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lec15

# lec15 - Scott Hughes Massachusetts Institute of Technology...

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Scott Hughes 7 April 2005 Massachusetts Institute of Technology Department of Physics 8.022 Spring 2005 Lecture 15: Mutual and Self Inductance. 15.1 Using induction Induction is a fantastic way to create EMF; indeed, almost all electric power generation in the world takes advantage of Faraday’s law to produce EMF and drive currents. Given the defining formula, E = - 1 c d Φ B dt where Φ B = integraldisplay S vector B · d vector A all that we need to do is set up some way to make a time varying magnetic flux and we can induce EMF and drive currents very happily. Given the definition of magnetic flux, there are essentially three ways that we can make it vary and thereby create an EMF: we can make the area vary; we can make the “dot product” vary; and we can make the magnetic field vary. Let’s look at these three situations one by one. 15.1.1 Changing area We already discussed exactly this situation when we looked at the rod sliding on rails in a uniform magnetic field: L x R v B +++ --- 136

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The changing flux in this case is due to the continually changing area. We get an EMF whose magnitude is E = BLv/c , and that (by Lenz’s law) drives a current which circulates counterclockwise in the loop. In principle, we could imagine using this as a power source. For example, we could replace the resistor with a lightbulb and use this arrangement as a lamp. However, you’d have to find some way to continually slide the bar: the energy radiated by the lamp has to come from somewhere! In this case, it would come from the poor schmuck who continually drags the bar back and forth. 15.1.2 Changing “dot product” Another way to generate an EMF is to make the “dot product” — i.e., the relative orientation of our area and the magnetic field — vary with time. Imagine we have a circular loop sitting in a uniform magnetic field: B n ^ The magnetic flux through the loop is given by Φ B = πr 2 vector B · ˆ n = πr 2 B cos θ where θ is the angle between the loop’s normal vector and the magnetic field. Now, suppose we make the loop rotate: say the angle varies with time, θ = ωt . The EMF that is generated in this case is then given by E = ωπr 2 B c sin ωt . This kind of arrangement is in fact exactly how most power generators operate: An external force (flowing water, steam, gasoline powered engine) forces loops to spin in some magnetic field, generating electricity. Note that the EMF that is thereby generated is sinusoidal — generators typically produce AC (alternating current) power. 137
15.1.3 Changing field Finally, in many situations we can make the magnetic field itself vary. Suppose the integral is very simple, so that it just boils down to field times area. If the magnetic field is a function of time, then the flux will be a function of time: Φ B ( t ) = B ( t ) A .

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lec15 - Scott Hughes Massachusetts Institute of Technology...

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