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Re: Confused by a derivation. (A long, multi-part quiz)



At 15:45 2002/02/03, Ludwik Kowalski wrote:

... I was referring to a situation in which
net charges on the original plate migrated to one side of the plate. The
field near that side of the plate doubles because all field lines (and
not only 1/2 of them) are now emerging from that side. The field on
the other side is zero because the local sigma there is zero.

Let's examine another viewpoint, one that I have not yet seen expressed in
this thread. Take this quiz to see if you can develop this viewpoint on
your own.

Question 1. Suppose you encountered a large but infinitesimally thin (with
no interior structure) planar layer of charge of total magnitude Q/2 (the
reason for this choice of magnitude should become clear below) and
one-sided area A. Further suppose that this charge layer could magically be
made to stay in place without the help of a supporting structure (e.g.,
without either a conducting or non-conducting plate to give it structural
support). What does Gauss's law predict for the electric field strength in
the space near either side of the planar layer? (Keep in mind that the
effective surface area is 2A). Hint: In MKS units, the denominator should
contain a factor of 4, derived from the pair of 2's given in the problem
statement.

Question 2. Suppose you had two !identical! planar charge structures of the
type described in Question 1, and arranged them so that they lay parallel
to one another at a very small, but finite, separation from each other.
Further suppose that they are magically constrained from moving away from
each other as a result of their mutual electrostatic repulsion. (a) What
would be the total charge of the combination? Hint: The answer may be
expressed without using a denominator, and is not zero. (b) What would be
the electric field strength in the narrow region between the charge layers?
[Use superposition.] (c) What would be the electric field strength in other
regions of space? (Do not consider any region where a singularity may
result.) (d) In answering parts (b) and (c), what did you assume about the
spatial extent of the flux lines emanating from either charge layer? In
particular, (i) do the flux lines from either layer extend into space a
finite or infinite distance away from that layer, and (ii) do they extend
into the region between the charge layers?

Question 3. Suppose that you inject into the region between the two charge
layers a copper plate whose thickness equals the original separation of the
planar layers. (a) Have you in effect constructed a single conducting plate
with a net charge of Q? (b) Does the presence of the copper change any of
the results or assumptions you determined in Question 2? (c) In
electrostatics, is it necessary that flux lines NEVER enter a conducting
region, or is it OK for electric field lines from nearby charge
distributions to enter/leave a conducting region as long as the
superposition of ALL flux lines entering/leaving the region results is
zero? (d) Are your results for the electric field within and around the
single conducting plate consistent with what textbooks predict?

Question 4. Place a second copper plate, identical to that constructed in
Question 3 except that it carries a total charge -Q, near and parallel to
the copper plate constructed in Question 3, with mutual separation d. (a)
Have you effectively constructed a charged parallel-plate capacitor? (b)
Consider the fact that the charges on each plate surface have likely
reconfigured themselves in response to the presence of opposite charges on
the neighboring plate. Does this reconfiguration change the fact that flux
lines from each individual charge layer will still penetrate each plate
(including the opposite plate!), as they did in Question 3(c)? (c) Despite
the response to question 4(b), is it still possible to use superposition to
show that the electric field within the plates and outside the capacitor is
zero? (d) Is it possible to use superposition to derive an expression for
the electric field strength between the plates that is consistent with
textbooks' predictions?

If you think my quizzes have lots of parts to them, you should see my
exams! ;-)

--MB