Any two conductors separated by an insulator or vacuum form a device called capacitor. It is a device to store charge and electric potential energy. To store energy in a capacitor we can simply charge the capacitor; charging a capacitor means transferring electrons from one conductor the other so that one conductor has positive charge and the other has an equal amount of negative charge.
An amount of work is done to transfer the charge from one conductor to the other against the resulting potential difference and that amount of work is stored as the electric potential energy in the capacitor. This is the same as charging a capacitor. To charge a capacitor you can simply connect the conductors of the capacitor to the opposite terminals of a battery. The figure below shows a parallel-plate capacitor whose two conductors are parallel plates separated at a distance in vacuum.
The fact that an amount of work done being stored in a capacitor is related to the electric field between the conductors which follows that the electric field itself is the storehouse of energy. This is one of the most interesting facts you can explore!
The total charge on a capacitor is zero (equal amount of positive and negative charge add to zero net charge) but it is common to represent the magnitude of charge on either conductor as the charge of the capacitor. If we say a capacitor has charge \(Q\), we mean that one conductor has charge \(+Q\) (higher potential) and other has charge \(-Q\)(lower potential) assuming \(Q\) is positive.
The ratio of the charge on either conductor \(Q\) to the potential difference \(V_{ab}\) (potential of conductor a with respect to conductor b) between the conductors is constant called capacitance.
\[C = \frac{Q}{V_{ab}} \tag{1}\]
The capacitance is always a positive quantity; the charge \(Q\) and the potential difference \(V_{ab}\) are always expressed as positive quantities. The capacitance of a capacitor is constant and depends on the shapes and sizes of the capacitor, the separation of the conductors and an insulator (dielectric) if present between the conductors. We'll talk about this later as well.
The SI unit of capacitance is one farad (\(1\text{F}\)), that is \(1\text{F} = 1\text{C}/\text{V}\). Please don't confuse between columb(\(\text{C}\)) and capacitance(\(\text{C}\)).
The electric field between the conductors is proportional to the charge of the capacitor and the potential difference is also proportional to the charge, so increasing the charge increases the electric field and increases the potential difference but the ratio of the charge to the potential difference remains the same.
The greater amount of capacitance for capacitor means the greater amount of charge for a given potential difference and therefore the capacitors with greater capacitance can store more charge and energy.
The electric field between two parallel plates is
\[E = \frac{\sigma }{\epsilon_0}\]
We know \(\sigma = Q/A\) and,
\[E = \frac{Q }{\epsilon_0 A}\]
The potential difference \(V_{ab}\) for the uniform electric field with separation \(d\) is
\[V_{ab} = Ed = \frac{Qd}{\epsilon_0 A}\]
And finally our capacitance is
\[C = \frac{Q}{V_{ab}} = \epsilon_0 \frac{A}{d} \tag{2}\]
It follows that the capacitance depends on the area of each plates and the separation distance. For a particular capacitor both \(A\) and \(d\) are constants and therefore the capacitance is constant for a capacitor. But the capacitance also depends on the material present between the conductors. If there is air instead of vacuum the difference in capacitance is negligible. If there is other material present between the conductors, the capacitance will be largely effected.
Determining the capacitance for a parallel-plate capacitor was quiet straightforward but the expression for the capacitance we determined for the parallel-plate capacitor is not the same for other capacitances of more complex shapes we are about to discuss below.
