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Prove a formula for the work required to place a charge on an uncharged condensor

Define V(q) to be the voltage necessary to place a charge q on the plates of a condensor. Then we define

    \[ \int_a^b V(q) \, dq \]

to be the work required to charge a condensor from q = a to q = b. Then, assuming the voltage is proportional to the charge, prove the work required to place a charge Q on an uncharged condensor is

    \[ \frac{1}{2} Q V(Q). \]


Proof. Since we are assuming the voltage is proportional to the charge we have V(q) = cq for some constant c. Also,

    \begin{align*}  W &= \int_a^b V(q) \, dq \\    &= \int_a^b cq \, dq \\    &= c \left. \frac{q^2}{2} \right|_a^b \\    &= \frac{1}{2} (cv(b) - cv(a)) \\    &= \frac{1}{2} Q V(Q).  \end{align*}

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