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So for the poynting vector, we have $\vec S=\vec E\times\vec B$, which would give us a vector (energy flux) that is perpendicular to both the electric field and magnetic field. The problem for me is that it reminds me of torque, which is a vector and it's direction follows the right hand rule, which is just a convention.

So in that case, would the energy flux direction be arbitrary?

I ask this because there is the case where a system with a static electric field that is perpendicular to a static magnetic field has an energy flux in a certain direction.

Would the energy flux look like a dipole?

Edit: I couldn't figure out how to type in the right form of the equation, so thanks!

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No, the direction of energy flux is physical, not arbitrary.

The direction of the magnetic field, on the other hand, is totally arbitrary. We follow the convention that a single equation, such as the Lorentz force law

$$ \vec F=q\vec v\times\vec B$$

follows the right hand rule, and this forces us to follow the same convention for all the other laws in order for them to be consistent.

We could redefine the magnetic field to follow the left hand rule instead, and the Poynting vector would then also follow the left hand rule.

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