> To clarify, claiming current is the derivative of charge is like claiming the current of a river is the derivative of it's water level.
My electro mag skills are a little rusty, but I don’t think that’s the right analogy. You wouldn’t say that the speed of a runner is the derivative of their height, similarly when someone is running you wouldn’t try to say that their speed isn’t a derivative of their height, as their height is more or less constant “for the length of the run”.
Similarly, it isn’t the height of the runner, but the distance they’ve covered, it isn’t the depth of the river, but how much water has passed through, it isn’t the static charge of the wire, but how much charge has passed through.
It's only a little bit off. To get the true hydrodynamic analog to a capacitor, connect the bars of two big pistons together so that volume accumulated in one comes at the expense of the other, then hit the bar with a spring so that its motion comes with some energy cost that can oppose a constant pressure.
The point is that this component alone ties flow to accumulation, whereas generally your other components (resistors=thin pipes, wires=thick pipes, batteries=Archimedes screws, inductors=turbines connected to flywheels) do not accumulate volumes of water inside of them. Flow needs to make sense even without accumulation due to flow-balance, just like force needs to make sense even in situations where velocity stays constant due to force-balance.
I think you're just rephrasing what I'm trying to say in several different ways.
We can probably agree that the amount of charge passing through is different from the change to the total charge.
And in my opinion something like a 'periodic system' can't just treat both the same. It's especially bad to derive identities by taking the time derivative (or anti-derivative) on one side and switching between current and charge on the other.
My electro mag skills are a little rusty, but I don’t think that’s the right analogy. You wouldn’t say that the speed of a runner is the derivative of their height, similarly when someone is running you wouldn’t try to say that their speed isn’t a derivative of their height, as their height is more or less constant “for the length of the run”.
Similarly, it isn’t the height of the runner, but the distance they’ve covered, it isn’t the depth of the river, but how much water has passed through, it isn’t the static charge of the wire, but how much charge has passed through.