CHAPTER 20 Induced Voltages and Inductance An electric current produces a magnetic field. B =  o I 2r Scientists in the 19 th century saw that electricity (a relatively new discovery) and magnetism (a relatively old discovery) were related. They hypothesized: “If electricity (current) produces a magnetic field, then a magnetic field should be able to produce electricity.” As with many initial hypotheses, this one was close to correct…but not exactly.

Michael Faraday (as in Farad) Faraday discovered the correct relationship between electricity and magnetism. The relationship came as a surprise to him and was discovered somewhat accidentally. - The iron in the illustration is not a magnet. - With the switch open, it contains no magnetic field. -When the switch is closed, the primary coil will have current flowing and a B-field is established. Use one of the 2 RHR’s to determine the direction of the B-field lines established in the iron. B

- Notice the B-field extends all the way around the inside of the iron. - A current in the secondary coil wire develops only momentarily when the switch is closed (indicated by the Galvanometer) and then returns to zero. - A current in the secondary coil also develops (in the opposite direction) momentarily when the switch is opened. Faraday: It is the change in a magnetic field that can produce current, not the magnetic field itself. The change in B-field strength produces an emf (voltage) in the wire that then causes current.

Application: Current produced by a battery producing current elsewhere seems at first like an interesting novelty, but of little practical value. However, a magnet alone moving through a coil of wire produces the same effect (current) for the same reason. This is essentially how electricity has been produced for most of the past 150 years and how it is still produced today. Demo: magnets, coil, galvanometer

Faraday’s Law of Magnetic Induction  = emf = voltage N = number of turns of wire in the loop  = change in flux (Webers)  t = change in time  = - N   t Lenz’s Law: “the no free lunch law” The direction of the induced emf (voltage) is such that it produces a current whose magnetic field opposes the change in magnetic flux through the loop. That is, the induced current tends to maintain the original flux through the loop.

Example #1 A Conducting Bar Moving Perpendicular to a Uniform Magnetic Field Size of loop increases as the bar is pulled right at a velocity v. Magnitude  Magnitude  =   t  = B  A  t  = B L  x  t  = B L v Direction of Current Downward B-field lines being added Current creates a B-field with upward pointing B-field lines Current goes up on right side of loop and down through resistor on left. (CCW)

Lenz Law Examples Example #2 Square Loop of Wire Pulled at Constant Velocity into a Magnetic Field v = xtxt A = Area exposed to B-field =  x  L  = B  x  L  t  t =  = B v L   t What will be the direction of current? Counter Clockwise B’-field lines pointed up canceling out the increase in B-field lines pointing down. NOTE: Current stops once loop is entirely in field. Field Strength = B (down) v L

Example #3 Stationary Loop I a Spacially Uniform Magnetic Field Whose Magnitude is Changing at a Constant Rate  o = B o A  = B A  = (B – B o ) A  =  B  A =  =  A   t ( ) BtBt Find Direction of Current Decreasing BIncreasing B Current produced B ‘ tends to restore B-field lines lost. I = Clockwise Current produced B ‘ tends to oppose B-field lines gained. I = Counter Clockwise t = t B A t = 0 BoBo A

Example #4 Loop of Wire Rotates at a Constant Rate about an Axis Perpendicular to a Uniform Magnetic Field  = 2 B l v  = 2 B l v sin  2 in formula because 2 wires of length l (left and right side of loop)  Remember: If loop is horizontal and moving up (or down) the velocity component of movement  to the B- field is at a maximum. I sin  = 1 v Rotation Right half of loop

Magnetic Flux (  ): Magnetic Flux (  ):A measure of the B-field lines that actually pass through a loop of given area (A) Flux Example Flux Example (Perpendicular Situation) B = 20 T = 20 Wb m 2 Wb (Weber) can be thought of as magnetic field lines. A =.25m 2 If B-field lines are perpendicular to the area of the loop, then the magnetic flux through the loop is:  = B  A  = 20 x.25m 2 Wb m 2  = 5 Wb A B

Flux Example Flux Example (Angled Situation) The B-field only “sees” an area of A cos  If  = 90 Area exposed to B-field view is 0. (cos(90)=0) If  = 0 Area exposed to B-field view is A. (cos(0)=1)  = B  A  = B A cos   = (20 )(.25m 2 )(cos30) Wb m 2  = 4.3 Wb Notice: As loop rotated in a stationary B-field, the  changed from 5.0Wb to 4.3Wb. This  will induce emf and current in the wire (as long as the loops is rotating and the flux through it is changing.) Area remains equal to.25 meters 2, but is at an angle of  (30  ) with the horizontal while B (20T) is perpendicular (  ) to the horizontal. B  A  Caution:  = B A cos  only when certain angles are referenced  = B A sin Think!

Faraday’s Law of Magnetic Induction  = emf = voltage N = number of turns of wire in the loop  = change in flux (Webers)  t = change in time  = - N   t Lenz’s Law: “the no free lunch law” The direction of the induced emf (voltage) is such that it produces a current whose magnetic field opposes the change in magnetic flux through the loop. That is, the induced current tends to maintain the original flux through the loop. As loop rotates CCW,  gets smaller (less B-field lines intersect the opening). The current sets up additional B-field lines going down through the loop. Find direction of current. Find direction of Force on the current carrying wire in a magnetic field. I B A  FBFB