# faraday's law - Chapter 10 Faraday s Law of Induction 10.1...

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Chapter 10 Faraday±s Law of Induction 10.1 Faraday±s Law of Induction. ............................................................................. 10-2 10.1.1 Magnetic Flux. ........................................................................................... 10-3 10.1.2 Lenz±s Law. ................................................................................................ 10-5 10.2 Motional EMF. .................................................................................................. 10-7 10.3 Induced Electric Field. .................................................................................... 10-10 10.4 Generators. ...................................................................................................... 10-12 10.5 Eddy Currents . ................................................................................................ 10-13 10.6 Summary. ........................................................................................................ 10-15 10.7 Appendix: Induced Emf and Reference Frames. ............................................ 10-15 10.8 Problem-Solving Tips: Faraday±s Law and Lenz±s Law . ............................... 10-16 10.9 Solved Problems . ............................................................................................ 10-17 10.9.1 Rectangular Loop Near a Wire . ............................................................... 10-17 10.9.2 Loop Changing Area. ............................................................................... 10-18 10.9.3 Sliding Rod . ............................................................................................. 10-19 10.9.4 Moving Bar. ............................................................................................. 10-21 10.9.5 Time-Varying Magnetic Field . ................................................................ 10-22 10.9.6 Moving Loop . .......................................................................................... 10-23 10.10 Conceptual Questions . .................................................................................. 10-24 10.11 Additional Problems . .................................................................................... 10-25 10.11.1 Sliding Bar . ............................................................................................ 10-25 10.11.2 Sliding Bar on Wedges . ......................................................................... 10-26 10.11.3 RC Circuit in a Magnetic Field. ............................................................. 10-26 10.11.4 Sliding Bar . ............................................................................................ 10-27 10.11.5 Rotating Bar. .......................................................................................... 10-27 10.11.6 Rectangular Loop Moving Through Magnetic Field. ............................ 10-28 10.11.7 Magnet Moving Through a Coil of Wire. .............................................. 10-28 10.11.8 Alternating-Current Generator. .............................................................. 10-29 10.11.9 EMF Due to a Time-Varying Magnetic Field. ....................................... 10-30 10.11.10 Square Loop Moving Through Magnetic Field . .................................. 10-30 10.11.11 Falling Loop. ........................................................................................ 10-31 10-1

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Faraday±s Law of Induction 10.1 Faraday±s Law of Induction The electric fields and magnetic fields considered up to now have been produced by stationary charges and moving charges (currents), respectively. Imposing an electric field on a conductor gives rise to a current which in turn generates a magnetic field. One could then inquire whether or not an electric field could be produced by a magnetic field. In 1831, Michael Faraday discovered that, by varying magnetic field with time, an electric field could be generated. The phenomenon is known as electromagnetic induction. Figure 10.1.1 illustrates one of Faraday±s experiments. Figure 10.1.1 Electromagnetic induction Faraday showed that no current is registered in the galvanometer when bar magnet is stationary with respect to the loop. However, a current is induced in the loop when a relative motion exists between the bar magnet and the loop. In particular, the galvanometer deflects in one direction as the magnet approaches the loop, and the opposite direction as it moves away. Faraday±s experiment demonstrates that an electric current is induced in the loop by changing the magnetic field. The coil behaves as if it were connected to an emf source. Experimentally it is found that the induced emf depends on rate of change magnetic flux through the coil. 10-2
10.1.1 Magnetic Flux Consider a uniform magnetic field passing through a surface S , as shown in Figure 10.1.2 below: Figure 10.1.2 Magnetic flux through a surface Let the area vector be , where A is the area of the surface and its unit normal. The magnetic flux through the surface is given by ± A A G n ± n cos B BA T ) ± G G (10.1.1) where is the angle between B and . If the field is non-uniform, G ± n B ) then becomes B S d ) ± ³³ B A G G (10.1.2) The SI unit of magnetic flux is the weber (Wb): 2 1 Wb 1 T m ± Faraday²s law of induction may be stated as follows: The induced emf H in a coil is proportional to the negative of the rate of change of magnetic flux: B d dt ) ² (10.1.3) For a coil that consists of N loops, the total induced emf would be N times as large: B d N dt ) ² (10.1.4) 10-3

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Combining Eqs. (10.1.3) and (10.1.1), we obtain, for a spatially uniform field
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