Understanding Gauss' Law: A Guide to Divergence and Charge Determination

[rockyshephear]

I think after weeks of study, I'm finally getting a handle on Gauss' Law.

A few ? though.

The equation does not specifically state that there is not charge inside the surface. One may think that it doesn't matter if there or isn't...it's alway zero. How come that is not state in the equation somehow?

Second ?. So from what I think, Gauss' Law can be used to determine if a charge exists at an infinitesimal point just like derivate can be used to determine the slope of the tangent line at a point on a curve. Why would I want to know if a charge exists at point x,y,z? It's not like the universe is set in amber charges are just sitting at discrete places. They move around.

 

[dx]

There are two Gauss' laws in EM. Which one are you talking about? If you're talking about the one involving E, then I don't know what you mean by "it's always zero" since divergence of E is not always zero. If you're talking about the one for magnetism, it doesn't involve any charge, so I still don't know what you're trying to ask.

 

[vin300]

I think after weeks of study, I'm finally getting a handle on Gauss' Law.

A few ? though.

The equation does not specifically state that there is not charge inside the surface. One may think that it doesn't matter if there or isn't...it's alway zero. How come that is not state in the equation somehow?

Second ?. So from what I think, Gauss' Law can be used to determine if a charge exists at an infinitesimal point just like derivate can be used to determine the slope of the tangent line at a point on a curve. Why would I want to know if a charge exists at point x,y,z? It's not like the universe is set in amber charges are just sitting at discrete places. They move around.

Once the charges are set such that all the components of force parallel to the surface are nullified, they do not move around like marbles.
Maybe you are meaning that the surface is impermeable to electric lines of force there is no intensity on the inside of an infinite surface.

 

[Tac-Tics]

The equation does not specifically state that there is not charge inside the surface. One may think that it doesn't matter if there or isn't...it's alway zero. How come that is not state in the equation somehow?

I'm not sure what you mean here. Guass's law states that if there's charge inside a volume, you can figure out how much by measuring the electric field on the surface of the volume. Nowhere is it implied the charge inside is always zero.

Second ?. So from what I think, Gauss' Law can be used to determine if a charge exists at an infinitesimal point just like derivate can be used to determine the slope of the tangent line at a point on a curve. Why would I want to know if a charge exists at point x,y,z? It's not like the universe is set in amber charges are just sitting at discrete places. They move around.

You can't understand how something moves until you understand how it stays in place.

It may be helpful to understand that the mathematics of electrodynamics was worked out *before* scientists knew that charge was quantized. Gauss's mathematics treats charge as a continuous quantity. Instead of discrete, charged little balls, you are working with charge densities of volumes. (It's similar to the relationship between a "sum" and an "integral" in math).

Suppose I have a box which takes up 1 liter of volume, and I know it holds 1 coulomb of charge inside. Also assume the charge is spread EVENLY throughout the interior of the box. How much charge does the point at the center of the box have? The answer is "practically zero". It's infinitesimal. If you moved the center point outside of the box, it would in fact have zero coulombs of charge.

So instead, we talk about charge densities. A the center point in our box has a charge density of 1 coulomb per liter.

The cool thing about charge densities is that if we know the charge density of EVERY point in a volume, we can integrate and figure out the total charge of that volume.

 

[rockyshephear]

Here's my reference:

The integral form of Gauss's law for magnetism states:

\oint_S \mathbf{B} \cdot \mathrm{d}\mathbf{A} = 0

where

S is any closed surface (a "closed surface" is the boundary of some three-dimensional volume; the surface of a sphere or cube is a "closed surface", but a disk is not),
dA is a vector, whose magnitude is the area of an infinitesimal piece of the surface S, and whose direction is the outward-pointing surface normal (see surface integral for more details).

The left-hand side of this equation is called the net flux of the magnetic field out of the surface, and Gauss's law for magnetism states that it is always zero.

The integral and differential forms of Gauss's law for magnetism are mathematically equivalent, due to the divergence theorem. That said, one or the other might be more convenient to use in a particular computation.

 

[dx]

Gauss' law for magnetism makes no reference to charge, so what did you mean by your first question?

 

[rockyshephear]

Yea, I'm getting the two mixed up. So is magnetic field affected by permeability as electric field is?

 

[vin300]

Here's my reference:

The integral form of Gauss's law for magnetism states:

\oint_S \mathbf{B} \cdot \mathrm{d}\mathbf{A} = 0

where

S is any closed surface (a "closed surface" is the boundary of some three-dimensional volume; the surface of a sphere or cube is a "closed surface", but a disk is not),
dA is a vector, whose magnitude is the area of an infinitesimal piece of the surface S, and whose direction is the outward-pointing surface normal (see surface integral for more details).

The left-hand side of this equation is called the net flux of the magnetic field out of the surface, and Gauss's law for magnetism states that it is always zero.

The integral and differential forms of Gauss's law for magnetism are mathematically equivalent, due to the divergence theorem. That said, one or the other might be more convenient to use in a particular computation.

Are you talking about gauss's theorem in electrostatics or something else?
There is no magnetic charge, it is caused by circulating electrons.