Electrostatic Potential and Capacitance

The energy stored in a sytem of charges due to their positions in the electric field is called the electrostatic potential energy. The energy for the two charges q1 and q2 separated by distance r is:

The energy is positive if both charges are of the same sign and it is negative if for the opposite signs. When a system has over two charges, the energy is calculated by the sum of energies of all unique pairs. This concept helps analyze behaviour and stability of charged systems.

When placed in an electric field, polarization is the alignment of dipole moments in a dielectric material. In a dielectric, molecules are randomly oriented. When an electric field is applied, it creates induced dipoles by slight shift in molecules from positive to negative. This weakend the net field, reduces the internal electric field and allow more charge storage. The polarization is quantified as:

Here, V is the volume, Pm is the total dipole moment. The polarization is a key concept in understanding the dielectric behavior in capacitors and it leads to increased capacitance.

In an electric field, energy density is the amount of energy stored per unit volume . For a linera or vacuum dieletric, it is given by:

E is the electric field magnitude and ?0  is the permittivity of free space. 

Where every point has the same electric potential, it is an equipotential surface:

The main characterstics include:

• Two equipotential surfaces never intersect.
• The electric field is always perpendicular to an equipotential surface.
• Equipotential surfaces are parallel planes in a uniform electric field.
• If the surfaces are closer, the field in that region will be stronger.

The negative gradient of the electric potential is called an electric field. The electric field points in the direction of the steepest decrease of potential. The electric field is defined in a simpler terms as equal to the change in potential per unit displacement.  The electric field is strong, if the potential decreases sharply over a small distance. The electric field can be calculated using this relationship if the potential distribution is known, and vice versa. It is important for the understanding of the energy transfer in electrostatics and charge dynamics.

At any point, the electrostatic potential is the amount of work done in bringing a unit positive test charge from infinity to that point in an electric field. It is done without acceleration. It is denoted by V and it is a scalar quantity.
? =? /Q
where? is the work done and q is the test charge. 
The electrostatic potential helps in quantifying the energy configuration of charges. The unit is volt (V), where 1 V = 1 J/C.