5.10: Energy Stored in a Capacitor
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Let us imagine (Figure V.10) that we have a capacitor of capacitance C which, at some time, has a charge of +q on one plate and a charge of −q on the other plate. The potential difference across the plates is then q/C. Let us now take a charge of +δq from the bottom plate (the negative one) and move it up to the top plate. We evidently have to do work to do this, in the amount of qCδq.
FIGURE V.10
The total work required, then, starting with the plates completely uncharged until we have transferred a charge Q from one plate to the other is
1C∫Q0qdq=Q2/(2C)
This is, then, the energy U stored in the capacitor, and, by application of Q=CV it can also be written U=12QV, or, more usually,
U=12CV2
Verify that this has the correct dimensions for energy. Also, think about how many expressions for energy you know that are of the form 12ab2. There are more to come.