Faraday's Law ds B B r r Φ B B S d dφ ε B = dt

Transcription

1 ds Faaday's Law Φ ε ds dφ =

2 Faaday s Law of Induction Recall the definition of magnetic flux is Φ = da Faaday s Law is the induced EMF in a closed loop equal the negative of the time deivative of magnetic flux change in the loop, ε = d da dφ = Constant field, no induced EMF in loop changing field, causes induced EMF in loop

3 Y&F Poblem 29.2 In a physics laboatoy expeiment, a coil with 200 tuns enclosing an aea of 122cm^2 is otated in a time inteval of 0.04s fom a position whee its plane is pependicula to the eath's magnetic field to one whee its plane is paallel to the field. The eath's magnetic field at the lab location is 6x10(-5)T. What is the total magnetic flux though the coil befoe it is otated? What is the total magnetic flux though the coil afte it is otated? What is the aveage emf induced in the coil? o Φ i = N da = N cos0 Φ f = 200(6x10 ( Φ ε = f Φ Δt i 5 = N cos90a = 0 T )(0.0122m da = 7 ) 7.31x10 Tm = s 2 2 NA ) = 1.46x10 4 Tm = 3.66x V

4 Slide Wie Geneato; wok and powe Last lectue we found S ε = Lv we now ask what is powe dissipated, P d and powe applied, P a assume total esistance is R, then I=/R and P d =I 2 R= 2 L 2 v 2 /R The powe applied, P a =Fv, is the ate of wok to the ba to the ight against the foce, F, which is pointed to the left. F=IL= (/R)L = 2 L 2 v/r. Thus P a = Fv = 2 L 2 v 2 /R = P d and we find the powe put into the moving the ba equals the heat dissipated in the cicuit. Powe needed to move ba = Powe dissipated in cicuit

5 Histoy of the Discovey of Electomagnetic Induction In 1825, J. Colladon (who fist measued sound speed), had a coil connected to a galvanomete and moved a lage magnet neaby, UT since his magnetic field would affect his galvanomete, he moved the mete to the next oom. So when he moved the magnet and then went to the othe oom to check on the mete, the effect disappeaed. In 1930, Joseph Heny of Pinceton (invented telegaph) obseved induction, convesion of magnetism to electicity, but did not publish and did not ecognize the vast impotance of this obsevation. Late howeve, he obseved and published evidence fo self induction. In 1831, Faaday obseved and published evidence of electomagnetic induction and also invented the homopola geneato that convets continuous DC. Note the unit of capacitance is Faad afte Faaday and the unit of inductance is Heny afte Heny.

6 Lenz s Law The diection of any magnetic induction effect is such as to oppose the cause of the effect. Convenient method to detemine I diection Example if an extenal magnetic field on a loop is inceasing, the induced cuent ceates a field opposite that educes the net field. Heinich Fiedich Emil Lenz ( ) Example if an extenal magnetic field on a loop is deceasing, the induced cuent ceates a field paallel to the that tends to incease the net field.

7 Lenz s Law; conside moving magnet towads a wie loop No cuent and no field stationay magnet Induced field is opposite moving N side of magnet into loop Induced field is opposite moving S side of magnet into loop Repulsive foce, like pushing two S poles togethe.

8 Slide Wie Geneato; use Lenz s Law to get I diection S Lenz s Law says diection ceates field that opposes change in magnetic flux. If we pull ba to ight, the net magnetic flux in ectangle inceases into sceen, hence the I diection must induce opposite field which is out of sceen and is coect in dawing. Suppose Lenz s law wee evesed, then I would be evesed and F would go ight and the ba would be acceleated to the ight, w/o need of extenal positive wok and heat would be dissipated at the same time. The is violates cons. of Enegy, so Lenz s law is coect.

9 Motional Electomotive Foce ε = d da = dφ In Faaday s Law, we can induce EMF in the loop when the magnetic flux, Φ, changes as a function of time. Thee ae two Cases when Φ is changing, 1) Change the magnetic field (non-constant ove time) 2) Change o move the loop in a constant magnetic field The slide wie geneato is an example of #2 and the induction of EMF via moving pats of the loop is called, motional EMF.

10 Slide Wie Geneato; evisited again Suppose we move a conducting ba in a constant field, then a foce F=q v moves + chage up and chage down. The chage distibution poduces an electic field and EMF,, between a & b. This continues until equilibium is eached. E F qv b = = = v ε = E dl = vl q q In effect the ba motional EMF is an equivalent to a battey EMF a

11 Slide Wie Geneato; evisited again If the od is on the U shaped conducto, the chages don t build up at he ends but move aound the U shaped potion. They poduce an electic field in the U shaped cicuit. The wie acts as a souce of EMF just like a battey. Called motional electomotive foce. b ε = E a dl = vl

12 Diect Cuent Homopola Geneato invented by Faaday Rotate a metal disk in a constant pependicula magnetic field. The chages in the disk when moving eceive a adial foce. The causes cuent to flow fom cente to point b. R ε = ω d = ω R 2 Faaday s oiginal appaatus had the disk fixed with a mete and then he spun a magnet unde the disk.

13 Faaday s Law (continued) What causes cuent to flow in wie? Answe: an E field in the wie. A changing magnetic flux not only causes an EMF aound a loop but an induced electic field. Can wite Faaday s Law: ε = E dl = d da = dφ Remembe fo a long staight wie of length l, V = El. Note: Fo electic fields fom static chages, the EMF fom a closed path is always zeo. Not tue hee. Thee ae two souces fo electic fields!

14 Induced Electic Fields Suppose we have electomagnetic that has an inceasing magnetic field Using Faaday s Law we pedict, E dl = d dφ da = If we take a cicula path inside and centeed on the magnet cente axis, the electic field will be tangent to the cicle. (E field lines ae cicles.) NOTE such an E field can neve be made by static chages N S E E field lines will look like an onion slice (w/o smell!!) NOTE thee ae no wie loops, the E fields can appea w/o loops If we place a loop thee, a cuent would flow in the loop

15 Induced Electic Fields; example If we have a solenoid coil with changing cuent thee will be cicula electic fields ceated outside the solenoid. It looks vey much like the mag. field aound a cuent caying wie, but it is an E field and thee ae no wies o loops. E Note the E fields ae pedicted by Faaday eqn. E dl = d da dφ =

16 Eddy Cuents Changing magnetic fields in metal induce eddy cuents. Example: Enegy loss in tansfomes. To educe use laminations. ut eddy cuents often useful.