Applying voltage to a capacitor is an incorrect mental model -- the voltage across a capacitor is a function of its charge [1]; it can't be both that and the voltage you apply to it.
In reality, when you apply voltage "to" a capacitor, it's to the series circuit comprising the capacitor and a (hidden or explicit) series resistance, thus causing a current to flow determined by the resistance and the net voltage across it.
The correct mental model is that you inject current into a capacitor. So long as you inject current, the voltage across the capacitor rises. viz. a flywheel -- so long as you apply a force, the flywheel speeds up.
> A flywheel speeds up as long as you apply a force to it; an inductor's current increases as long as you apply a voltage to it.
Likewise, a spring gains tension so long as you keep elongating it.
Inductors and capacitors are dual to each other, so -- ignoring the mechanical linkages themselves -- there's no inherent advantage for one mechanical analog or its dual. We are thus free to choose either to make a sound analogy of the physics of linkages:
> And, current and velocity are both motions, while force and voltage are both pressures.
This grammatical correspondence is not relevant to the mathematics. Kirchoff's laws [2] only hold for mechanical linkages when current is held analogous to force, and voltage is held analogous to velocity.
A more grammatically satisfying analogy can be found in hydraulic systems, with current : flow :: voltage : pressure. Flow, unlike motion, sums at nodes; pressure, unlike force, is equal at nodes.
If one is insistent on treating springs like capacitors and flywheels like inductors, you can do so, but only if you also take the dual of the full circuit, as seen here [3]. This is then a mathematically correct analogy, but personally I find it much less straightforward than simply swapping springs and flywheels.
Applying voltage to a capacitor is an incorrect mental model -- the voltage across a capacitor is a function of its charge [1]; it can't be both that and the voltage you apply to it.
In reality, when you apply voltage "to" a capacitor, it's to the series circuit comprising the capacitor and a (hidden or explicit) series resistance, thus causing a current to flow determined by the resistance and the net voltage across it.
The correct mental model is that you inject current into a capacitor. So long as you inject current, the voltage across the capacitor rises. viz. a flywheel -- so long as you apply a force, the flywheel speeds up.
> A flywheel speeds up as long as you apply a force to it; an inductor's current increases as long as you apply a voltage to it.
Likewise, a spring gains tension so long as you keep elongating it.
Inductors and capacitors are dual to each other, so -- ignoring the mechanical linkages themselves -- there's no inherent advantage for one mechanical analog or its dual. We are thus free to choose either to make a sound analogy of the physics of linkages:
> And, current and velocity are both motions, while force and voltage are both pressures.
This grammatical correspondence is not relevant to the mathematics. Kirchoff's laws [2] only hold for mechanical linkages when current is held analogous to force, and voltage is held analogous to velocity.
A more grammatically satisfying analogy can be found in hydraulic systems, with current : flow :: voltage : pressure. Flow, unlike motion, sums at nodes; pressure, unlike force, is equal at nodes.
If one is insistent on treating springs like capacitors and flywheels like inductors, you can do so, but only if you also take the dual of the full circuit, as seen here [3]. This is then a mathematically correct analogy, but personally I find it much less straightforward than simply swapping springs and flywheels.
[1] https://en.wikipedia.org/wiki/Capacitor#Current%E2%80%93volt...
[2] https://en.wikipedia.org/wiki/Kirchhoff%27s_circuit_laws
[3] https://www.ams.org/publicoutreach/feature-column/fc-2019-05