Gauss's Law

[Yarn]

Gauss's law is as follows (ignore the second 2 lines because they are case specific):


Hence, if the net enclosed charge is equal to 0, then it follows that E must also be 0. This means that Gauss's law predicts that the electrical field of an electrical dipole or capacitor is 0. Since that is not the case, how can Gauss's law be valid?

[Angry Citizen]

Because the net electric field of a dipole is 0. If you wish to determine the "internal" electric field induced by the dipole (similar to finding the internal forces in a statically determinant rigid body undergoing a load), you must take a limiting case. The separation of the dipole is non-zero and finite; it's just very very small. Enter your calculus.

Think back to the visual interpretation of Gauss's Law as the flux through a field (note the definition of Gauss's Law as a surface integral of the electric field vector dotted with the differential area vector, which is the definition of a flux quantity). Where you draw your field of interest is important. So, draw your field of interest around one end of the dipole moment, and you'll find the electric field induced by that point charge. Any kinematic forces induced can be found using the Gauss's Law results from each of the two charges. At a sufficiently large distance, the two charges become inseparable, and the net effect on a distant charge thus becomes zero***.

*** Note: Again, I want to stress that this is net effect. The net field may be zero at large distance, but that doesn't mean the field isn't there.

[Yarn]

Dipoles have net electric fields possesing a coefficient that varies from 1 to 2 depending on your location relative to the dipole axis. Regardless of where you are relative to the axis, the electric field strength decreases with 1/r^3.

Conceptually, in all locations relative to a dipole, you are either closer to one monopole than the other (per instance, this is true on the axis of the dipole) or some of the components of the monopoles' electric fields sum rather than negate each other (per instance, this is true on a perpendicular-bisector axis).

It is true that the electric field falls faster with distance than that of a point charge does, but it exists nonetheless.

[Angry Citizen]

What is the limit as r approaches infinity of 1/r^3? Everything I said is true.

[Yarn]

Gauss's law would say the electric field is 0 at any distance so long as both monopoles were enclosed by the Gaussian surface. That isn't true.

Gauss's law is said to be universally true, meaning that somehow or other the equation in the OP must solve for a non-zero E. I don't get how that can be done.

[Angry Citizen]

Again, the matter is trivial. Construct an enclosure such that one pole is isolated from the other. At r values greater than some large, arbitrarily defined limiting case, the enclosure cannot isolate one particle from the other, and so the dipole becomes a point charge from a geometric and mathematical perspective. If I recall correctly, the dipole equation can be formed by such an analysis and summing the resulting (SIGNED) electric field vectors.

I'm not sure what you're majoring in, but if you want to be an engineer or a physicist, you need to understand the idea of vanishing and inconsequential quantities, and to construct geometric interpretations whenever possible. At a sufficiently small r, E_net is 'significant'. At a sufficiently large r, E_net is inconsequential.

Think of it like this. You have a dipole. One monopole is going to be closer to our test charge than the other, so you split the problem into two: One Gauss Law for one monopole, another for the other. Working the math out for that problem yields that the electric field is inversely proportional to the cube of the radius. And it makes sense, does it not? At small r, the distance between one monopole and the other compared to the distance between the dipole and the test charge is quite comparable - within an order of magnitude, let's say. At large r - for instance, three orders of magnitude larger than the radius of the dipole system - the ratio of the two is much larger. Hence the inverse-cube law.

[gallo]

Wow! you guys are awesome.

When I came home from college on a break, a subject came up in a discussion with my father where I realized that I knew more than he did. When my two sons were in high school we were discussing something and I realized that they knew more than I. It was a strange feeling.

I again have that feeling. I'm a biologist and never really studied what you guys are talking about. My sons did. One is a nuclear engineer (could have opted for a degree in mechanical engineering, math, nuclear physics, or physics). The other has a degree in math with a minor in chemistry (one of two Bachelor degrees he earned in four years). He's back from Afghanistan where he served with the 1st Infantry Division. Now at Ft. Riley.

I wish I had even the vaguest idea of what you are talking about. Carry on.

[Angry Citizen]

Don't worry, gallo. If this were a topic onbiology, you would smoke us all. And even ignoring that.. one of the wisest guys I know is my father. Wanna know what he does for a living? Drives a truck. Neither I nor your sons will have your life experience for decades to come. There is value in that.