Magnetic Properties of Materials.

Ampere's theorem written for a vacuum, $$ \overrightarrow {rot} _P \overrightarrow B = \mu _0 \overrightarrow j $$ , shows how $$ \overrightarrow {rot} _P \overrightarrow B $$ is only zero at points (P) which are without current, and therefore is nonzero elsewhere. The $$ \overrightarrow B $$ therefore cannot be derived from a uniform scalar potential (V), and in effect, if we could write for all P that $$ \overrightarrow B = \overrightarrow {grad} _P $$ V, we will end up with $$ \overrightarrow {rot} _P \overrightarrow B = 0 $$ , which is not true as we have just seen. Nevertheless, we can define a pseudoscalar potential (V*) such that $$ \overrightarrow B = - \overrightarrow {grad} _D $$ V* where D represents points without current, as at these points $$ \overrightarrow {rot} _D \overrightarrow B = 0 $$ (a pseudoscalar is a scalar that is defined by its being limited to certain points). [ABSTRACT FROM AUTHOR]