# Maxwell's equations for electricity and magnetism

## Main Question or Discussion Point

I was reading Michio Kaku's book Prallel Worlds recently and I believe I saw it say that Maxwell's equations for electricity and magnetism become the same for electricity and magnetism when monopoles are introduced.

My question is, if the equations become the same then why don't we say that electric charge is an example of a monopole? I may have misread, but if the equations for monopoles and electric charge are the same then why not?

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jtbell
Mentor

We've never observed a magnetic monopole experimentally. That's why we don't include them in Maxwell's equations.

Electric charges are monopoles: electric monopoles!

Born2bwire
Gold Member

An electric charge is an example of a monopole, like jtbell has explained. In their original state, Maxwell's Equations are almost symmetrical with respect to E and H. Sometimes, for purposes of making the equations symmetrical and/or to aid in computation, we will introduce a "magnetic current." This requires us to change the divergence of the magnetic flux density to give rise to a nonzero magnetic charge.

Now these magnetic currents and charges are completely ficticious, however, if we are solving for fields in a source free region, then the answers will be completely equivalent. In addition, it allows us to make use of the Duality Principle. The Duality Principle uses the symmetry of Maxwell's Equations to switch between the expressions for the E and H fields easily. The magnetic currents also can be used to solve certain problems more easily through the ability to interchange waves and currents (it is sometimes easier to use an equivalent magnetic current across the opening of a cavity to represent the resulting EM waves).

So what you read is true. The introduction of, what we currently consider ficticious, magnetic monopoles and currents, the Maxwell Equations for E and H are the same. For most practical purposes, the results that we get using these adjusted equations are also the same.

Thanks for the replies:)

But, for a magnetic monopole, the lines of force wouldn't come back to the monopole because that would make it have two poles, so its lines of force would go on for ever, right? Please correct me if I am wrong.
If magnetic fields interact with electric charges, then wouldn't a monopole, that has lines of force that go forever like an electric fields line of force, be exactly the same as an electric charge, as in irrecognizable from one another?

Thanks for the replies:)

.. so its lines of force would go on for ever, right? Please correct me if I am wrong.
The force lines will never comeback to the same monople, but that doesnt imply that they have to go on and on for ever..they can easily ''crash'' onto a different (and opposite) monopole.
Cheers.

Do electric field lines crash into other electric field lines too?

Born2bwire
Gold Member

Do electric field lines crash into other electric field lines too?
If you have two equal but opposite charges in close proximity they form a dipole which has closed field lines. Electric fields do not "crash" into each other though. Fields do not interact with other fields, they just add up in linear superposition.

If magnetic monopoles are included in Maxwell's equations, there in a nice symmetry that results. If you take Maxwell's 4 equations, and replace E with B, and B with -E, then the electic charge density becomes magnetic charge density. As well, magnetic charge density becomes negative electric charge density, etc. It's a cyclic symmetry.

I think Wikipedia has table of Maxwell's equations where magnetic charge is allowed.

edit: here it is http://en.wikipedia.org/wiki/Magnetic_monopole" [Broken]

It's been often said that Maxwell's equations cannot be stated in terms of a vector potential if the magnetic charge can be nonzero (All exact forms are closed.). This is not exactly true. To both allow for magnetic charge, and maintain a (4)vector potential, mandates that generic charge be vectoral. Within such a model, in the low energy regime, only one species of charge would be observed. (So one can have one's cake (magnetic monpoles) and eat it too (no magnetic monopoles have been observed).)

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