Now work on the right side of (4) and show that d / dt E middot dS Ampere's Law The last of Maxwell's Laws is Ampere's Law with a crucial generalization first proposed by Maxwell. The French physicist Andre-Marie Ampere (1775-1836) discovered that an electrical current (moving charges) produces a magnetic field. If C is a closed curve that encloses a current I, then Ampere's Law states that B middot dr = mu0 I, where mu0 is the magnetic permeability of the medium. The law says that the magnetic field induced by the current, when integrated along the closed loop C is proportional to the current enclosed by the loop. Maxwell understood that the Ampere's Law does not apply to all situations and saw the need for another term - called the displacement current - that also contributes to the induced magnetic field. Maxwell's Law, which extends Ampere's Law, is where S is any surface with C as its boundary. Notice the important parallel between the changing flux of the magnetic field in Faraday's Law and the changing flux of the electric field in this law. Suppose the enclosed current I can be expressed in terms of a current density J as I Theorem to show that the differential form of this law is
