Maxwell's eqn in invariant form

In summary, Maxwell's equations in invariant form state that the derivative of the electromagnetic tensor equals the four-current, and the sum of the contractions of the electromagnetic tensor with the Levi-Civita tensor equals zero. If there is a magnetic charge and current, the second equation can be modified to include a proportionality constant and the four-current. The dual tensor is defined as the product of the electromagnetic tensor and the Levi-Civita tensor.
  • #1
mathfeel
181
1
Maxwell's eqn, in invariant form reads:

[tex]F^{\mu \nu}{}_{;\nu} = J^{\mu}[/tex]

and

[tex]F_{\alpha \beta ;\gamma} + F_{\beta \gamma ;\alpha}+F_{\gamma \alpha; \beta} = 0 [/tex]

Can someone give Maxwell's eqn if there is magnetic charge and current? I do not believe the form (matrix element) of F change, however, if it does, please state that as well.
 
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  • #2
Originally posted by mathfeel
Maxwell's eqn, in invariant form reads:

[tex]F^{\mu \nu}{}_{;\nu} = J^{\mu}[/tex]

and

[tex]F_{\alpha \beta ;\gamma} + F_{\beta \gamma ;\alpha}+F_{\gamma \alpha; \beta} = 0 [/tex]

Can someone give Maxwell's eqn if there is magnetic charge and current? I do not believe the form (matrix element) of F change, however, if it does, please state that as well.

The second can be written in terms of the electromagnetic duel tensor [tex]D^{\mu \nu}[/tex] as
[tex]D^{\mu \nu}{}_{;\nu} = 0[/tex]
Instead of setting that equal to zero try setting it proportional to your hypothetical magnetic four current [tex]M^\mu[/tex] like:
[tex]D^{\mu \nu}{}_{;\nu} = kM^{\mu}[/tex]
(Normally I would explicitely put in the constants determied by your system of units for both sets of equations)
I haven't checked into this, but off the top of my head I think this would work. Of course your next job will be to go out and find a magnetic monopole in order to justify having done this.
 
  • #3
How do you define the dual tensor?
 
  • #4
Originally posted by mathfeel
How do you define the dual tensor?

The electromagnetic duel tensor [tex]D_{\mu\nu}[/tex] is related to the electromagnetic tensor [tex]F^{\mu\nu}[/tex] and the rank 4 Levi-Civita tensor [tex]\epsilon_{\alpha\beta\mu\nu}[/tex] by
[tex]D_{\mu\nu} = \frac{1}{2}F^{\alpha\beta}\epsilon_{\alpha\beta\mu\nu}[/tex].
 

1. What is Maxwell's equation in invariant form?

Maxwell's equation in invariant form is a set of four equations that describe the fundamental laws of electricity and magnetism. They are used to mathematically describe the relationship between electric and magnetic fields and their sources, such as charges and currents.

2. Why is it called "invariant" form?

It is called "invariant" form because the equations remain the same in all inertial reference frames, meaning they are consistent regardless of the observer's velocity. This is a fundamental concept in the theory of relativity.

3. What is the significance of Maxwell's equations in invariant form?

Maxwell's equations in invariant form have significant implications in the fields of physics and engineering. They not only provide a complete description of the behavior of electric and magnetic fields, but they also led to the development of important technologies such as radio, television, and telecommunications.

4. How do Maxwell's equations in invariant form relate to the speed of light?

One of the most significant implications of Maxwell's equations in invariant form is the prediction of the existence of electromagnetic waves, which travel at the speed of light. This was a major breakthrough in understanding the nature of light and its relationship to electricity and magnetism.

5. Are Maxwell's equations in invariant form still valid today?

Yes, Maxwell's equations in invariant form are still considered to be valid today as they have been extensively tested and have consistently been shown to accurately describe the behavior of electric and magnetic fields. However, they have been modified and extended in certain situations, such as in the theory of quantum electrodynamics, to account for more complex phenomena.

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