Voltage for Nonuniform Fields

The calculus-savvy reader will have no difficulty generalizing the field- voltage relationship to the case of a varying field. The potential energy associated with a varying force is

so for electric fields we divide by q to find

Applying the fundamental theorem of calculus yields

Voltage associated with a point charge

Self-Check

Show that you can recover the expression for the field of a point charge by evaluating the derivative E = -dV/dx.