Potential is a scalar. Equipotential lines, for example, represent the voltages at different points. Until now, we have considered only field lines and equipotentials in a plane (consider the equipotentials found in lab using the cupric acetate solution). The basic relation is

How can we turn this relationship around - obtain the field as a function of V? We need a new operator. Since the magnitude of the field caused by a one-dimensional potential is given by

(The vector arrow is not usually written over "del", by custom, although it is a vector operator.)

Here's the example, finally...

Problem: Suppose the potential is given by  Find the Electric field.

Solution: We use the operator; i.e., we take the gradient of the potential. Note that this is NOT a dot or a cross product, since V is not a vector. "del" is an operator...

At the origin (0,0,0), the field is undefined because of the z-coordinate in the denominator. At point (0,0,1) the field is

To choose one more point as a demonstration, the field at (1,0,1) is