Faraday's Law da B B r r Φ B B A d dφ ε B = dt

Transcription

1 Faraday's Law da Φ ε = r da r dφ dt

2 Applications of Magnetic Induction AC Generator Water turns wheel rotates magnet changes flux induces emf drives current Dynamic Microphones (E.g., some telephones) Sound oscillating pressure waves oscillating [diaphragm + coil] oscillating magnetic flux oscillating induced emf oscillating current in wire Question: Do dynamic microphones need a battery?

3 More Applications of Magnetic Induction Tape / Hard Drive / ZIP Readout Tiny coil responds to change in flux as the magnetic domains (encoding 0 s or 1 s) go by Nobel Prize!!!!!!!! Giant Magnetoresistance Credit Card Reader Must swipe card generates changing flux Faster swipe bigger signal

4 More Applications of Magnetic Induction Magnetic Levitation (Maglev) Trains Induced surface ( eddy ) currents produce field in opposite direction Repels magnet Levitates train S N eddy current rails Maglev trains today can travel up to 310 mph Twice the speed of Amtrak s fastest conventional train! May eventually use superconducting loops to produce -field No power dissipation in resistance of wires!

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

6 Getting the sign EMF in Faraday s Law of Induction Define the loop and an area vector, A, who magnitude is the Area and whose direction normal to the surface. A The choice of vector A direction defines the direction of EMF with a right hand rule. Your thumb in A direction and then your fingers point to positive EMF direction.

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8 Lenz s Law easier way! The direction of any magnetic induction effect is such as to oppose the cause of the effect. Convenient method to determine I direction Example if an external magnetic field on a loop is increasing, the induced current creates a field opposite that reduces the net field. Heinrich Friedrich Emil Lenz ( ) Example if an external magnetic field on a loop is decreasing, the induced current creates a field parallel to the that tends to increase the net field.

9 Lecture 16, Act 2 y A conducting rectangular loop moves with constant velocity v in the +x direction through a region of constant magnetic field in the -z direction as shown. 2A What is the direction of the induced current in the loop? (a) ccw (b) cw (c) no induced current A conducting rectangular loop moves with constant velocity v in the -y direction and a constant current I flows in the +x direction as shown. What is the direction of the induced 2 current in the loop? X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X vx X X X X X X X X X X X X X X X X x (a) ccw (b) cw (c) no induced current y v I x

10 1A 2A Lecture 16, Act 2 y A conducting rectangular loop moves with constant velocity v in the +x direction through a region of constant magnetic field in the -z direction as shown. What is the direction of the induced current in the loop? X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X vx X X X X X X X X X X X X X X X X x (a) ccw (b) cw (c) no induced current There is a non-zero flux Φ passing through the loop since is perpendicular to the area of the loop. Since the velocity of the loop and the magnetic field are CONSTANT, however, this flux DOES NOT CHANGE IN TIME. Therefore, there is NO emf induced in the loop; NO current will flow!!

11 Lecture 16, Act 2 A conducting rectangular loop moves with y constant velocity v in the -y direction and a constant current I flows in the +x direction as shown. What is the direction of the induced 2 current in the loop? (a) ccw (b) cw (c) no induced current v I x The flux through this loop DOES change in time since the loop is moving from a region of higher magnetic field to a region of lower field. Therefore, by Lenz Law, an emf will be induced which will oppose the change in flux. Current is induced in the clockwise direction to restore the flux.

12 Y&F Problem 29.2 In a physics laboratory experiment, a coil with 200 turns enclosing an area of 122cm^2 is rotated in a time interval of 0.04s from a position where its plane is perpendicular to the earth's magnetic field to one where its plane is parallel to the field. The earth's magnetic field at the lab location is 6x10(-5)T. What is the total magnetic flux through the coil before it is rotated? What is the total magnetic flux through the coil after it is rotated? What is the average emf induced in the coil? r r o Φ i = da = cos0 da = A Φ f = (6x10 ( Φ ε = f 5 Φ Δt T )(0.0122m = cos90a = 0 i 7 ) 7.31x10 Tm = s 2 ) = 7.31x Tm 2 = 1.83x10 5 Tm 2

13 Incredible shrinking loop: a circular loop of wire with a magnetic flux is shrinking with time. In which direction is the induced current? (a) There is none. (b) CW. (c) CCW X X X X X X X X X X X X X X X X X X X X X X X X

14 Determine the direction of the induced current in a b when 1.) Switch is opened after having been closed for a long time. (a) No current. (b) left to right. (c) right to left. 2.) The resistance R is decreased while switch remains closed. (a) No current. (b) left to right. (c) right to left.

15 Slide Wire Generator S The area is LS(t) and ds/dt equals velocity of slider d d r r d ε = Φ = da = da dt dt dt d d d = A = LS = L S = Lv dt dt dt Direction of current and ε as shown from Lenz s Law.

16 Slide Wire Generator; work and power we found S ε = Lv we now ask what is power dissipated, P d and power applied, P a assume total resistance is R, then I=Ɛ/R and P d =I 2 R= 2 L 2 v 2 /R The power applied, P a =Fv, is the rate of work to the bar to the right against the force, F, which is pointed to the left. F=IL= (Ɛ/R)L = 2 L 2 v/r. Thus P a = Fv = 2 L 2 v 2 /R = P d and we find the power put into the moving the bar equals the heat dissipated in the circuit. Power needed to move bar = Power dissipated in circuit

17 Slide Wire Generator; use Lenz s Law to get I direction S Lenz s Law says direction creates field that opposes change in magnetic flux. If we pull bar to right, the net magnetic flux in rectangle increases into screen, hence the I direction must induce opposite field which is out of screen and is correct in drawing. Suppose Lenz s law were reversed, then I would be reversed and F would go right and the bar would be accelerated to the right, w/o need of external positive work and heat would be dissipated at the same time. This violates Conservation of Energy, so Lenz s law is correct.

18 Motional Electromotive Force ε = d dt r r da = dφ dt In Faraday s Law, we can induce EMF in the loop when the magnetic flux, Φ, changes as a function of time. There are two Cases when Φ is changing, 1) Change the magnetic field (non-constant over time) 2) Change or move the loop in a constant magnetic field The slide wire generator is an example of #2 and the induction of EMF via moving parts of the loop is called, motional EMF.

19 Alternator Generator Wire loop area A rotates with respect to constant magnetic field. r r Φ = da = Acosφ d dt If the angular frequency is ω, then Φ Φ = Acos ( ω ) and EMF in the loop is t = Aω sin ωt ( ) ε = d dt Φ = Aω sin ωt ( )

20 Slide Wire Generator; revisited again E E Suppose we move a conducting bar in a constant field, then a force F =q v moves + charge up and charge down. The charge distribution produces an electric field, E, force, F E, and EMF, Ɛ, between a & b. This continues until equilibrium is reached: F E = -F. r r r r b r FE F qv r r r r E = = = = v ε = E dl = vl q q q In effect the bar motional EMF is an equivalent to a battery EMF a

21 Slide Wire Generator; revisited again If the rod is on the U shaped conductor, the charges don t build up at the ends but move around the U shaped portion. They produce an electric field in the U shaped circuit. The wire acts as a source of EMF just like a battery. Called motional electromotive force. b r ε = E a r dl = vl

22 Direct Current Homopolar Generator invented by Faraday Rotate a metal disk in a constant perpendicular magnetic field. The charges in the disk when moving receive a radial force. The causes current to flow from center to point b. R ε = ω r dr = ω R 2 Faraday s original apparatus had the disk fixed with a meter and then he spun a magnet under the disk.