# What does capacitance physically mean?

so im reading a bunch of textbooks on static electricity(high school level). and the capacitance concept keeps bugging me.

C=q/V right.

lets say we have a conductor. we place excessive charge on it. the charge creates some potential . is it correct to understand this potential as work that we have to do to move a unit charge from a 'zero potential energy level' to any spot in this net field, created by this excessive charge? does that mean that the potential of this field is a constant everywhere? is there a better way to understand capacity? i dont understand what does voltage have to do with it.... why do we divide the charge by it? what does capacity physically mean?

The potential is linearly increasing between the two 'plates' of the capacitor, which is equivalent to a homogeneous electric field (because it is the derivative of the potential with respect to the position between the plates).

And yes, potential is work (per charge) you have to do to move the charge from the reference level (ground): $$W=q\cdot U$$. Since the potential is linearly increasing, it is much like moving a carriage onto a mountain with constant slope, the downward force will be constant, the height/potential energy is increasing on your way up.

On the other hand, the reason for dividing voltage by charge for a capacitor, is because if you add a certain amount of charge to the capacitor, the work you have to do for the next amount of charge increases proportionately. The constant of proportionality is called (by convention) the (inverse) capacitance: $$\Delta U=constant\cdot \Delta Q=:\left(\frac{1}{C}\right)\cdot \Delta Q$$

It is much similar to when you add some water to an artificial lake, the water level rises, requiring you to surmount a greater height (and hence, higher potential energy) when you want to add the next bucket of water.

Note, that this is not always so. A rechargeable battery for example is typically keeping the voltage constant as more charge is added to it. This is much more useful than a capacitor, because circuits often need constant voltage. Hence, a capacitor is more suitable for short-term storage, as opposed to a battery, which is long-term storage.

Note also, that capacitance has nothing at all to do with a "maximum amount" of charge a capacitor can hold. This quantity (which could be called "capacity"; but in German for example the naming would be ambiguous) is more related to the maximum voltage the capacitor can stand. But it is rarely referred to.

Take a metal spring, the force $$\mathcal{F}$$ it takes to balance it depends on its extension $$x$$, where the extension is measured from the zero force point, and the so-called constitutive equation that describes their relationship is $$\mathcal{F} =kx$$ where $$k$$ is the spring parameter that depends on the metal, its size, shape, etc. The amount of work it takes to stretch it from $$x$$ to $$x+\delta x$$ is $$\delta W = \mathcal{F}\delta x = kx\delta x$$. In this sense, the spring parameter $$k$$ describes what it takes to stretch/compress/ the spring, $$k$$ is the capacitance of the spring. (By very ancient convention we use the reciprocal of the capacitance as the analogue of the spring parameter, just as the current carried is negative by ancient convention that will never change).

For a dielectric between metal plates the role of voltage is the "force" that moves the charges on and off the plate, the electric charge has the role of the "length" of the spring as it is stretched or compressed, and the spring parameter, or by ancient convention the reciprocal of the spring parameter, is like the electric capacitance.