> But it's not at all obvious to me why the exponent a is 1 in nature
i think we can rule out any exponent just by dimensional analysis if you allow powers of q, then K has ambiguous units. Same reason you cant exponentiate unitful quantities
More specifically I think we can rule out even exponents by anti-symmetry of charge. That is q^2n = (-q)^2n which know is ruled out by experiment.
Why would the universe care about dimensional analysis? Besides, the outside constant would do the unit conversion from whatever the right-hand side produces to units of force.
This is making assumptions about the nature of charge and its relation to force that are what is under discussion! We get no guarantee from nature that the force is linear in the amount of charge; if it turned out to be sqrt(charge) then we'd still have to deal with it. We could redefine charge to the sqrt of the previous quantity, but then other aspects of linearity (like conservation of charge) would have to take a different form.
We're lucky (and maybe only in the linear domain, which is more accurately the case) that these ended up being the same.
But if it were sqrt(charge) then we'd define Culoumbs constant to have commensurate units to translate sqrt(charge) * sqrt(charge) / distance^2 to force units.
Partitioning a charge can't change the physics. If i have a bar of charge Q generating a force F on q which is proportional to sqrt Q then by partition i have n bars of charge Q_i = Q/n producing forces F_i
sqrt Q ~ F <> sum F_i = sum sqrt(Q_i) = sum sqrt(Q/n) = n * sqrt (Q/n) = sqrt (Qn)
Partitioning charge would lead to infinite forces.
I thought it was "obvious" based on the principle that two charges at the same location should have the same force as one combined charge at that location. Of course this immediately brings up the question of the self-force of a point charge...
Show by experiment that the force of charge0 against charge1+charge2 equals the force of charge0 against charge1 plus charge0 against charge2. Induce an additive-homomorphic property F(r,q0,q1+q2) = F(r,q0,q1)+F(r,q0,q2). Then, exponent 1 follows.
I figured this out by listing a bunch of mathematical properties. I couldn't see how the author jumps from zero-preserving to multiply-charges, and I still don't know how, but we can call it out of scope lol
r : distance between p and q
q0 : charge 0
q1 : charge 1
F : coulomb force function
charge-commutative: F(r,q0,q1) = F(r,q1,q0)
zero-preserving: 0 = F(r,q0,0)
additive-homomorphic: F(r,q0,q1+q2) = F(r,q0,q1)+F(r,q0,q2)
homogenous-degree-1: F(r,q0,n*q1) = n*F(r,q0,q1)
multiplicative-separability: F(r,q0,q1) = K*R(r)*Q(q0,q1)
multiply-charges: F(r,q0,q1) = K*R(r)*(q0*q1)^a
Given F(r,q0,q1) = K*R(r)*(q0*q1)^a, charge-commutative, zero-preserving, additive-homomorphic.
Induction using additive-homomorphic proves homogenous-degree-1. (For example, F(r,q0,2*q1) = F(r,q0,q1+q1) = 2*F(r,q0,q1))
Equational proof follows from homogenous-degree-1:
K*R(r)*(q0*n*q1)^a = n*K*R(r)*(q0*q1)^a
(q0*n*q1)^a = n*(q0*q1)^a
n^a*(q0*q1)^a = n*(q0*q1)^a
n^a = n
n = 0 or a = 1
n≠0, therefore a=1.
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.
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.
nh23423fefe · · focus · HN ↗
i think we can rule out any exponent just by dimensional analysis if you allow powers of q, then K has ambiguous units. Same reason you cant exponentiate unitful quantities
More specifically I think we can rule out even exponents by anti-symmetry of charge. That is q^2n = (-q)^2n which know is ruled out by experiment.
andrewla · · focus · HN ↗
mitthrowaway2 · · focus · HN ↗
andrewla · · focus · HN ↗
We're lucky (and maybe only in the linear domain, which is more accurately the case) that these ended up being the same.
But if it were sqrt(charge) then we'd define Culoumbs constant to have commensurate units to translate sqrt(charge) * sqrt(charge) / distance^2 to force units.
nh23423fefe · · focus · HN ↗
Partitioning a charge can't change the physics. If i have a bar of charge Q generating a force F on q which is proportional to sqrt Q then by partition i have n bars of charge Q_i = Q/n producing forces F_i
sqrt Q ~ F <> sum F_i = sum sqrt(Q_i) = sum sqrt(Q/n) = n * sqrt (Q/n) = sqrt (Qn)
Partitioning charge would lead to infinite forces.
mhh__ · · focus · HN ↗
NooneAtAll3 · · focus · HN ↗
any dimensional analysis gets consumed by its unknown dimensionality
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> q^2n = (-q)^2n which know is ruled out by experiment.
doesn't mean equation can't be using absolute values ("number of electrons/protons") and just applying needed sign at the end
nh23423fefe · · focus · HN ↗
it can't use absolute values, you can't write that equation down
jeremysalwen · · focus · HN ↗
amavect · · focus · HN ↗
I figured this out by listing a bunch of mathematical properties. I couldn't see how the author jumps from zero-preserving to multiply-charges, and I still don't know how, but we can call it out of scope lol
nh23423fefe · · focus · HN ↗
F(r,q0,q1) = F(r,q0,q1+0) = F(r,q0,q1)+F(r,q0,0) = F(r,q0,q1) + 0
amavect · · focus · HN ↗
nh23423fefe · · focus · HN ↗
amavect · · focus · HN ↗
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.