Examples of invariant quantities

[BookWei]

In SR, we know that $\vec E \cdot \vec B$ and $E^{2}-B^{2}$ are invariant.
Although I can prove those two invariant physical quantities mathematically, I do not know how to find at least
one example to demonstrate that $\vec E \cdot \vec B$ and $E^{2}-B^{2}$ are invariant.
Many thanks!

 

[PeterDonis]

I do not know how to find at least
one example to demonstrate that $\vec E \cdot \vec B$ and $E^{2}-B^{2}$ are invariant.

Here are two that you can try:

(1) In one particular frame, the EM field is a static electric field, say the field due to a point charge at the spatial origin. The values of the two invariants are obvious in this frame. Then transform to another frame and verify that the invariants stay the same.

(2) An electromagnetic wave in vacuum.

 

[sweet springs]

Say
$$\mathbf{B}\cdot\mathbf{E}=0,$$
B and E are perpendicular in all IFRs.

Say
$$\mathbf{E^2}-\mathbf{B^2}=0,$$
|B|=|E| in all IFRs.

Electromagnetic wave in vacuum is a good example that has these properties in any IFR.

 

[sweet springs]

As for electrostatic cases where all charges are still and B=0 everywhere, the invariant says that |E|>|E0| where E0 is original electric field and E is one Lorentz transformed. I assume moving charges make more electric field because multiple signals are sent from the past charge positions, or Lorentz contract of distance enlarge Coulomb force. I will appreciate it if someone could show me a precise explanation.

 

[PeterDonis]

I assume moving charges make more electric field

They also make a magnetic field; in any frame other than the rest frame of the charge, B is nonzero.