# Electric Potential Due To A Point Charge

## Electric Potential-

Electric potential at a point in an electric field is defined as-

It is the amount of work done by an external agent in moving a unit positive charge from infinity to that point against the electrostatic forces.

#### OR

It is the amount of work done by an external agent per unit charge in moving an electric charge from infinity to that point against the electrostatic forces.

## Electric Potential Due To A Point Charge-

Consider-

• A positive point charge Q is placed at point O.
• We wish to calculate the electric potential at any point P located at a distance r from it.

To find the electric potential at point P, we bring a test charge q0 from infinity to point P very-very slowly and calculate the work done in this process.

Then, electric potential at point P would be given by-

We know, electric field is a conservative field. So, work done in a closed path is zero. So, we must have-

Thus, the above relation can be re-written as-

(Equation-01)

Now, we will calculate the work done by the external agent in taking the charge q0 from point P to infinity and then substitute in equation-01.

Consider the charge q0 to be present at some point O at a distance x from the charge Q during its motion from P to infinity.

#### NOTE

It is important to note that-

The external agent exerts an equal amount of force as exerted by the charge Q but in opposite direction so that the net force on charge q0 at any instant is zero and the charge moves towards infinity very-very slowly with a constant velocity.

The work done by the external agent in causing a small displacement dx is given by-

To calculate the total work done by the external agent in taking the charge from point P to infinity, we integrate the above calculated small work done as-

Substituting in equation-01, we get-

Thus, Electric potential due to a point charge Q at a distance r from it is given by-

(Important Formula)

### Graph Showing The Variation of Electric Potential-

The following graph shows the variation of electric potential due to a point charge with distance from it-

### Characteristics of Electric Potential-

• Electric potential is a scalar quantity.
• The SI unit of electric potential is volt (V).

1 V = 1 JC-1

• Electric potential is positive due to a positive charge & negative due to a negative charge.
• The electric potential due to a point charge is spherically symmetric.
• Its dimensional formula is [M1L2T-3A-1].

### Deriving Dimensional Formula of Electric Potential-

The dimensional formula of electric potential can be derived as-