Relation between Electric field intensity and potential?

Or

Prove that the electric field at a point is equal to the negative gradient of electrostatic potential at the point.

E = − dv / dr

The relationship between electric field intensity and electric potential is fundamental in understanding electrostatics. The electric field (E) at a point in space is directly related to how the electric potential (V) changes in that area.

Understanding Electric Field and Potential

Electric potential is the amount of work done to move a unit positive charge from a reference point to a specific point in an electric field without any acceleration. The electric field, on the other hand, represents the force experienced by a unit charge placed in that field.

Mathematical Relationship

The relationship can be expressed mathematically as:

E = -dV/dr

Here, dV represents the change in electric potential, and dr is the change in distance. The negative sign indicates that the electric field points in the direction of decreasing potential.

Visualizing the Concept

• If you move against the electric field, you are moving to a region of higher potential.
• If you move with the electric field, you are moving to a region of lower potential.

Proving the Relationship

To prove that the electric field is equal to the negative gradient of the electrostatic potential, consider a small displacement in the electric field:

1. The work done (W) in moving a charge (q) through a distance (dr) in the field is given by:

W = q * E * dr

2. The change in potential energy (U) is related to the electric potential (V) as:

U = q * V

3. The change in potential energy when moving through a distance is:

dU = -dW

4. Substituting the expressions gives:

dU = -q * E * dr

5. Dividing by the charge (q) leads to:

dV = -E * dr

6. Rearranging this equation results in:

E = -dV/dr

Conclusion

This relationship shows that the electric field is indeed the negative gradient of the electric potential, illustrating how potential changes in space relate to the force experienced by charges in that field.