Why q = CV?

This post will be brief. Why is it that in some textbooks, the definition of capacitance is taken to be the equation $q = CV$ ? (Related: $V = iR$ )

To me, this is just as bad as stating the equation for density as $m = \rho V$ (or that for velocity as $d = vt$ ). It might be correct, but we write $\rho = \frac{m}{V}$ firstly to emphasize the importance of the quantity we are actually defining (the density) and secondly because it explains what density really is: a quantity that is higher when you can stuff more mass into a smaller space.

Now, the first time that I read through Fundamentals of Physics by Halliday and Resnick (3 years ago?), I skipped the chapter on capacitance just because I couldn’t understand what the equation $q = CV$ was really defining. In retrospect, I was probably just being dim or lazy that day; I should have made more of an effort. But I maintain that what confused me is that the very term capacitance, plus the fact that capacitors store charge, implies that capacitance should express how well a structure can hold charge, and that’s not obvious at all from $q = CV$ , because it’s not clear where the $V$ is coming from.

If we write the equation as $C = \frac{q}{V}$ , it becomes much more clear. Not only does this actually define $C$ , but it also shows what high capacitance really means: it’s the ability to hold a lot of charge without building up too much of a potential difference (after all, when charging a capacitor, once the potential difference builds up to the level of the EMF, the capacitor stops charging). And we generally tend to think of the $V$ as being set up by the $q$ , rather than the $q$ being a consequence of the capacitance $C$ and some potential difference $V$ that’s already present, which is what $q = CV$ suggests.

Of course, the equation $q = CV$ , as presented, has its value too — to cast capacitance as a conversion factor between the stored charge and the established potential. Just as $m = \rho V$ and $d = vt$ have their value. Just not as definitions.

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This entry was posted on March 3, 2011 at 9:15 pm and is filed under Uncategorized. You can follow any responses to this entry through the RSS 2.0 feed. Responses are currently closed, but you can trackback from your own site.

One Response to “Why q = CV?”

A Silent Soliloquy Says:
March 23, 2011 at 10:41 am

Hey,

I feel you raise a very valid point here that really affects how well we can learn material. Although you aren’t altering the equations, altering their presentation matters as it affects logical flow. I always thought it was only me that was irked by this, (especially with the density equation). Now if only professors and authors caught on to this…

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Why q = CV?

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