Re: Etesla6 math questions

["Tesla list" <tesla-at-pupman-dot-com>]

Original poster: "Paul Nicholson by way of Terry Fritz <teslalist-at-qwest-dot-net>" <paul-at-abelian.demon.co.uk>

Further to Terry's answers,

Peter Lawrence wrote:

 > 3. The E-field at any point of the enclosing surface can be computed
 > as the (vector) sum of the E-fields from all the points on the
 > surface of the object(s) inside the enclosing surface.

To be more precise, it is the electrostatic potential which E-Tesla
is adding up, which is a scalar quantity.   Each bit of charge on
the coil/topload contributes a potential

   Q/(4*pi*epsilon*r)  (volts)

to the potential at your test point, where Q is the bit of charge, and
r is the distance of it from your test point.  This is Coulomb's law
which applies when the field is constant, or in our case, only slowly
varying with time.

So the actual potential at your test point is the scalar sum of all
these contributions.   The E-field is then the gradient of this
potential.

 > 4. The E-field at a point on the object(s) inside the enclosing
 > surface depends on the charge density at that point.

Not necessary to mention the enclosing surface here.  Simply that as
your test point approaches the object's surface, its potential (and
potential gradient, ie the field strength) becomes more and more
dominated by the bit of charge at the point on the surface which you
are approaching.  In the limit, the field strength approaches
(surface charge density)/epsilon, which is the surface field strength,
and varies from place to place as the charge density does.

Electrostatic potentials (and a lot of other things (*)) satisfy
Poisson's equation, and it is this equation which E-Tesla6 is solving
for the space around the coil.  A numerical technique for doing this
is called 'relaxation' and is a great intro to numerical techniques of
field calcs.  Poisson's equation mathematically describes a certain
simple kind of 'smoothness' to the field.  If you make a guess of the
field in a region of space, it will probably not satisfy Poisson's
equation too well, but if you refine your guess by setting the
potential at each point of your region to the average potential of
its neighbours, then that will improve your estimate.  This process
is repeated iteratively until the solution is sufficiently good.

 > In statement (3) this is independent of whether the line from a
 > point on the object's surface to the measurment surface crosses
 > through the object or not (we're assuming the object(s) is
 > conductive).

Remarkably, yes!  Coulomb's law always applies, even if the path
along which 'r' is measured passes through a conductor.  Doesn't the
intervening conductor 'shield' the test point, you might ask?  Yes, it
does so, by taking on its own surface charge.  Free charges within
the shield take up a distribution such that the potential over the
shield is a constant.  *Their* Coulomb contribution to the total
potential at your test point subtracts from the 'direct' contribution,
to give the familiar shielding effect.  So conductors don't actually
block the Coulomb field Q/(4*pi*epsilon*r), instead they add in their
own contributions.

 > In statement (3) what if the line crosses significant amounts of
 > dialectric.

This is similar to the above case.  Coulomb's law continues to work
in the presence of dielectrics along the path 'r', and the permittivity
(epsilon) continues to have its free space value!  In this case, there
are no free charges - all that happens is that the bound charges are
pulled apart a little.  Positive charges are drawn one way, negative
the other, but the material as a whole remains neutral, at least on
the inside [+].  Only on the surface of the dielectric (or on the
boundary between two dielectrics) is there any net charge.  So the
dielectric can be replaced by a volume of empty space with some charge
distribution over its surface.  A conductor sorts out its surface
charge distribution to make the potential the same all over.  It turns
out that a dielectric sorts out its surface charge distribution to
make both the field potential and the flux density be continuous
functions across the dielectric surface.

It's as well to remember that the free space permittivity applies also
inside materials (conducting and dielectric).  But the free and bound
charges add their own bits to the field and the overall effect within
the body of the material is that it behaves as if the permittivity is
some other value.  But if you don't care about the innards of the
material, you can just represent a blob of dielectric by some suitable
surface charge distribution and work entirely with the free space
permittivity. (Google for something like 'equivalent charge method').

 > It seemed to us that even though the voltage on the toroid is
 > constant across all points on its surface, the charge density would
 > not be,

Yes, true for any conductor.  Free surface charges fall down through
the field to find their own level, which occurs at uniform potential.
Thus uniform potential goes hand in hand with non-uniform charge
density.  The business of calculating the charge distribution over a
given conductor is known in physics as 'the capacitance problem',
which should give lots of hits on Google.

 > ditto for the secondary solenoid.

Worse for the secondary, because the potential is not uniform to begin
with.  Different parts of the coil are joined by a wire immersed in
an alternating magnetic field, so there must be a non-uniform
potential.  Free charges now move to their most relaxed positions with
respect to this potential gradient.

'Capacitance' is just a measure of the total amount of charge
displacement required to meet some given potential distribution.
We normally take this to be a uniform potential when we're talking
about capacitors, but it needn't be.  When the system we're dealing
with inherently has a non-uniform voltage on it, it makes sense to talk
about C with respect to this, because this is the capacitance which is
'in effect', in other words it properly quantifies the total charge
displacement in the system.   This is roughly what we're doing when we
talk about 'equivalent capacitance', and this is the value computed
by E-Tesla.

 > we have a major problem computing it (and I think that is the
 > "trick" in Etesla6, but I cannot remember what it is),

Yes.  We have a vicious circle: The H field tries to set up an E-field
potential distribution (by induction). Charges move in accordance with
it, and their movement (currents) sets up the H field!  Breaking this
circle is what solving the field equations is all about:  Find a
charge distribution, whose assembly involves currents which create a
magnetic field which is just right to create your charge distribution!

Turns out that for any given TC, only certain patterns of charge
distribution actually fit into this circle of dependency.  We call
these 'resonances', and it requires a fair amount of computational
effort to calculate them.

E-Tesla6 avoids this obstacle by using an estimated voltage
distribution, which is quite reasonable since the approximate shape
of the distribution can be estimated on fairly general grounds.
Estimates have been refined as the program evolved, using results
from more long winded modeling.  As far as I know, ET was the first
program which computed TC resonances based on physical principles
rather than empirical formulas.

[+] A +ve charge shifting to the left is replaced by another coming
in from the right, which is shifting to the left of *its* equilibrium
position, and so on, throughout the body of the material, which thus
remains neutral.  Only on the surfaces does this 'cancellation' fail
to work (there is nothing on the 'right').

[*] The magnetic vector potential also satisfies Poisson's equation,
separately and independently for each of the 3 components.
--
Paul Nicholson,
Manchester, UK
--