# Relation Between Electric Field & Potential

## Electric Potential-

Before you go through this article, make sure that you have gone through the previous article on Work Done In Moving Charge In Electric Field.

We have learnt-

• An external agent has to do some work against the electrostatic forces while moving a charge in an external electric field.
• The work done by an external agent in moving a charge qo from initial point ‘i’ to final point ‘f’ in an electric field is given by the formula-

In this article, we will learn the Relation Between Electric Field & Electric Potential.

## Relation Between Electric Field & Electric Potential-

Consider-

• A positive point charge Q is fixed at a point.
• We wish to find the relation between electric field and electric potential.
• Another point charge qo is being moved by an external agent from point A to point B in the electric field of source charge Q.

Consider at any instant, the charge qo is present at some point P at a distance r from the source charge Q during its motion from A to B.

The work done by external agent in causing a small displacement dr from point P is given by-

(Equation-01)

To calculate the total work done by the external agent in taking the charge qo from point A to point B, we integrate the above calculated small work done dW in Equation-01 as-

Thus, the relation between electric field and electric potential is given as-

(Very Important)

## Special Scenario-

If the electric field in a region is everywhere uniform, then the above general relation between electric field and electric potential can be written as-

Thus, in a region of uniform electric field, on moving from point A to point B, the change in electric potential is given by-

(Very Important)

## Special Cases-

### Case-01: When θ = 0°

On moving in the direction of electric field, the angle between electric field & displacement vector is 0°.

We have-

VB – VA = -Edcos0°

VB – VA = -Ed(1)

∴ VB – VA = -Ed

This shows that-

• On moving in the direction of electric field, the electric potential decreases.
• On moving a distance d, the electric potential decreases by the amount Ed.

### Case-02: When θ = 180°

On moving in the opposite direction of electric field, the angle between electric field & displacement vector is 180°.

We have-

VB – VA = -Edcos180°

VB – VA = -Ed(-1)

∴ VB – VA = Ed

This shows that-

• On moving in the opposite direction of electric field, the electric potential increases.
• On moving a distance d, the electric potential increases by the amount Ed.

### Case-03: When θ = 90°

On moving perpendicular to the direction of electric field, the angle between electric field & displacement vector is 90°.

We have-

VB – VA = -Edcos90°

VB – VA = -Ed(0)

∴ VB – VA = 0

Or VB = VA

This shows that-

• On moving perpendicular to the direction of electric field, the electric potential remains constant.
• This gives rise to the concept of Equipotential Surface.

## Another Important Relation

### E = -dV/dr

From Equation-01, we have-

From here, we conclude that-

Electric Field is the negative of potential gradient which can be mathematically expressed as-

Here, negative sign shows that electric potential decreases in the direction of electric field.