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# 329lect14 - 14 Faradays law and induced emf Michael Faraday...

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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, induces 1 (i.e., 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 cur- rent distributions with time-varying components 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 also be expressed in integral form as C E · d l = - S B t · d S Faraday’s law for all surfaces S bounded by all closed paths C (with the directions of C and d S related by right hand rule). 1 Relativistic derivation of static B given in Lecture 12 can be extended to show that Coulomb interactions of charges in time-varying motions require a description in terms of time-varying B and E — see, e.g., Am. J. Phys. : Tessman, 34, 1048 (1966); Tessman and Finnel, 35 , 523 (1967); Kobe, 54 , 631 (1986). Thus, the cause of induced E is not really the time-varying B , but rather the time-varying current J producing the variation in B . Still, we find it convenient to attribute the induced E to time-varying B mainly because Ampere’s law provides an explicit link of a time-varying J to B (see Lect’s 12 and 16). 1

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The right hand side of the integral form equation above 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, then 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 is independent of S (so long as bounded by C , by Stoke’s theorem).
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329lect14 - 14 Faradays law and induced emf Michael Faraday...

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