A non-conducting square has a fixed surface charge distribution. Make a rectangle with the same area and total charge by cutting off a slice from one side of the square and gluing it onto an adjacent side. The energy of the rectangle is lower than the energy of the square because we have moved charge from points of high potential to points of low potential. An even lower energy results if we let the charge of the rectangle rearrange itself in any manner that keeps the total charge fixed. By definition, the rectangle is now a conductor. The electrostatic energy U_E = Q²/2C of the rectangle is lower, so the capacitance of the rectangle is larger than the capacitance of the square. Maxwell made this argument in 1879 in the course of editing the papers of Henry Cavendish. Much later, the eminent mathematician Gyorgy Polya observed that the conclusion is correct but that “Maxwell’s proof is amazingly fallacious.”

(a) Find the logical error in Maxwell’s argument.
(b) Make a physical argument which shows that C_rect > C_sq.