Why does capacitance depend on medium?

Question

Capacitance does not depend on charge or the potential difference due to it fundamentally being a constant which we have defined. I only have trouble understanding why does it then change if we change dimensions say of a parallel plate capacitor or if we change the medium between the two plates?

Answer 1

Capacitance is a measure of the ability of a capacitor to "store charges". The charges are "stored" on the capacitor plates by accumulating there until their total electric force balances out the electromotive force from the battery. In essense, the battery "pushes" charges to the plate until the charges are enough to "push back" with equal "strength". Cut the wires after having reached this steady state, and the "stored" charges stay "stored". The reason that they build up a larger and larger counteracting electric force, that eventually balances out the battery's effect and stops all further charges from arriving is, that multiple charges at the same region build up a larger total electric field.

Larger capacitor plates

With larger capacitor plates the charges can spread more out, hence allowing for more charges to accumulate before the electric field is large enough to balance out the battery's effect. So, more charges can be stored, i.e. the capacitance is larger.

In-between medium (dielectric)

If you place an insulating material in-between the two plates, then you risk that any dipole pairs inside this material will arrange themselves in such a way that they reduce the accumulated electric field a bit, thus allowing for even more charges before the battery is balanced out, so an even higher capacitance. Such in-between material is called a dielectric.

To understand how this works, we must keep in mind that when a charge is arriving at a plate, then it repels away a corresponding (equal) charge in the other plate due to their closeness. A charge missing in this other plate corresponds to a charge of the opposite sign arriving at this other plate. The total negative charge accumulated on one plate will thus be equal to a total positive charge that has accumulated simultaneously on the other plate. The plates now carry the same numerical charge but of opposite sign, so the plates attract each other. When you cut the connection to the battery, then it is this attraction that keeps the charges in place on the plates.

A dielectric messes with this mutual attraction between the accumulated charges on the plates. Inside a dielectric, although it is an insulator, it still might be made of particles that due to their electron arrangements are slightly more positive on one side and negative on the opposing side. They can't move since they are stuck in a non-conductive material, but when a strong electric field appears, they might rotate to align with thus field. These are called dipoles. Their positive ends will then point towards the negatively charged plate, and their positive ends towards the positive plate.

All in all, the negative charges arriving on one plate now feel a greater attraction towards the other side, because not just the opposite plate with induced positive charge but also the positive ends of many dipoles in the in-between medium attract them. They feel a greater attraction than if the dielectric was now there. And this greater attraction more strongly attracts negative charges to the negative plate, thus essentially "helping" the battery so that more can be located on that plate, before the battery's effect is balanced out. You can also think of the built up electric field being reduced slightly by the dielectric so that more charges are needed on the plate to establish the same electric field as before that was necessary to balance out the battery.

From this walk through of how a capacitor works, it should also be clear why the distance between plates matters, and more proporties can be understood in this way.