Introduction

Gauss was a prolific guy, you will find him all over math and physics. It makes sense that he created a few theorems for vector calculus, namely the Divergence Theorem and the more famous Gauss' Law. We shall get an intuition for both.

Gauss' Divergence Theorem

This follows naturally from our intuitive explanation of flux and div. We know that div is the flux density, or flux/volume. How do we find the flux given the div? It's simple: multiply by volume. This is just like calculating the mass of an object given its density and volume.

Flux = div * volume

Integration is usually required because we are considering regions of varying div over a region, just like calculating the mass of an object with varying density. To do this, we take a tiny volume, multiply by its divergence, take another tiny volume, multiply by its div, and keep adding these tiny mass elements up. Thus,

Flux = Integral(div dv)

This is the Divergence Theorem... it's pretty easy once you know how to look at flux and div.

Math

The equation is straightforward, it follows from Flux = div * volume:

Gauss' Law

This is the big boy, you'll find it in a lot of unexpected places. Gauss' Law is an application of a conservation law: you do not lose flux. Makes sense, right? If you have a hose shooting water and you have a surface completely covering the hose, then the amount of flux will equal the amount of water coming out.

However, vector fields are different. They are not particles that fly through surfaces (like water drops). Vector fields have magnitudes that can change with distance; in fact, most depend on r, the radius vector, in some way.

Thus, we have a vector field that changes with r, and we want to maintain the property of flux conservation. Why? Good question: it just seems to be the way that nature operates. Conservation, conservation, conservation.

Ok, so we have a point that is creating this vector field. We want the property that any closed surface surrounding the point has the same flux. Remember the garden hose: any closed surface that goes around it shall capture the same amount of water.

Now, notice that as we take a larger surface (larger sphere), our surface area increases. If the vector field did not change, then the larger surface has a larger area, therefore more flux. But this can't be: the larger surface has to have the same flux as the smaller one.

To counter act this, we need to have the magnitude of the field decrease to offset the larger area (remember that strength1 * area1 = strength2 * area2). Notice that the surface area of a sphere increases as r^2. Thus, the field should decrease as 1/(r^2) to cancel the increase in area. If the sphere doubles its radius, its surface area goes up 4 times. However, by doubling its radius, the field drops to 1/4 its strength. 4 * 1/4 = 1, so the flux does not change.

Thus, we see that for flux conservation (Gauss' Law) to work, we need an inverse square field.

We used spheres because they are easy to work with. But would it matter if we had chosen a different shape? Even crazy shapes that shapes that twist, turn, and bend back on themselves do not really matter.

Imagine a cone with its point at the source. Divide the shape into a series of panels, with each panel being the base of the cone. You will see that as the shape gets closer, its area decreases (as r^2) and its the strength of its flux increases as r^2. The reverse happens when we move the panel away, and the two effects cancel so the flux is constant. Thus, closed surface surrounding the source has the same flux.

Ok, here's the equation for those interested:

[INSERT EQ HERE]

Other fields

Wait, what if we don't have an inverse square field? It is easy to see what happens.

The right hand side is the "Gauss' Law" for the field in the first column. The first is the real Gauss' Law, which says that the flux at each point is the same. Thus, any surface surrounding the source will have the same flux. For the other fields, the flux will depend on the distance of the surface from the source.

Why is Gauss' law so special if it only holds for one type of vector field? Well, nature likes conservation, so you'd expect it to have the flux of its fields conserved. Well, some fields in nature are the gravitational and electric fields. Guess what? To conserve flux, they follow inverse square laws! Thus, Gauss' Law tells us about the flux of gravity (the source is mass) and or the flux of electric field (the source is charge).

So, while any schmoe could do what I did above, and create a "Gauss' Law" for any old field, Gauss already tagged the important ones.

Observations

You can use Gauss' Law "in reverse" and come up with some interesting facts. If you have a closed surface in an inverse square field and notice no net flux, that means there is no source enclosed. Using Gauss' Law this way is helpful when studying electric charges.

Recall that we are always working within a constant factor. In my book there is a factor of 4*pi (due to the surface area of a sphere), in some physics book you might see some other constants there. These simply depend on the units you are using to measure flux (cgs vs. mks, for example). The underlying idea is the same.

Iron Shell Theorem

We don't talk about gravitational flux very much, but it exists, and we'll look at something cool as a result of it. Suppose you have a hollow iron sphere floating in space. You open it up, fly in with your jetpack, and seal it again. What, if any forces do you feel?

Suppose you jet over to the center: would you feel any forces? Probably not, due to the symmetry.

Now suppose you move away from the center and stop. Do you feel any force? Hmm. Do you get pulled to closer wall? Or do you get pulled to the wall across from you, because there is "more" of it pulling you?

Hey, we've seen this situation before. If we consider a point inside the sphere, the mass on the sides pull it with the inverse square field. However, the point sees a different amount of opposite walls: imagine two cones, point to point; they meet up at our location. Just as before, the surface area increases as r^2, and the field falls as 1/r^2. Therefore, the flux stays the same from both sides!

This means all sides are pulling you equally, so you feel no net force anywhere inside the sphere. This is something Newton proved centuries ago, but we can see how it makes sense intuitively. As you approach one wall, a small amount of mass is pulling you with great force. But, on the opposite side, a huge amount of area is pulling you with a smaller force. It all cancels, so there is no net pull from the sphere! This works for any inverse square field, such as electric fields.

Note: This does not mean you can make an anti-gravity machine by walking into a huge sphere. The Iron Shell Theorem says you will feel no force from the shell itself. Fields from other bodies, such as the Earth, will still affect you.

Also, suppose you dig a tunnel down into the earth. As you get lower, there is a layer of the earth that forms a shell around you. By our new theorem, we know this layer does not effect you at all. In addition, if we have a hazy sphere of charge and we are located somewhere inside, we only feel a pull from the sphere between us and the center. The shell outside us does not pull us at all. These results are very useful when studying charge distributions.

(Question: Suppose a proton is a point charge surrounded by "shells" of moving electrons [the electrons move quickly enough that there is a smear of charge around the proton]. Does the proton feel any force from the electron? Does the electron feel any force from the proton?

My Answers: The proton feels no force from electron. The electron feels force from the proton, which pulls it radially inward. This gives it the centripetal acceleration to keep orbiting).

Revelations

• Gauss' Law only works for inverse square fields! (1/r^2). Remeber that Gauss' Law works because of our surface area argument (field falls off at same rate as area increases).
  
  
• The concept of a field diminishing as area increases is very powerful, and will be used to intuitively find other relations.
  
  
• A closed surface in an inverse square field with no net flux contains no source.
  
  
• When measuring flux, we are working within a constant. This constant may vary depending on the units used.
  
  
• Iron Shell Theorem: if surrounded by a spherical shell of source that has a inverse square law (mass or charge), then you will feel no net force inside the shell.