Invariant quantities in the EM field

In summary, the quantities E^2 - B^2 and \vec{E} \cdot \vec{B}, where E and B are the electric and magnetic components of an EM wave, are invariant under Lorentz/Poincare transformations. The second expression may not always be zero, as an EM wave is just a special case of a radiated EM field. The invariants have limited physical significance, but the first one is useful for a lagrangian density. There are other solutions to Maxwell's equations besides EM waves, and it may be difficult to understand the mixing of E and B fields under boosts.
  • #1
Mentz114
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292
I understand that the quantities

[tex]E^2 - B^2[/tex]

[tex]\vec{E} \cdot \vec{B}[/tex]

(the dot is vector inner product).
where E and B are the electric and magnetic components of an EM wave,
are invariant under Lorentz/Poincare transformations.
Can someone explain the physical significance of this ? Is either quantity related to the velocity of light ( or the invariance of the velocity of light ) ?

The second expression must be zero at all times surely ?
 
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  • #2
Mentz114 said:
I understand that the quantities

[tex]E^2 - B^2[/tex]

[tex]\vec{E} \cdot \vec{B}[/tex]

(the dot is vector inner product).
where E and B are the electric and magnetic components of an EM wave,
are invariant under Lorentz/Poincare transformations.
Can someone explain the physical significance of this ? Is either quantity related to the velocity of light ( or the invariance of the velocity of light ) ?

The second expression must be zero at all times surely ?

Not necessarily wave. An EM wave is just a particular case of a radiated EM field. That's why the scalar product is not always 0, because the radiated EM field is not always a wave.

There's not too much physical significance of the invariants, just that the first one is good for a lagrangian density since it leads to field equations second order in time.

Daniel.
 
  • #3
Thanks, Daniel.

I didn't know there are solutions to Maxwells equations other than the EM wave.

It's hard getting my head around the idea that the E and B fields 'mix' like space and time, when boosted.
 

1. What are invariant quantities in the EM field?

Invariant quantities in the EM field refer to physical properties or measurements that remain unchanged under certain transformations or changes in the electromagnetic field. These quantities are important in understanding and predicting the behavior of the EM field.

2. What are some examples of invariant quantities in the EM field?

Some examples of invariant quantities in the EM field include electric charge, magnetic flux, and total energy. These quantities are conserved and do not change even when the EM field undergoes transformations such as rotation or translation.

3. How are invariant quantities useful in studying the EM field?

Invariant quantities provide a way to simplify and understand the complex behavior of the EM field. By focusing on these quantities, scientists can make predictions and calculations without having to consider the effects of every small change in the field.

4. Can invariant quantities be measured in real-world experiments?

Yes, invariant quantities can be measured in real-world experiments. For example, the electric charge of an object can be measured using an electrometer, while the magnetic flux can be measured using a gaussmeter. These measurements can then be used to calculate other invariant quantities.

5. Are invariant quantities unique to the EM field?

No, invariant quantities are not unique to the EM field. Invariant quantities can also be found in other areas of physics, such as in mechanics and thermodynamics. However, in the EM field, these quantities play a crucial role in understanding the behavior of electromagnetic waves and their interactions with matter.

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