# How to get electric field from potential field

If you know the potential at a point then you can get the electric field. For this you have to know the formula for potential field and the formula of the relation between electric field and potential field.It is also easier to remember these things if you know how they are actually derived.

The potential difference between two points is the line integral of electric field. Let E be some stationary electric field distribution. Consider a path near the E field and let two points P1 and P2 in the path.

The line integral of this field E along the path from P1 to P2 is,

Since the field E is stationary, this integral gives us a scalar quantity. This scalar quantity is named the potential difference between points P1 and P2.

This potential difference is the work done in moving a positive charge from P1 to P2. φ is a scalar quantity also called scalar field which is due to the electric field E. E is a vector field.

From the above equation we can guess the E is the derivative of φ. That is the above equation says equivalently that the electric field is the negative gradient of potential.

Thus using the above two equation we can calculate potential field if the electric field is given or we can calculate the potential field if the scalar field is given.

For example, consider a charge distribution shown in gray color below.  We want to find the field at point (x,y,z) due to the charge distribution.

The potential field due to a small volume element at x',y',z' is given by-

or in case of discrete distribution,

From this scalar field or potential field we can apply negative gradient to get electric field at the point (x,y,z).

See the Purcell Electromagentism pdf book for more detailed explanation.