Re: Electrostatics (was Charge distribution on a Toroid (was spheres vs toroids)

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Original poster: Paul Nicholson <paul-at-abelian.demon.co.uk> 

Robert Jones wrote:
 > 1. ... 5.

Hi Bob, all ok on those.

 > However due to 2 the force is only a function of the charge on
 > an object where as 3 and 5 suggest it's also a function of
 > the capacitance???

 > Were have I gone wrong?

You haven't.  The potential field in (3) is consistent with the
force field in (2), and with the material polarisation in (5).

Let's go through these points.

1.
 > 1. Charge is the fundamental quantity and can be positive and
 > negative

Yes.  Indeed, we don't actually have to say what 'charge' really is
in order to understand EM.  All we need is the simple fact that you
can take a bag of assorted particles and sort them into two piles,
those that react with each other with the 'electric' force, and
those that don't.  Discard the particles that don't.   Call the
other ones 'charged' particles and carry out another sort: this time
forming two piles so that within each pile all the particles repel,
and so that any two particles picked, one from each pile, will
attract.

Since charge can be defined and recognised 'operationally' like this,
Bob's definition is pretty much all we need to go on!  And it's easy
to go on to define ways to quantify the 'amount' of charge each
particle has - invent any units you like!

Then measure how the force varies with distance and amount of charge
to reach...

2.  The inverse square law for force between two particles.  This
introduces the 'force' field, otherwise known as the electric field.
Should we think of the 'E' field as something 'real'?  Other areas of
physics suggest that fields do indeed have much the same 'ontological
status' as material particles, but for our purposes just treat E as
a mathematical device to help with calculating the forces that charges
apply to each other.

3. Drag a 'test' particle through the field and it will gain or lose
energy, with the total energy change being the integral of force
times distance along the path.  For the electrostatic field it doesn't
matter what path was chosen - the energy only depends on the location
of the end points.   This allows us to define a 'potential energy'
field.  Integrating the inverse square 'force' law gives a related
law for the potential energy: the Coulomb law which, thanks to the
integral, is now only inverse not inverse square.

4. Flux: I'll come back to this one...

5.
 > The energy required to move a charge is a property of the material
 > surrounding the location that the charge will reside at.

Yes, the material is some collection of charged particles, which
sets up a more or less complicated electric field inside the material,
along with its associated potential field.  Your moving charge,
struggling through this potential landscape, collects or loses energy
in accordance with (3), and experiences a force in accordance with
(2).  So no conflict there.

In (5) you also mention stored energy and permittivity.  This first
is a measure of the energy required to set up this collection of
material charges - call it chemical energy if you like, and it
tends to be negative.  The second is a measure of how much the
'internal' field raised by the arrangement of particles will subtract
from an incident field.  I'll come back to this when I talk about 4.

As for the capacitance in (5), this is just a measure of how much
charge is needed to raise a given conductor to a given voltage. In
this sense the charge on the conductor is a function of its
capacitance, and therefore other charges in the neighborhood will
experience a force from the conductor dependent on its capacitance.
But this is just another example of the force law (2) in action.

As for 4. that's a trickier one.  The trouble with the force field E
is that it's not a conserved quantity.  Imagine a source charge
and lines of E radiating outwards from it, steadily weakening by the
inverse square law.  Ideally if we integrate E over any surface
enclosing the source we should get a constant proportional to the
amount of source charge.  And indeed this is true in free space.
But lets say our radial field passes through a layer of dielectric.
Then the field E drops by a factor 1/(dielectric constant), only to
rise again as it passes out beyond the layer.  Clearly this screws
up the conserved quantity - if our surface integral took in a bit
of the dielectric, the total would fall short!  The reason E drops
inside the material is because the material charges have displaced
in response to it, thus putting in their own contribution to the
field - and partly cancelling our incident E.

Fortunately the solution is easy.  We just define another vector
field: roughly, D = E * dielectric_constant, and you can probably
see that this quantity always remains conserved.  This we call 'flux'
until somebody can think of a better name.  Of all the menagerie of
EM quantities, this one is probably the least intuitive.

 > ...I know the above are not rigorous definitions but I think they
 > will do.

They did just fine.  Hope this little monologue helped!
--
Paul Nicholson
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