N. 3 - 21 aprile 2001 - Giano Bifronte

182
qE = -(q/c) d
dt Am
acts on the stationary charge but no opposite force acts on the solenoid.
It is convenient to reconcile, within the standard relativistic
interpretation, the force on the charge with the expression of the
opposite force on the solenoid or line of magnetic dipoles of moment
m .
When both q and m are at rest we have a stationary configuration, and
the em momentum in can be written, without approximations, as
Pem = -Ph = (1/c) ∫ ρAdτ = (1/c 2 ) ∫ ΦJdτ = (1/c) ∫ (E×M)dτ =
= -(1/c) m×E = -(q/c) m×r/r 3 = (q/c)Am , (6)
where Ph is the hidden momentum or momentum due to the internal
stresses, [7], Φ is the scalar electric potential, M the magnetization
density and r = xm - xq the distance between m and q. In (6) E has to be
evaluated at the position xm of the dipole and the vector potential
Am = -m×(xm-xq)/⎜xm-xq⎜ 3
of the dipole has to be evaluated at the position xq of the charge. Using
the result that for stationary system the total momentum is zero [7],
P = Pem + Ph = 0
we conclude, as mentioned above, that the total system cannot acquire a
net mechanical momentum when the current is switched off, and find
that in correspondence of the action force on q
fq = - d
dt Pem = -(1/c) d
dt ∫ ρAdτ = -(q/c) d
dt Am ,
when the current of the dipole varies with time, there is an equal and
opposite reaction force fm = -fq ,
fm = - d
dt Ph = (1/c2 ) d
dt ∫ ΦJdτ = -(1/c) d
(m×E) (7)
dt
acting on m , that solves paradox of Shockley and James.