Unit 1 - Potential

Vector problems have their uses; they often correspond directly to reality. But scalar problems are often easier to handle. So we will discuss the standard approach to writing electric field problems as scalar problems.

As Griffiths shows (Sec. 2.2.4), in electrostatics the curl of any E is zero. Now, it is a straightforward combination of definitions and a standard result on partial derivatives to show that the curl of the gradient of any function is zero; it is more challenging, but true, to show that if a vector function has zero curl, then that vector function is the gradient of some scalar function. Applying this last result, we conclude that every valid electrostatic field function equals the gradient of some other function, and we can write that function as the line integral of E. If we also insert a minus sign, then we have the "electric potential", a new, scalar, field, which, in its totality, is as complete a statement of an electrostatic situation as the electric field is. The minus sign needs to be remembered if potential is to have the properties one has come to expect. (In a sense, the minus sign is a historical residue, but it is also required by the connection between this function and energy concepts.)

Warning: the potential is a complete statement of the electrostatic situation only in its entirety. The value of the potential at a single point actually tells (in itself) nothing about the situation, because of the arbitrary integration constant involved in potential. The value on a line or surface tells nothing about electric field effects in directions that leave the line or surface.

E-M Home Page/Unit 1/

Format for this page adapted with permission from pages
constructed by Dr. Lionel D. Hewett for his course
Modern Physics 1.