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Comment by nh23423fefe

3 days ago

Can you use the homomorphism to get that?

F(r,q0,q1) = F(r,q0,q1+0) = F(r,q0,q1)+F(r,q0,0) = F(r,q0,q1) + 0

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nh23423fefe

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amavect  2 days ago

Actually, yes! We can use homogenous-degree-1 (provable from the homomorphism) to prove it.

  F(r,q0,q1)
  = F(r,q0*1,q1*1)
  = q0*q1*F(r,1,1)

Very simple! It remains to show that F(r,1,1) = K/r^2, as intended.

  • nh23423fefe  2 days ago

    I feel like that should be gettable from rotational symmetry and somehow mentioning L2 norm to get the square. Either that or conservation of E-field flux.

    • amavect  2 days ago

      Rotational symmetry or L^2 norm only matter in vector formulations. I assumed a scalar formulation.

      Conservation of E-field flux certainly implies additive-homomorphism (addition of charges equals addition of forces). But that seems a bit ahistorical because Maxwell would develop field theory 70 years after Coloumb. Either way, the axiom you choose requires empirical justification, and I think a home hobbyist could more easily demonstrate by experiment that adding charges will add the forces.

      Or, if you meant F(r,1,1) = K/r^2, then yeah, conservation of E-field flux could give you an inverse square law. But again, that requires an experiment to justify the axiom.