmeter.
5. Click the right button to draw a field line. The color along the line indicates the strength of the field; red is strong, and blue is weak.
6. Click the middle button (or alt-left) to draw an equipotential.
7. Draw a (green) surface for Gauss's law:
   • Shift-left drag to draw a rectangle.
   • Shift-control-left drag to draw a circle. The applet calculates and prints the amount of charge within the surface. Click the left button again to erase the surface.
8. Drop an array of indicators:
   • To drop field arrows, hit "A".
   • To drop divergence meters, hit "D".
9. Hit "E", backspace, or delete to erase the lines and laws.

Confused

Questions

1. Where is the center of the cylinder?

2. How does the charge density change with the distance from the center? It is a polynomial. (Use the circle-drawing tool.)

3. Is there any point where the divergence of the electric field is equal to zero?

4. Verify the Divergence Theorem.

Confusion to Avoid

Sometimes, particularly in math textbooks, you will see the Divergence Theorem referred to as "Gauss's Theorem". This is confusing but not incorrect. Be sure you do not confuse Gauss's Law with Gauss's Theorem. The Law is an experimental law of physics, while the Theorem is a mathematical law that depends only on the definitions of field, divergence, and surface and volume integrals.

Gauss, like Euler, was a little too prolific for his own good. He discovered many more things than can be named for him without creating confusion.