My water analogy for a capacitor is a piston with pipes attached to both ends, with a spring system that pushes the piston towards the center position. Is that not a mathematically correct equivalent? (in an idealized system with no water resistance/inertia and disregarding that the piston is of fixed length - not that real capacitors have zero resistence, inductance or can store infinite charge)
That's what I started out with! Fill the entire universe with solid rock (since air and vacuum are insulating.) Bore out a cylinder, fill it with water (electricity), and add a piston and spring. Wires are water-channels added to either end.
But note that, if we use a constant-force spring, then the voltage remains the same until just before the "capacitor" is totally discharged. So, it acts like a battery! To get a "capacitor," the spring must have an unchanging spring-constant, so that the potential-difference rises in proportion to how much water has been pumped from one terminal to the other.
Its risky to speculate on fuzzy analogies stacked on fuzzy analogies but another one I've seen is the water balloon, or balloon over an open piece of pipe. So some voltage pressure pushes the current into the balloon and the balloon kinda sorta works to keep a constant-ish pressure/voltage on the pipes.
Now if that was too easy try an inductor analogy. Something like a waterwheel hooked up to a very big flywheel so it works hard to keep the watercurrent flow constant.
The best thing about learning by analogies is eventually they get so hairy and crazy that reality is simpler in comparison...
From an energetic standpoint, a capacitor is something that takes a trickle over a long time, and releases a flood over a short time. So it would be a water tower with a small input and a large gated output.
But a water tower only has one terminal. A better "capacitor" would be a pair of water towers side by side.
To "charge" this double-water-tower capacitor, pump some water from one to the other. And, when the water-tower capacitor is entirely "discharged," the towers both have the same water level inside. (As with a real capacitor, the total amount of water never changes.)
