N. 3 - 21 aprile 2001 - Giano Bifronte

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because the gradient term gives zero contribution to ΔV in a closed
loop.
Equivalently, (2) reads
∫ Eeff.dl = -(1/c) ∫ [∂tA-(v.∇)A].dl . (4)
We see from the integral forms (2) and (4) that the velocity dependent
term is somewhat arbitrary as it can be defined to within the gradient of
a scalar function. The correct expression of the force must be
determined by the internal consistency of electrodynamics and by
experimental verification.
The velocity dependent effective field term of (2), E⊥ = (1/c)v×B ,
exhibits the correct structure of the force field perpendicular to v as
verified experimentally in charged particle accelerators. On the other
hand, the parallel component of the velocity dependent effective field
term of (4), E⎢⎢ = -(1/c)(v.∇)A⎢⎢ = -(1/c)∇⎢⎢(v.A) , exhibits the correct
structure of the force field, parallel to v , that can be shown to be
consistent with the conservation of energy [12].
If the charge moves with velocity v in a circuit defined by the element
dl = vdt , then
∫ (v.∇)A.dl = ∫ (v.∇)A⎢⎢.dl = ∫ ∇⎢⎢(v.A).dl = 0 ,
so that to the standard velocity term of (2) we can add the term ∇⎢⎢(v.A)
without altering the emf of the Faraday Law in integral form. With the
new longitudinal term the velocity terms read
v×B - ∇⎢⎢(v.A) .
Thus, in this case, the standard Lorentz force
∫ [ ρE + (1/c)J×B ]dτ ,
may be implemented by an extra term that takes the form
f = -(1/c) ∫ ∇⎢⎢(J.A)dτ . (5)