Determining Electric Field From Electric Potential

For a system of charged particles, the integral of the dot product of the electric field with the displacement of a test charge gives the difference in potential between the initial and final positions.

This change in potential will be infinitesimally small for an infinitesimally smaller displacement of a test charge.

Simplifying and expressing the above equation in component form gives the electric field as a partial derivative of the change in potential along a particular axis.

In vector notation, the electric field is expressed as a gradient of the electric potential. Here, the operator is called a del operator.

At each point in space, del operators give the electric field pointing in the direction along which the steepest decrease in the potential is present.

For a system with spherical or cylindrical symmetry, an appropriate del operator can be used. 

For example, the electric potential of a positive point charge has spherical symmetry, and the corresponding electric field can be calculated using a del operator in spherical symmetry.