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.