What are the applications of Gauss's law in technology? [closed]

Question

Freshmen physics textbooks use Gauss's law plus symmetry to calculate the electric field. I was wondering if this method of finding the electric field using a symmetry is used in real applications in life, science, research, or technology. For example do researchers use symmetry to find the electric field due to a charged virus that is spherically symmetric, or a charged DNA that has some kind of symmetry or charged symmetrical objects in nanotechnology. I am looking for such specific applications.

Answer 1

do researchers use symmetry to find the electric field due to a charged virus that is spherically symmetric?

The answer is no. Why they need to reinvent the wheel? They just use the known formula that exists for such situations that can be proved using Gauss Laws (though they all can be proved without using Gauss laws)

Similar answer can be given to your other question. Though note that since sizes are of atomic level, other forces can come into play.

Theoretically we don't need Gauss law. Only law we need in electrostatics is Coulomb's law to find $\vec E$ and $\vec F = m\vec E$. Gauss Laws and all other tools only help to simplify the calculations. So there is no application of Gauss Law per se but I think you must appreciate the power of Gauss Laws in simplification of many tough situations,many of which have profound practical applications. However theoretically all situations can be solved without using Gauss Laws or symmetry arguments.

Though I recommend checking out http://en.wikipedia.org/wiki/Faraday_cage a really outstanding application of electrostatics and its working can be explained using Gauss' Laws along with properties of conductors.

Answer 2

Gauss theorem is a law relating the distribution of electric charge to the resulting electric field.

So if scientist knows the distribution of charge on some DNA or the surfaces of some virus then they can calculate the electric field.

If you know that charge distribution is symmetrical, you can expect same result for electric field. For example if the charge distribution has spherical symmetry then the field will depend only on the distance:

$$E = k\frac{q}{r^2} \sim \frac{1}{r^2}.$$

More examples of charge distributions you could find here:

http://en.wikipedia.org/wiki/Gaussian_surface

http://iweb.tntech.edu/murdock/books/v4chap2.pdf

And this post may be useful:

In which cases is it better to use Gauss' law?

Answer 3

The freshman physics textbooks use the integral forms of Gauss's law, which (in the vacuum) look like this:

$$ \int \mathbf{E} \cdot d\mathbf{A}=Q_{enc}/\epsilon_0 $$ $$ \int \mathbf{B} \cdot d\mathbf{A}=0 $$

These laws also have differential forms, which look like this:

$$\nabla \cdot \mathbf{E}=\rho/\epsilon_0 $$ $$\nabla \cdot \mathbf{B}=0 $$

These equations don't really require symmetry to be useful, and can be solved with standard partial differential equation techniques on a computer. For problems involving surfaces with variable charges (like cavities with conductive sides), these differential forms (and the PDEs for the scalar and vector potentials that they imply) are practically the only way to solve for the $\mathbf{E}$ and $\mathbf{B}$ fields everywhere.