Friday, March 13, 2015

Interesting Physics Problem that seems to require a Mathematics Answer

Consider the situation shown. There are four small conductors, each with charge either +q or the opposite, -q. There are arranged as shown in (a). One +q is arbitrarily close to one -q. Likewise for the other pair of opposite charges. And the two sets of opposite pairs are arbitraily far apart. Now suppose (b) I connect the distant opposite charges with thin wires. What will happen? There are two possibilities:

1) Nothing happens, because (recall opposites attract) each charge +q will be much more strongly attracted to the opposite charge -q that is close by, because the strength of the attractions drops as the square of the distance. If I make the opposite charge at the end of the wire a thousand times farther away than the neighboring opposite charge, then the neighbors will have a million times stronger attraction. The charge on the end of the wire simply cannot overcome that attraction of the neighbor, and so everyone stays put, fat dumb and happy.


2) The charges will run along the wire, canceling each other out.

In the first case (as in (a)) there will be varying electric potential (voltage) and electric fields, the potential and field being very, very big near the charges. The second case is featureless: there will be no electric fields and constant (taken to be zero) electric potential. As if nothing is there.

These are two very different possibilities.

I don't know how to do "the answer is below the fold," or I would!

The answer is (2). The charges will run along the wire and cancel leaving us with no fields. I can't come up with a good physics explanation--to me the wrong answer (1) has a plausible physics explanation. But mathematics answers the problem unambiguously.  It does so via a uniqueness theorem that applies to the partial differential equations (from physics) that apply to this situation. A uniqueness theorem says something like this: If you find a solution, by any means, then you have found the one and only, i.e., unique solution. A problem cannot have two solutions.      

The relevant uniqueness theorem is this: In a situation such as that shown, the electric field is uniquely determined if the total charge on each conductor is given.

But when we connect the charges by wires we effectively covert the two separate conductors, one with +q and one with -q, into a single conductor with zero total charge (c). So our four conductors become two conductors with no charge. Such an arrangement has no field and a featureless potential.

This is an unusual problem in that (at least to me) the "obvious" physical solution is trumped by the mathematics. A uniqueness theorem from the field of partial differential equations say the answer, physics tuition be damned, must be (2), and so it is.

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