Energy stored in a capacitor

Series combination of capacitors

The series combination of capacitors is shown in the following figure

From the figure, V1=Q/C1;
V2=Q/C2;
V3=Q/C3;
The potential difference across the series combination as
V=V1+V2+V3;
V=(Q/C1)+(Q/C2)+(Q/C3);
=Q(1/C1 + 1/C2 + 1/C3)
If C is the equivalent or effective capacitance of the series combination, and has the same charge Q, then C=Q/V and V=Q/C;
From the above equations,
Q/C =Q(1/C1 +1/C2 + 1/C3)
1/C=1/C1 +1/C2 +1/C3)
For 'n' number of capacitors,
1/C= 1/C1 +1/C2+.........+ 1/Cn
For two capacitors, the effective capacitance is
C=(C1C2 /C1+C2 )

Effect of dielectric on the capacitance of a capacitor

For a parallel plate capacitor,

Dielectric made of polar molecules

Dielectrics such as water, hydrogenchloride and alcohol made of molecules that have non uniform distribution of electric charges. In such materials, molecules in which the positive charge centre will not coincide with the centre of the negative charge, even in the absence of electric field. This shown in the following figure.
In the absence of external electric field, these molecules shows the polar characterstic, therefore when these molecules placed in an external electric field, the field will try to orient the positive charge centres in the direction of the field and negative charge centres in the opposite direction.
When polar molecules placed between the plates of a charged capacitor, the effect will be similar to that of non-polar molecules, the effect will be more pronounced because the molecules are already polarized even without field. The capacitance will increase C=KCo

Dielectric materials

Any materials that do not allow the electric charges to easily pass through the materials is called insulator or dielectric material.
Any material is made up of molecules but the molecules have symmetric charge distributions. So, such kind of molecules are called non-polar molecules. In the absence of external electric field, non-polar molecule has its centre of positive charge conincides with the centre of negative charge.
But, when the dielectric placed in an external electric field Eo, in each molecule the centre of positive charge will be displaced in the direction of the field, and the centre of negative charge will be displaced in the opposite direction. We called it as molecule gets polarized.



In the electric field Eo existing between the plates, each molecule in the dielectric gets polarized. Therefore, the entire dielectric gets polarized. Due to this reason, there become opposite charges to the charges on molecules of dielectric are induced. Inside the volume of the dielectric, the near by positive and negative charges cancel each other. Altogether, dielectric becomes neutral. Due to induced charges on the surface of the dielectric, the electric field Ei appears in the opposite direction to the actual electric field direction Eo.

So that, the net electric field E=Eo-Ei. Therefore, the potential difference Vo(without dielectric) reduces to V(with dielectric)

E=Eo/K
V=Vo/K
The capacitance of the capacitor with dielectric will be

C=Q/V=Q/(Vo/K)=K(Q/Vo)=KCo