Derive an expression for the capacitance of a parallel-plate capacitor filled with a dielectric.

Explanation

Consider a parallel-plate capacitor, consisting of two parallel plates A and B separated by a distance d as shown in the figure. Let A be the area of each plate and k the dielectric constant of the medium between the plates. Plate B is connected to the Earth. Suppose that the capacitor is charged such that the charges on plates A and B are +Q and -Q, respectively.

If the area A is very large and the distance between the plates is very small, the electric field in the region between the plates is almost uniform except at the edges. The magnitude of the electric field `vecE` at a point between the plates and the potential difference V between the plates are related by `E=V//d`.
But, `E=(sigma)/(epsilon_(0)k)`, where `epsilon_(0)` is the permittivity of free space and `sigma` is the surface charge density.
`therefore (sigma)/(epsilon_(0)k)=(V)/(d) " " ` ...(1)
Since `sigma=(Q)/(A), E=(Q)/(epsilon_(0)kA)=(V)/(d) " " ` ...(2)
The capacitance of a capacitor is `C=(Q)/(V)`
`therefore C=(epsilon_(0)kA)/(d) " " ` ...(3)