Adding up the energy stored in a capacitor

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A capacitor stores a separation of charge. Since to separate the charges on a capacitor, you have to move charges against where the E field wants to push them, charging takes work. That work becomes stored energy — just like carrying water up a hill so that you can let it roll down at a later time and turn a generator.

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A. Suppose the charges on the capacitor plates are equal to $+q$ and $-q.$ How much work does it take to carry a small positive charge $Δq$ from the negative plate to the positive one, thereby increasing the charge on the plates?

B. If you start with no charge on the plates and start to carry small charges over a bit at a time, the total charge $Q$ will increase.  Since the potential difference is proportional to the charge on each plate, the potential difference is a function $V(q)$ of how much charge, $q$ has been moved. The graph of the potential difference vs. the amount of charge on the plate will be a straight line, as shown by the dotted line on the graph at the right.   rom the geometry of this figure, or by doing the integral $U = \int^{\Delta V}_0 {V(q) dq}$ find an expression for the total amount of work it takes to charge the plates up to a value of $+Q, -Q.$

Joe Redish 4/19/08