# Questions about the relationship between capacitance and permittivity

I am trying to understand the definition of capacitance, specifically, the relationship between permittivity and capacitance intuitively.

Starting with definitions,

$$C = \frac{Q}{V_{applied}}$$

$$C = \epsilon\frac{A}{l}$$

the second one is for plate capacitors.

To my knowledge, when an electric field applied, within a material there will be a charge separation.

Eventually, there will be two opposing forces on every charged particle: one from the applied field and the other is from the induced field due to the charge separation.

At equilibrium, those two forces are balanced and canceling each other thus a net force on a charge will be zero.

Mathematically, for the charge q

$$qE_{applied} + qE_{induced} = 0$$

$$|E_{applied}| = |E_{induced}|$$

By using the Coulomb force equation, I suppose that

$$|E_{applied}| = |E_{induced}| = \frac{Q}{4\pi\epsilon{d^{2}}}$$

where $$d$$ is the distance between the charge pair.

Eventually, I can get

$$4\pi\epsilon{d} = \frac{Q}{E_{applied}d} = \frac{Q}{V_{applied}}$$

as you can see, this kind of relationship is similar to the definition of capacitance;

$$C = {\frac{Q}{V}} = \epsilon\frac{A}{l}$$

So my question is, first, do I correctly derive (qualitatively) the relation between the permittivity and the capacitance?

Second, the permittivity sounds like to me that it is a degree about how an electric field is permeable through the space (like within a material).

But I am not sure how this property is directly connected to the number of separated charges like in the definition of capacitance.