DrRocket

2009-Dec-09, 08:08 PM

In a related thread, now in the ATM forum and closed at this time, Tusenfem posted the following in response to my assertion that div B =0

I have to disagree here, the general equation in the presence of possible monopoles would be:

div B = 4 π ρm,

where ρm would be "magnetic charge" inside a volume.

However, as, up to now, there has never been any real confirmation of the existence of magnetic monopoles (there once was one stray observation in a superconducting thingamajik, but can't find the reference right now, which was never repeated afterward) we can safely assume that ρm=0 and thus div B = 0.

Basically, what div B = 0 means is that (loosely defined) if we have a volume V with magnetic particles in it, then there are equal amounts of field lines that go out through the surface of the volume as come in. (Actually, it is the surface integral over the surface of the volume of the normal component of the magnetic field to that surface, if you understand what I mean).

I both agree and disagree with Tusenfem. This post is intended to make clear the maintstream position with regard to monopoles within the context of classical electromagnetism.

For a concise summary of Maxwell's equations, and the necessary modifications if magnetic monopoles were to be included it the theory the reader is invited to refer to this Wiki article (http://en.wikipedia.org/wiki/Magnetic_monopole).

Maxwell's equations as currently accepted in mainstream physics, and as presented in standard text books such as J.D. Jackson's Classical Electrodynamics include Gauss's law (for magnetism), div B = 0. This simply asserts the nonexistence of point magnetic charges, or monopoles. Classical electrodynamics is sufficiently mature as to be amenable to an axiomatic treatment, and Maxwell's equations plus the Lorentz force equation are the axioms.

Tusenfem's interpretation of div B = 0 is correct, and is simply an application of the generalized Stokes Theorem (often called the Divergence Theorem in texts on electromagnetism), resulting in the integral form of Gauss's Law. Whether you prefer the point form or the integral form, the result is still the denial of the existence of magnetic monopoles in classical theory.

There are reasons stemming from quantum electrodynamics to think that monopoles might exist. They make the quantization of electric charge an elegant result rather than an ad hoc "add on". But no reliable experimental evidence has been found for the existence of magnetic monopoles.

If magnetic monopoles were to be found, then Maxwell's equations would be modified. The equation suggested by Tusenfem, and found in the table in the Wiki article, is what would replace div B = 0. That equation is the analog of div D = rho, which is Gauss's law for the electric field, for which it is known that point charges do exist (electrons and protons). One would also have to modify the Lorentz force equation to reflect the existence and effect of magnetic charge. Simply stated, physicists know how to formulate a classical electrodynamic theory that would accomodate magnetic monopoles. That theory would be mathematically consistent, and it would be an axiomatic construct using the altered Maxwell and Lorentz equations. But mathematical consistency is only part of the story. A valid theory would also have to reflect reality and the experimental data base. That data base does not at this time support the existence of magnetic monopoles, but rather denies it. This situation will change if and only if magnetic monopoles are confirmed experimentally.

The importance of this is that any claim to a proof of the existence of magnetic monopoles based on phenomena known to be very accurately described by Maxwell's equations -- currents, wires, solenoids, moving charges, antennas, etc. -- is doomed. Maxwell's equations simply forbid monopoles. Adding in the Lorentz force equation will not result in a valid argument either. The Lorentz force equation is compatible with Maxwell's equations, and simply describes the force exerted on a text particle by the fields governed by Maxwell's equations. It cannot negate div B = 0. In addition the existence of magnetic monopoles would necessitate a change in the Lorentz force equation -- see the table in the Wiki article for the form in the presence of magneetic monopoles.

So, bottom line: Mainstream classical electrodynamics is based on the established form of Maxwell's equations, including div B = 0, and monopoles are forbidden. That will not change unless and until magnetic monopoles are shown to exist in reliable experiments.

I have to disagree here, the general equation in the presence of possible monopoles would be:

div B = 4 π ρm,

where ρm would be "magnetic charge" inside a volume.

However, as, up to now, there has never been any real confirmation of the existence of magnetic monopoles (there once was one stray observation in a superconducting thingamajik, but can't find the reference right now, which was never repeated afterward) we can safely assume that ρm=0 and thus div B = 0.

Basically, what div B = 0 means is that (loosely defined) if we have a volume V with magnetic particles in it, then there are equal amounts of field lines that go out through the surface of the volume as come in. (Actually, it is the surface integral over the surface of the volume of the normal component of the magnetic field to that surface, if you understand what I mean).

I both agree and disagree with Tusenfem. This post is intended to make clear the maintstream position with regard to monopoles within the context of classical electromagnetism.

For a concise summary of Maxwell's equations, and the necessary modifications if magnetic monopoles were to be included it the theory the reader is invited to refer to this Wiki article (http://en.wikipedia.org/wiki/Magnetic_monopole).

Maxwell's equations as currently accepted in mainstream physics, and as presented in standard text books such as J.D. Jackson's Classical Electrodynamics include Gauss's law (for magnetism), div B = 0. This simply asserts the nonexistence of point magnetic charges, or monopoles. Classical electrodynamics is sufficiently mature as to be amenable to an axiomatic treatment, and Maxwell's equations plus the Lorentz force equation are the axioms.

Tusenfem's interpretation of div B = 0 is correct, and is simply an application of the generalized Stokes Theorem (often called the Divergence Theorem in texts on electromagnetism), resulting in the integral form of Gauss's Law. Whether you prefer the point form or the integral form, the result is still the denial of the existence of magnetic monopoles in classical theory.

There are reasons stemming from quantum electrodynamics to think that monopoles might exist. They make the quantization of electric charge an elegant result rather than an ad hoc "add on". But no reliable experimental evidence has been found for the existence of magnetic monopoles.

If magnetic monopoles were to be found, then Maxwell's equations would be modified. The equation suggested by Tusenfem, and found in the table in the Wiki article, is what would replace div B = 0. That equation is the analog of div D = rho, which is Gauss's law for the electric field, for which it is known that point charges do exist (electrons and protons). One would also have to modify the Lorentz force equation to reflect the existence and effect of magnetic charge. Simply stated, physicists know how to formulate a classical electrodynamic theory that would accomodate magnetic monopoles. That theory would be mathematically consistent, and it would be an axiomatic construct using the altered Maxwell and Lorentz equations. But mathematical consistency is only part of the story. A valid theory would also have to reflect reality and the experimental data base. That data base does not at this time support the existence of magnetic monopoles, but rather denies it. This situation will change if and only if magnetic monopoles are confirmed experimentally.

The importance of this is that any claim to a proof of the existence of magnetic monopoles based on phenomena known to be very accurately described by Maxwell's equations -- currents, wires, solenoids, moving charges, antennas, etc. -- is doomed. Maxwell's equations simply forbid monopoles. Adding in the Lorentz force equation will not result in a valid argument either. The Lorentz force equation is compatible with Maxwell's equations, and simply describes the force exerted on a text particle by the fields governed by Maxwell's equations. It cannot negate div B = 0. In addition the existence of magnetic monopoles would necessitate a change in the Lorentz force equation -- see the table in the Wiki article for the form in the presence of magneetic monopoles.

So, bottom line: Mainstream classical electrodynamics is based on the established form of Maxwell's equations, including div B = 0, and monopoles are forbidden. That will not change unless and until magnetic monopoles are shown to exist in reliable experiments.