Electrostatic Potential | Electricity & Magnetism - Physics PDF Download

Electric potential, in simple terms, is a way to describe how much electric energy a charged object has based on its position relative to other charged objects. It's similar to how we describe a ball's gravitational potential energy based on its height. Electric potential helps us understand how charges interact and move in electric fields. In this document, we will understand the mathematical interpretation of electric potential. 

Curl of Electric field

Consider a point charge at the origin, then the electric field at a distance r is given by,
 Now we will calculate the line integral of this field from some point a to some other point:
In spherical coordinates,

 
where r_a is the distance from the origin to point a and r_b is the distance to b.
Then integral around a closed path is zero i.e.
This line integral is independent of the path. It depends on two endpoints.
Applying Stokes theorem, we get  The electric field is not just any vector but only those vectors whose curl is zero. If we have many charges, the principle of superposition states that the total field is the vector sum of their fields:  

So,

Since the line integral is independent of the path.
So, we can define a function
where ϑ is some standard reference point V and then depends only on the point r. It is called the electric potential.
The potential difference between two points a and b is

Now, the fundamental theorem for gradients states that  

Potential obeys the superposition principle. 

The potential of localized charges

The potential of a point charge q is  where R is the distance from the charge.

The potential of a collection of point charges is  
For continuous volume charge distribution 

The potential of line and surface charges are  and 

Example 1: Which one of these is an impossible electrostatic field?

(a)
So,  is an impossible electrostatic field.
(b)  so   is a possible electrostatic field. 

Example 2: Find the potential inside and outside a spherical shell of radius R, charge q. 

From Gauss's law the field  

Potential outside ( r >R) is:

Potential inside ( r <R) is:

So potential inside the spherical shell is constant.  
Thus

Example 3: Find the potential inside and outside a uniformly charged solid sphere whose radius is R and whose total charge is q.

From Gauss's law the field  
 
Potential outside ( r >R) is:

Potential inside ( r <R) is:

Example 4: Find the potential distance r from an infinitely long straight wire that carries a uniform line charge λ.

Since  In this case, we cannot set the reference point at ∞ , since the charge 

itself extends to ∞ . Let’s set it at r =a

Then,