Chapter 17.7, Problem 42E

Ampère’s Law The French physicist André–Marie Ampère (1775–1836) discovered that an electrical current I in a wire produces a magnetic field B. A special case of Ampère’s Law relates the current to the magnetic field through the equation ${\displaystyle {\oint }_{C}B\cdot dr=\mu I}$, where C is any closed curve through which the wire passes and μ is a physical constant. Assume that the current I is given in terms of the current density J as $I={\displaystyle {\iint }_{S}J•ndS}$, where S is an oriented surface with C as a boundary. Use Stokes’ Theorem to show that an equivalent form of Ampère’s Law is ▿ × B = μJ.