Electric Potential

Amount of work done in moving unit positive charge from infinity up to the point under consideration against the field of a given charge q
In other words, it is negative of the work done by the internal forces

It is a scalar quantity
(i)    Potential at a point due to several charges

(ii)    Potential of a point due to an electric dipole

Potential due to a uniformly charged disc
(i)    At a point on its axis

σ = surface charge density on disc
(ii)    At the centre of the disc

Potential at the edge of a uniformly charged disc

Potential at the apex of a cone having charge Q distributed uniformly on its curved surface and having slanting length L

Potential of a conducting sphere of radius R at a distance r from the centre

Potential of a dielectric sphere

Torque on a dipole in presence of external electric field
Torque = qE. 2a sin θ
τ¯= PE sin θ
Potential Energy U =  -P¯.E¯

Example  8.     Two conducting spheres having radii a and b charged to q₁ & q₂respectively.  Find the potential difference between 1 & 2 ?

Solution :         The potential on the surface of the sphere 1 is given by

Electric Potential Energy
The electric potential energy of a system of point charges is the amount of work done in bringing the charges from infinity in order to form the system. For point charges q₁ and q₂  separated by a distance r₁₂, Electric potential energy of the system q₁ and q₂ is given by

For three particle system q₁, q₂ and q₃

We can define electric potential (V_P) at any point P in a electric field as

where U_p, is the change in electric potential energy in  bringing the test charge q₀from infinity to point P

Example 9.    Determine the interaction energy of the point charges of the following set- up