Electricity and Magnetism Eddy Currents Faraday s Law and Electric Field

Transcription

1 Electricity and Magnetism Eddy Currents Faraday s Law and Electric Field Lana heridan De Anza College Mar 8, 2018

2 Last time Lenz s law applying Faraday s law in problems technological applications

3 Overview eddy currents changing magnetic field and electric field

4 Faraday s Law Faraday s Law If a conducting loop experiences a changing magnetic flux through the area of the loop, an emf E F is induced in the loop that is directly proportional to the rate of change of the flux, Φ B with time. Faraday s Law for a conducting loop: E F = dφ B dt

5 Lenz s Law The magnet's motion creates a magnetic dipole that opposes the motion. Additional examples, video, and practice available at Lenz s Law 30-4 Lenz s Law An induced current has a direction such that the magnetic fieldoon dueafter to the Faraday current proposed opposes his the law of induc devised a rule for determining the direction of an change in the magnetic flux that induces the current. An induced current has a direction such that the ma opposes the change in the magnetic flux that induces t Furthermore, the direction of an induced emf is th a feel for Lenz s law,let us apply it in two different where the north pole of a magnet is being moved to N Basically, Lenz s 1. Opposition law let s us to Pole interpret Movement. the minus The approach N Fig increases the magnetic flux through µ sign in the equation we write to represent current in the loop. From Fig , we know t Faraday s Law. netic dipole with a south pole and a north po moment : is directed from south to north. i increase E = being dφ caused B by the approaching mag thus : ) must dt face toward the approaching no 30-4). Then the curled straight right-hand rul Fig Lenz s law at work.as the the current induced in the loop must be counte magnet is 1 moved toward the loop, a current Figure from Halliday, Resnick, Walker, 9th If we ed. next pull the magnet away from th

6 esult n the e sysnsisfore, Faraday s and Lenz s Laws What about this case? We found the current should flow counterclockwise. ire in to I v Figure (Quick Quiz 31.3)

7 esult n the e sysnsisfore, Faraday s and Lenz s Laws What about this case? We found the current should flow counterclockwise. ire in to I v The flux from Figure the wire is into the (Quick page and Quiz increasing. 31.3) The field from the current in the loop is out of the page. There is an upward resistive force on the ring. (cf. HW3, #3.)

8 31.4 Induced emf and Electric Fields 947 Loop moving into and out of a B-field,, v a w 0 3w B in x B v B v c e x B F x B w B 2 2 v R. b 0 w 3w 4w x d 0 w 3w 4w x

9 Eddy Currents If the wire is replaced by a solid conducting plate, circulations of ON AND current INDUCTANCE form in the plate. - y l B Eddy current loop Pivot n- B (a) ince the cross section of the plate is larger than that of a similar Eddy wire, Currents the resistance will be low, but the current can be high. uppose we replace the conducting loop of Fig with a solid conducting plate. If we then move the plate out of the magnetic field as we did the loop (Fig a), The plate will heat. the relative motion of the field and the conductor again induces a current in the conductor. Thus, we again encounter an opposing force and must do work because of the induced current. With the plate, however, the conduction electrons making (b)

10 Induced Electric Fields If moving a conductor in a magnetic field causes a current to flow, if must be because the process has created an electric field across the conductor. The fact that in a conducting plate circulations of current appear tells us that the electric field lines must also makes these circles. Another way to cause a current and electric field is to change the flux by increasing or decreasing the magnetic field INDUCED ELEC Copper ring Circular path E (a) R i B r (b) E R B E r E

11 Induced Electric Fields Electric field lines R R B 3 (c) (d) The circulation Fig E-field(a) occurs If the whether magnetic or field not aincreases conductor at a issteady rate, a present: rent it isappears, the direct as shown, result of in the changing copper ring magnetic of radius flux. r.(b) An induce even when the ring is removed; the electric field is shown at four po picture of the induced electric field, displayed as field lines. (d) Four Faraday s Law of Induction (in words) enclose identical areas. Equal emfs are induced around paths 1 and 2 A changing the region magnetic of changing field gives magnetic rise to field.a an electric smaller field. emf is induced aroun partially lies in that region. No net emf is induced around path 4, whic

12 Induced emf and the Electric Field For a closed path, s, E = E ds Notice that by definition V = E ds = 0. Emf does not have this property. When a charge is moved around a closed path in an electrostatic electric field (E = V ) the work done is zero: W E = q( V ) = 0

13 d a current in a o Induced an electric emf field If B changes in time, an electric and the Electric Field field is induced in a direction, we can For relate the induced an E-field tangent from to the a circumference changing magnetic of flux, the ing that associated an elecagnetic flux. force F the = qe loop. is not conservative. n electric field is B in nce suggests that E E ield generates an ostatic field proconducting loop ular to the plane ith time, an emf in the loop. The ced electric field tion in which the work done by the al to qe. Because e electric field in E Figure A conducting loop The work done tracing of radius around r in a the uniform grey loop magnetic is not zero. r E field perpendicular to the plane of the loop. W E 0

14 Non-Conservative Fields Non-conservative vector fields cannot be represented with a topological map. 1 Lithograph in the mathematically-inspired impossible reality style, by M.C. Escher.

15 Induced emf and the Electric Field For the induced E-field from a changing magnetic flux, the associated force F = qe is not conservative. We say the E-field is nonconservative. We now write the electric field in a more general way: E = V a t B = a

16 Induced emf and the Electric Field For the induced E-field from a changing magnetic flux, the associated force F = qe is not conservative. We say the E-field is nonconservative. We now write the electric field in a more general way: (Note that a transformation: E = V a t B = a V V λ t a a + λ where λ is any twice-differentiable function of position and time, does not change the E and B fields. Gauge invariance.)

17 g that an elecetic Induced flux. emf and the Electric Field the loop. lectric field is suggests that E E generates an atic field pronducting loop r to the plane time, an emf the loop. The d electric field in which the k done by the o qe. Because lectric field in tangent to the circumference of E r E B in Figure A conducting loop We can also write Faraday s Law as: of radius r in a uniform magnetic field perpendicular to the plane of the loop. E ds = dφ B dt

18 ummary of Material in Ch Chapter 29 Magnetic Fields force on a charge from a magnetic field motion of a charge in a magnetic field particle accelerators particle in crossed electric and magnetic fields velocity selector mass spectrometer / discovery of the electron the Hall effect

19 ummary of Material in Ch Chapter 29 Magnetic Fields force on a charge from a magnetic field motion of a charge in a magnetic field particle accelerators particle in crossed electric and magnetic fields velocity selector mass spectrometer / discovery of the electron the Hall effect force on a wire carrying current in a B-field torque on a wire loop in a B-field magnetic moment torque and potential energy of a magnetic moment in a B-field

20 ummary of Material in Ch Chapter 30 ources of the Magnetic field B-field around a moving charge B-field around a steady current (Biot-avart law) B-field from a long, straight wire B-field around a loop of wire

21 ummary of Material in Ch Chapter 30 ources of the Magnetic field B-field around a moving charge B-field around a steady current (Biot-avart law) B-field from a long, straight wire B-field around a loop of wire Gauss s law Ampère s law B-field inside soleniods magnetism of bulk matter

22 ummary of Material in Ch Chapter 31 Faraday s Law and Induction Motional emf Faraday s law Lenz s law generators and applications nonconservative electric field

23 Review Your questions.

24 ummary motional emf Faraday s law Lenz s law applications Next Test this Friday, Mar 9. Homework study erway & Jewett: PREVIOU: Ch 31, Obj. Qs: 1, 3, 5, 7; Conc. Qs: 3, 5; Problems: 1, 5, 9, 13, 21, 27, 31, 33 PREVIOU: Ch 31, Problems: 39, 40, 41, 45, 50 (for each eddy current show is it correct or incorrect?).