At 10:58 PM 12/8/4, Stephen A. Lawrence wrote:
>The basic formula for A at a particular point, from Rindler, 2nd
>edition, p. 111, or Griffiths, 3rd edition, p. 423 is just
>
> A = (1/4pi)integral([J]dV/r)
>
>where the integral is taken over all space, [J] is the retarded value of
>the 4-current density, and r is the distance from the point where one is
>evaluating A. Since J is time invariant in this case, [J] = J. Each
>component of J is integrated separately, which means
>
> phi = (1/4pi) integral(rho dV/r)
>
>where phi = electric potential and rho = charge density.
>
>To look at it yet one more way, if you're looking at a case where the
>current is not varying, then you're in the domain of magnetostatics and
>you don't need anything beyond simple E&M to analyze it. Fancier
>approaches, such as the pancaking model, must agree with a simple
>analysis in simple cases.
I have such uneasy feelings about the vector magnetic potential A,
especially in the context of relativity, due to my lack of understanding I
assume.
Please excuse this momentary lapse into humor.
Humor shield on. ********
If the vector magnetic potetnial were real in any sense, then it seems to
me we should build a National Vector Magnetic Field Facility (VMFF). Using
wire made of twisted pair superconductors, there is no limit other than
financial to the intensity of the VMF that can be created by a coil of
such, due to the lack of force when A is not changing. The VMF shows up
in a real way quantum phenomena, so maybe unusual things would happen at
the center of the coil. The facility would also have political advantages
for the operators. Unlike operators of future great tokamaks that happen
to not break even, the VMFF operators would have a handy excuse. "Someone
changed the guage, so we dont know where the VMF went." 8^)
Humor shield off. ********
If it is true that the definition:
@f/@t = lim dt->0 ( f(x,y,z,t+dt) - f(x,y,z,t) )/dt
holds in a vector point field, for f a vector at point (x,y,x,t), then in
an FEA simulation one can simply approximate:
@A/@t = (1/dt) ( A(x,y,z,t+dt) - A(x,y,z,t) )
where (1/dt is now a finite scalar and ( A(x,y,z,t+dt) - A(x,y,z,t) ) is a
vector subtraction. We thus use a finite number of values for x,y,z,t to
simulate the results, based on starting conditions at time 0.
SIDE NOTE: To compute A(x,y,x,t), given the set of current filaments
comprising the torus, we can use:
A = line integral of [(mu I dL)/(4 Pi R)]
where R is the distance from the current element to the point at which the
vector magnetic potential is is being calculated. This sets the guage, by
assuming div A = 0, but the guage is arbitrary.
Consider now a toroidal coil (or just "torus") carrying constant current
moving in space along its major axis, which is also the x axis, toward an
electron at point (x,0,0). Since the current is constant we should be
able to assume both B and E outside the torus is zero.
<- O Torus cross section
(x,y,z)
(e-)....................... Main axis of torus
<- O Torus in motion
Now, since the coil approaches (x,y,z)==(x,0,0) for some fixed x, the
magnitude of the magnetic vector potential A(x,0,0,t+dt) is larger than
the magnitude of A(x,0,0,t). Using the definition of @A/@t above would
imply that the electric field E = @A/@t imposed at (x,0,0) by the moving
torus is non-zero.
Now, consider a toroidal coil carrying constant current located at the origin
with its major axis the x axis. An electron at point (x,0,0) has initial
velocity v towards the origin. Since the current is constant we can
assume E outside the torus is zero.
O Torus cross section
(x,y,z)
(e-)...........>..........X Target "X" at origin
O
Since the electron at (x,y,z)==(x,0,0) approaches the origin, i.e. dx/dt =
v, then x diminishes with time, and the magnitude of the magnetic vector
potential A(x+v*dt,0,0,t+dt) is larger than the magnitude of A(x,0,0,t).
In other words dA/dt is non-zero for the electron, even for a fixed
current toroidal coil. We can not assume @A/@t is non-zero because dA/dt
is non-zero? Using the definition of @A/@t = dA/dt here implies that the
electric field E = @A/@t imposed at (x,0,0) by the stationary fixed
current torus at the origin upon a moving charge is non-zero. Yet we see
that A(x,0,0,t) = A(x,0,0,t+dt), and A(x+v*dt,0,0,t) =
A(x+v*dt,0,0,t+dt), so @A/@t at (x,0,0,t) = 0, and @A/@t at
(X+v*dt,0,0,t+dt) = 0. At no time should there be an E = @A/@t
experienced by the electron. Yet, as we saw above, if we change reference
frames to that of the electron, both dA/dt and @A/@t are non-zero. This
seems to indicate that the E experienced depends on velocity relative to
the source of the A.
If it is true that E depends on relative velocity with respect to the
source of A, then we merely aim an electron beam at target "X" and it
accelerates there with no energy applied. The torus could be a circular
permanent magnet (with field fully enclosed) for that matter.
Alternatively, if we assume that there is only one E at a point, then all
motion can not be relative, because the results of the torus moving toward
the electron differ from those of the electron moving toward the torus.
Any idea how is this might be simply resolved?
Regards,
Horace Heffner
