Physics Puzzle: Where has the Energy Gone?
Here’s a classic physics textbook braintwister. Take a look at the following circuit:
It has two capacitors (with capacitance C each) and two switches. The left capacitor is charged and the right is uncharged. The energy of a capacitor with capacitance C and charge Q is given by Q²/2C, so the total energy in this system is Q²/2C. Now imagine the switches are closed. Symmetry and charge conservation dictate that the charge now distributes itself equally among the two capacitors:
The total energy of the system is now the sum of energies of the individual capacitors: (Q/2)²/(2c) + (Q/2)²/(2c) = Q²/4C.
It seems half the energy has disappeared! How can this be?
EDIT, 21/Apr/2009: A solution has been added in the comments section. Just read the comments to see it!
Solution, added 21/Apr/2009: the “catch” is that you’ve assumed the charge would distribute evenly between both capacitors once the switches are closed. Sure, eventually, due to friction and whatnot, it will; but who says the charges will neatly flow to their equilibrium position? In fact, if it weren’t for the inherent dissipation in the system - the resistance of the wires, the fact that closing the switch instantaneously might trigger a spark, etc. - the charges would never reach equilibrium. It’s like pulling a pendulum back and letting it swing. Eventually it would settle at the bottom and stop, but that only happens because of friction involved in its motion; an ideal pendulum, without any friction, would continue swinging forever, converting energy from potential to kinetic and back again. Thus, what we’ve done is assumed implicitly all of the kinetic energy of the charges has already been dissipated. The precise method of dissipation depends on the precise form of the circuit, e.g., how resistive are the wires? How ideal/nonideal are the switches? And so forth.
Challenge: suppose the only source of friction is due to the resistive wires. Can you estimate the amount of time it would take the system to reach equilibrium after closing the switches?