Chapter 32 Faraday?s Law of Electromagnetic Induction Topics of Chapter 32 ? Induced Electromotive Force ? Magnetic Flux ? Faraday?s Law of Induction ? Lenz?s Law ? Generators and Motors ? Transformers ? Inductance ? Energy Stored in a Magnetic Field Motional emf Let?s consider a U-shaped conductor with a metal slider, all immersed in a magnetic field B. If we push the slider at velocity v, a (+) charge q has a downward force on it F = qv?B. The force does work W = Fl moving q a distance l. + The work increases the PE and ?PE =W So ?(PE) = qv?Bl. But emf = ?(PE)/q emf vBl? ? This ?motional emf? can drive a current I around a closed loop. Motional emf We can rewrite this equation, During time ?t, the slider moves distance ?x = v(?t) So Now, the quantity BA counts the number of magnetic field lines passing through surface A + The emf is ? how fast the number of field lines is changing inside the loop! emf vBl? ()x lx A BA emf B l B B tttt ???? ???? ???? v?t = ?x Magnetic flux SI unit of magnetic flux is T-m 2 = Weber = [Wb] Note that B = flux/area, so B is ?flux density.? Magnetic flux is useful in the calculation of the induced emf. Quantitatively, the magnetic flux ? B ?counts? the number of magnetic field lines. We define (cos ) B BA B A? ? ?? ? ?B?A A Motional emf We can rewrite this equation, as follows. During time ?t, the slider moves distance ?x = v(?t) So Now, the quantity BA counts the number of magnetic field lines passing through surface area A + The emf is ? how fast the number of field lines is changing inside the loop! emf vBl? ()x lx A BA emf B l B B tttt ???? ???? ???? v?t = ?x () B emf t ?? ?? ? Induced emf: Faraday?s Law of Induction Experimentally, the creation of an emf by changes relative to a magnetic field is much more general than the motional emf just described. As Faraday discovered and we?ll demonstrate, any change in magnetic flux generates an emf. B emf N t ?? ?? ? Faraday?s Law of Electromagnetic Induction: The (-) sign comes from Lenz?s Law. The factor N is the number of turns, where each turn has flux ? B . Induced emf Faraday?s experiment: closing the switch in the primary circuit induces a current in the secondary circuit, but only while the current in the primary circuit is changing. Induced emf Note the motion of the magnet in each image. Lenz?s Law An induced current always flows in a direction that opposes the change that caused it. Therefore, if the magnetic field is increasing, the magnetic field created by the induced current will be in the opposite direction; if decreasing, it will be in the same direction. Lenz?s Law When the slider moves to the right, the number of field lines (magnetic flux) pointing out of the U-shaped loop increases. The induced current flows in a CW direction, making flux into the page and opposing the change that creates it. + Lenz?s Law The force due to the induced current is upward, slowing the fall. Lenz?s Law Currents can also flow in bulk conductors. These induced currents, called eddy currents, can be powerful brakes. Generators and motors An electric generator converts mechanical energy into electric energy. If the coil rotates at angular rate ?, then ? = ?t. The flux ? B in 1 turn = cos cos( ) B B ABA BA t? ? ? ?? ? ? [cos()] sin( ) BAt emf N t emf NBA t ? ?? ? ? ?? ? ? Generators and motors The induced emf in a rotating coil varies sinusoidally with time t. In the US, the period (repeat time) is (1/60) s, i.e., the frequency = 60 Hz. In some other places in the world, it is 50 Hz. This makes alternating current ? AC. Generators and motors An electric motor does the opposite of a generator ? it uses the magnetic torque on a current loop to create mechanical energy. Transformers N 1 ?V 1 ?V 2 N 2 A transformer is a device that changes voltage in an AC circuit from one value to another. 11 2 2 Since and , we have one one turn turn VN VN??? ??? 11 22 VN VN ? ? ? Transformers The input power I 1 ?V 1 must be equal to the output power I 2 ?V 2 . Therefore, if the voltage is increased, the current must be reduced. For long distance transmission of electrical power, transformers ?step-up? the voltage and ?step- down? the current to reduce resistive losses in the lines. Self-inductance When the switch is closed in this circuit, a current is established that increases with time. The increasing current makes an increasing B-field and increasing flux, which in turn induces an emf. By Lenz?s Law, this back emf opposes the change; from Faraday?s Law, ()/ ()/emf t I t???? ???? ? = I emf L t ? ? ? ? where L = ?self inductance.? [L] = V/(A/s) = Henry = [H] Self-inductance; stored energy Inductance is the proportionality constant that tells us how much emf will be induced for a given rate of change in current. An inductor tends to maintain the status quo, with electrical inertia analogous to mechanical mass. It takes work to pump current into an inductor; that work is stored as magnetic PE, where 2 =/2 B PE LI The PE is stored in the magnetic fields; the stored energy/volume (energy density) is given by 2 0 vol 2 B B PE B u ? ?? jrt Ch 32 Electromagnetic Induction
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