Spell out all physical assumptions used in your solution
Given that there is unbalanced charge on the two plates,
that charge must have come from somewhere ... perhaps
some sort of "chassis ground".
Therefore I assume this is really a /three-terminal/ device.
I then assume that the third terminal is so far that its
higher-order details don't matter much. Still, though, the
zeroth-order existence of the third terminal is important.
Otherwise we would have field lines with a beginning and
no end.
I should have documented this assumption when I first posted
my solution.
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Remark: This "parallel plate" system may seem like a
busywork exercise with no relevance to the real world.
The proverbial spherical cow in the ivory tower.
The original question is rather similar to twinax cable
with +2 volts on one conductor and -3 volts on the other.
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As another category of examples, a lot of sensors have
two symmetrical capacitor plates (plus ground). For example,
the accelerometer in your smartphone is almost certainly in
this category.
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Every so often somebody asks me to talk about "What is physics?"
I always start by saying I don't know where the far edge of
physics is, but I can tell you roughly where the center is.
One thing that is very near the core of physics is symmetry
and conservation laws. Noether's theorem tells us there is
a huge overlap between the idea of symmetry and the idea of
conservation laws, but let's not worry about that right now.
The twinax has a high symmetry that we can uses to our
advantage.
Rather than thinking in terms of signal A and signal B,
it pays to think in terms of the differential signal A-B
and the common-mode signal (A+B)/2. A purely differential
signal will produce an antisymmetric solution, while a
purely common-mode signal will produce a symmetric solution.
The general signal has no particular symmetry, but you
can always pick it apart into a symmetric piece and an
antisymmetric piece. Solve the two sub-problems separately,
then combine the answers. This is always possible for a
linear system.
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If you think one objective of the course is to teach students
to "think like a physicist" you should call attention to the
importance of symmetries.
-- If there's a symmetry, find a way to use it.
-- If the symmetry is hidden, it's worth your trouble
to find it anyway.
The symmetry-based solution is likely to be easier, more
powerful, more memorable, and more elegant.