Gauss' Law -- Simplified

[zking786]

I have been trying to learn college-level electrostatics at home. I've understood charges, electric fields, and voltage, but haven't quite grasped Gauss' law and some of the capacitance theorems that base off of it.

Every site I have gone to seems to explain Gauss' law in a tricky, vector-related form. Since I only have a basic understanding of vectors, I can't completely understand how the formulas and calculations work. Not to mention my confusion with the various symbols in the equations. Is there anyone who can simply explain the law and any necessary prerequisites to understanding it?

Any help you can provide will be valuable in my pursuit of knowledge.

Thanks!

[patcalhoun]

I have been trying to learn college-level electrostatics at home. I've understood charges, electric fields, and voltage, but haven't quite grasped Gauss' law and some of the capacitance theorems that base off of it.

Every site I have gone to seems to explain Gauss' law in a tricky, vector-related form. Since I only have a basic understanding of vectors, I can't completely understand how the formulas and calculations work. Not to mention my confusion with the various symbols in the equations. Is there anyone who can simply explain the law and any necessary prerequisites to understanding it?

Any help you can provide will be valuable in my pursuit of knowledge.
Thanks!

Sure, the total flux of a the electrical field of a sphere of radius of constant charge density \rho is \Phi = EA= \frac{1}{\epsilon_0} \rho V = \frac{Q_A}{\epsilon_0}, where A = 4 \pi r^2 is the surface area of a sphere with radius r and V = \frac{4}{3} \pi r^3 is its volume, Q_A is the total charge within the sphere and \epsilon_0 is a constant (the permitivity of free space).

[zking786]

To clarify my requests:
1. Firstly, what is flux?
2. Secondly, I've seen various equations that include (3 cos^2 (x) +1 )^.5
How does this fit in?
3. How does torque relate to this equation and Gauss' laws.
4. How were Gauss' laws derived. I don't want to simply memorize the equations, I'd prefer to understand how they are derived and what they signify.
5. Lastly, what is a Gaussian Surface?


Thanks for your prompt assistance!

[patcalhoun]

To clarify my requests:
1. Firstly, what is flux?

In electromagnetism, flux is the measure of net electrical or magnetic field flow through a surface enclosing an electrical or magnetic field source. It has units of that boil down to W/m^2. The case I just gave equations for is for the flux at the surface of a sphere of uniform charge density. This, obviously, gets more complex as we pick more difficult geometries and demand that charge density vary.

You should note that Gauss' law for magnetic fields holds that the net magnetic field flow through an enclosed surface is zero. If you think about it using Farady's "lines of force," this makes sense, as the poles of a magnetic begin and terminate each line.

2. Secondly, I've seen various equations that include (3 cos^2 (x) +1 )^.5
How does this fit in?

Those would be solutions to Gauss'ls Law for geometries far less trivial than the sphere solution I gave you. But you'll want to get a better handle on calculus and vectors before you tackle that.

3. How does torque relate to this equation and Gauss' laws.

I don't know how to address this without resorting to vectors, but there are plenty here who probably can.

4. How were Gauss' laws derived. I don't want to simply memorize the equations, I'd prefer to understand how they are derived and what they signify.

That would require some knowledge of the Divergence Theorem, which is where you'll need some vector calculus to follow. For now, let's just say that d \Phi \equiv dE \cdot dA, is defined as first principle.

5. Lastly, what is a Gaussian Surface?

Why, its any surface that we can use to Gauss's Law to calculate the flux through. :D In short, its gotta be closed, and its gotta be enclosing whatever it is that's "fluxing."

[zking786]

I guess the moment I dreaded is here... I guess I must learn about vector calculus to better understand electrostatics. I know first-year university-level Calculus and first-year mechanical physics. So, I have a basic understanding of vectors. Could you, introduce vectors to the equations in a simple way? Maybe with an explanation, along with the symbols.

Once again, thanks for all your help. These concepts are actually starting to make sense!!!

[swansont]

Why, its any surface that we can use to Gauss's Law to calculate the flux through. :D In short, its gotta be closed, and its gotta be enclosing whatever it is that's "fluxing."

And to be useful as a calculational shortcut, it has to be of a shape that all of the flux lines are perpendicular to the surface, so you use it for problems with the right symmetry. e.g. for a point charge, the Gaussian surface of choice is a spherical shell.