Proving Gauss's Law for Magnetism

[LucasGB]

As you probably know, there is a formal proof of Gauss's Law for electric fields based on Coulomb's Law and the concept of solid angles. How can one prove Gauss's Law for magnetic fields? Is there a similar proof based on solid angles?

 

[espen180]

There are no proofs in physics. The "proof" you cite is a proof that Coloumb's law and Gauss's Law are equivalent.

Gauss's Law from magnetism is as it is because a magnetic monopol has never been observed. Is one is observed, the equations will have to change.

 

[LucasGB]

There are no proofs in physics. The "proof" you cite is a proof that Coloumb's law and Gauss's Law are equivalent.

OK, sort of a semantics issue.

Gauss's Law from magnetism is as it is because a magnetic monopol has never been observed. Is one is observed, the equations will have to change.

I know that. I would like to know, as you put it, if there is a way of rigorously showing whether Ampere's Law and Gauss's Law are equivalent.

 

[espen180]

You can take a look at common symmetric charge configurations (point charge, line charge, plane change) and calculate the field of each using Gauss's Law and then integrating using ampere's law. You'll get the same result in every case.

 

[LucasGB]

You can take a look at common symmetric charge configurations (point charge, line charge, plane change) and calculate the field of each using Gauss's Law and then integrating using ampere's law. You'll get the same result in every case.

I see, but that's not a formal proof. The proof for Gauss's Law for electric fields is quite rigorous and general.

 

[Born2bwire]

I would say that Gauss' Law is proof of Coulomb's Law. In the end though, it is completely circular as both are derived via empirical reasoning. However, Gauss' Law is the more general law from which you can derive Coulomb's Law.

As for Gauss' Law for magnetism, that is simply derived from the fact that we currently do not allow for magnetic monopoles. The basic unit for magnetic fields is the dipole.

Maxwell's equations are the basic equations for classical electrodynamics. Coupled with Lorentz force you have the groundwork for it all. So there is no proof for these equations alotted using classical electrodynamics because electrodynamics is derived from them in the first place. They were primarily derived via experimentation. If you go up to quantum electrodynamics, then we can find more basic theories that predict Maxwell's equations.

 

[LucasGB]

I see. The reason I ask is because I'm trying to write a text which starts with simple empirical facts (the simplest being Coulomb's Law and Ampere's Law) and gradually builds towards the complete equations of classical electromagnetism. I have found that Gauss's Law can be achieved from Coulomb's Law quite rigorously and beautifully through the solid angle proof. I am ready to establish the nonexistence of magnetic monopoles as the experimental foundation for Gauss's Law for magnetism, but what I would really like to know is how can one prove rigorously (and I'm sure this must be quite simple) that the nonexistence of magnetic monopoles leads to zero divergence of the magnetic field.

PS.: I know you can use the field line arguments, but I think field lines are a rather crude way of proving things.

 

[jtbell]

The mathematical basis of the "field line argument" is the divergence theorem of vector calculus, which says that the net flux of a vector field through a closed surface equals the integral of the divergence of the field inside the surface.

If there is charge at a point, then the divergence of the associated field is nonzero at that point; if there is no charge at that point, then the divergence of the field is zero at that point.

If there is non "magnetic charge" anywhere, then the divergence of B is zero everywhere, and so the flux of B through any closed surface must be zero.