Parallel Plate Capacitance

A parallel plate capacitor consists of two conducting plates separated by an insulating material called a dielectric. Its capacitance measures how much charge it can store on each plate for a given potential difference between them.

For plates of area $$A$$, separated by a distance $$d$$, with a uniform dielectric filling the space between them, the capacitance is:

Here $$C$$ is capacitance, $$\varepsilon_0$$ is the permittivity of free space, and $$\varepsilon_r$$ is the relative permittivity, or dielectric constant, of the material. The equation assumes that edge effects are negligible: the field near the edges of the plates does not significantly affect the result.

Effect of plate geometry

Capacitance is directly proportional to plate area: increasing $$A$$ gives the plates more area on which to store charge. Capacitance is inversely proportional to separation: increasing $$d$$ decreases the capacitance. For example, doubling the plate area doubles the capacitance, while doubling the separation halves it, provided the other quantities stay fixed.

Relative permittivity

Relative permittivity compares the permittivity $$\varepsilon$$ of a material with that of free space. For the same plate area and separation, it also compares the capacitance with the dielectric to the capacitance with a vacuum between the plates:

Relative permittivity is a ratio and has no units. For fixed plate geometry, a greater $$\varepsilon_r$$ gives a greater capacitance. The dielectric reduces the electric field between the plates for a given charge, so the capacitor can store more charge at the same potential difference.

Alignment of Polar Molecules in a Dielectric

A polar molecule has a permanent separation of positive and negative charge. Its oppositely charged ends form a permanent electric dipole.

When the molecule is placed in an electric field, the field pushes its positive end in the direction of the field and its negative end in the opposite direction. These forces produce a turning effect that rotates the molecule towards alignment, with its positive end pointing along the field.

In a dielectric containing many polar molecules, the applied field causes a partial overall alignment of their dipoles. The alignment is not complete: random thermal motion prevents all the molecules from pointing in exactly the same direction.

Effect on a capacitor

The aligned dipoles polarise the dielectric. Their combined effect produces an electric field inside the dielectric that opposes the applied field, reducing the resultant field.

For a capacitor with a given charge on its plates, this weaker resultant field means a smaller potential difference between the plates. Capacitance is the charge stored per unit potential difference, $$C = Q/V$$. Since the same charge now corresponds to a smaller potential difference, the dielectric increases the capacitance. Equivalently, at a given potential difference, the capacitor can store more charge.