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14 Faraday’s law and induced emf Michael Faraday discovered (in 1831, less than 200 years ago) that a changing current in a wire loop induces current flows in nearby wires — today we describe this phenomenon as electromagnetic induction : current change in the first loop causes the magnetic field produced by the current to change, and magnetic field change, in turn, produces electric fields which drive the currents in nearby wires. Definitions of E and B have not changed: recall that E is force per unit sta- tionary charge B gives an additional force v × B per unit charge in motion with velocity v in the mea- surement frame. While static electric fields produced by static charge distributions are unconditionally curl-free, time-varying electric fields produced by static as well as time-varying sources are found to have, in accordance with Faraday’s observations, non-zero curls specified by ∇ × E = - B t Faraday’s law at all positions r in all reference frames of measurement. Using Stoke’s theorem, the same constraint can be expressed in integral form as C E · d l = - S B t · d S for all surfaces S bounded by all closed paths C — as in Ampere’s law, the circulation direction C and flux direction specified by d S should be related by the right hand rule. 1

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In the integral form equation above, the right hand side includes the flux of rate of change of magnetic field B over surface S . If contour C bounding S is stationary in the measurement frame, the equation can also be expressed as C E · d l = - d dt S B · d S , where the right hand side includes the rate of change of magnetic flux Ψ S B · d S that can only depend on time t for all S bounded by a stationary C . This modification (the exchange of the order of integration and time derivative on the right side) would not be permissible if path C were moving within the measurement frame or being deformed in time — but in such cases we could still express Faraday’s integral form equation with d Ψ dt
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