M8   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, ...
 

  1. 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'.

  2. We divide both matrices F and M by c and ( with the new definition of E ) we have

    No factor c needed any longer.

  3. c disappears in the law of the electromagnetic force, too: ( german 'statt' means 'instead of' )
  4. Maxwell's equations get slightly simplified:
  5. The determinants of F and M get rid of the factor c2 : Newly we have det(F) = det(M) = – (E·B)2 

  6. The poduct of our matrices reduces to F·M = (E·B) · Id  

  7. The other invariant is simplified to E2 – B2 , and we have det (F + M) = det (F – M) = – (E2 – B2)2 

  8. 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 )

  9. 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).

  10. The redefinition of E following point 1 above is just the most minimal operation to achieve the intended goal.