Parallel Plate Capacitor and its Capacitance

Parallel Plate Capacitor : Introduction

A capacitor is a device to store electric charge and hence electrostatic potential energy. The simplest example of a capacitor consists of two conducting plates of identical geometry, which are parallel to each other and seperated by a small distance. The arrangement is shown in the figure below and it is known as 'Parallel Plate Capacitor'.

The two plates M and N are generally thin i.e. they are of negligible thickness and of rectangular or circular shape and also of equal size to maintain a symmetry for the setup.

The seperation 'd' between the plates is kept small compared to the dimensions of the plates. This is done to minimise the edge effect, to make the system compact and also to increase the capacitance. The space between the plates may be filled with a dielectric material. This again improves the capacity to store electric energy and also serves as a physical support to the plates.

Energy Storage Mechanism in Parallel Plate Capacitor

When the two plates are given some charges say Q₁ and Q₂, equal and opposite charges induce at the inner surfaces (i.e. at the facing surfaces) and like charges appear at the outer surfaces.

If the plates are provided with equal and opposite charges i.e Q₁ = -Q₂, equal and opposite charges induce at the two facing surfaces while the remaining two surfaces i.e the outer surfaces carry no charge (neglecting edge effect). In this case the electric field produced by the charge distribution is confined uniformly in the region/volume between the two plates and hence electrostatic potential energy too is stored in the same limited volume.

Symbol for showing Parallel Plate Capacitor in Circuits

One of the following two symbols is generally used to represent a parallel plate capacitor in electric circuits.

Now let's proceed to obtain the expression for capacitance of a parallel plate capacitor.

Capacitance of a Parallel Plate Capacitor

Consider the digram of a parallel plate capacitor as shown in the figure below.

M and N are the two identical metallic plates, each of area 'A', kept at a small seperation 'd' from each other. Also assume that the space between the plates is filled with air or vacuum.

We put a charge +Q on plate M and a charge -Q on plate N. This can be done either by connecting plate M with the positive terminal and plate N with the negative terminal of a battery or by connecting one plate (here plate N) to the earth and by giving a charge +Q to the other plate only. This charge will induce a charge -Q on the earthed plate. The charges will appear only on the facing surfaces i.e. the inner surfaces. The surface charge density on each of these surfaces has a magnitude σ = Q/A. 

Electric field in the region between the plates is uniform and directed perpendicular to the plates except for small region near the edges. At any point P in the space between the plates, resultant electric field can be obtained by superimposing the individual fields produced by each of the plates. Both positive and negative plates produce electric field in the same direction (from positive plate towards negative plate). Electric field between the plates is 

Outside the plates, the field due to positively charged plate and that due to negatively charged plate are in opposite directions and of equal magnitude. Therefore, net field at these points is zero. 

Since, electric field is nothing but the negative gradient of electric potential, the potential difference between the plates can be written as