A Capacitor Paradox
Kirk T.
McDonald
Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544
(July 10, 2002)
1 Problem
Two capacitors of equal capacitance C are connected in parallel by zero-resistance wires and
a switch, as shown in the...
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A Capacitor Paradox
Kirk T.
McDonald
Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544
(July 10, 2002)
1 Problem
Two capacitors of equal capacitance C are connected in parallel by zero-resistance wires and
a switch, as shown in the lefthand figure below.
Initially the switch is open, one capacitor
is charged to voltage V0 and the other is uncharged.
At time t = 0 the switch is closed.
If
there were no damping mechanism, the circuit would then oscillate forever, at a frequency
dependent on the self inductance L and the capacitance C.
However, even in a circuit with
zero Ohmic resistance, damping occurs due to the radiation of the oscillating charges, and
eventually a static charge distribution results.
1.
How much energy is stored in the system before and (a long time) after the switch is
closed? The paradox is that these two energies are not equal.
2.
Verify that the “missing” stored energy has been radiated away by the transient current after the switch was
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