8. Further Improvements
Our presentation of Maxwell's equations can be enhanced by rescaling the units of the electric field.
Everything gets even simpler, more beautyful, of higher symmetry, ...
- We define E = Enew := Eold / c . So E and B are dual, no longer E and c·B .
E and B now have the same units 'Tesla'.
- We divide both matrices F and M by c and ( with the new definition of E ) we have
No factor c needed any longer.
- c disappears in the law of the electromagnetic force, too: ( german 'statt' means 'instead of' )
- Maxwell's equations get slightly simplified:
- The determinants of F and M get rid of the factor c2 : Newly we have det(F) = det(M) = – (E·B)2
- The poduct of our matrices reduces to F·M = (E·B) · Id
- The other invariant is simplified to E2 – B2 , and we have det (F + M) = det (F – M) = – (E2 – B2)2
- The equations describing the transformation of the components of E and B under a Lorentz boost get completely symmetric now:
Earlier we had v instead of beta on the left side and v/c2 instead of beta on the right side! The 'duality' of the E and B field is completely established now ! ( beta = v/c )
- Of course, all this simplifications and enhancements of symmetry follow, if you change the units of space and time in a way the speed of light reduces to 1 (without any units). Long ago C.F. Gauss has made an even more radical suggestion: Change the units of the electric and the magnetic field in a way that both constants e0 and µ0 reduce to 1 ! Hence c will be equal to 1, too, and in our notation of Maxwell's equations µ0 will disappear (point 4 above).
The redefinition of E following point 1 above is just the most minimal operation to achieve the intended goal.