iclicker: which statements are correct?

Transcription

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2 iclicker: which statements are correct? 1. Electric field lines must originate and terminate on charges 2. Magnetic field lines are always closed A: 1&2 B: only 1 C: only 2 D: neither 2

3 Inductive E-field: work along closed path 3

4 Inductive E-field: work along closed path 3

5 Inductive E-field: work along closed path v 3

6 Inductive E-field: work along closed path v E 3

7 Inductive E-field: work along closed path v E-field lines are closed! This is very different from E-field from a point charge! E 3

8 Faraday s Law of Induction Potential drop along the closed contour is minus the rate of change of magnetic flu. We can change the magnetic flu in several ways including changing the magnitude of the magnetic field, changing the area of the loop, or by changing the angle the loop with respect to the magnetic field 10

9 Lenz s Law (1) Lenz s Law defines a rule for determining the direction of an induced current in a loop An induced current will have a direction such that the magnetic field due to the induced current OPPOSES the change in the magnetic flu that induces the current Meaning, if the flu increases, the induced B-field will be directed against eternal B-field 25

10 Changing magnetic flu through the area generates non-zero circulation of E-field Eds = 0 Recall, that for electrostatic field this was 0! V = dφ/dt = vbl l B v E = vb E = v B 6

11 iclicker A metal bar is moving with constant velocity through a uniform magnetic field pointing into the page, as shown in the figure. Think where the Lorentz force will push electrons F L = qv B Which of the following most accurately represents the charge distribution on the surface of the metal bar? a) distribution 1 b) distribution 2 c) distribution 3 d) distribution 4 e) distribution 5 FL,+ E

12 Changing magnetic flu through the area generates non-zero circulation of E-field Eds = 0 Recall, that for electrostatic field this was 0! V = dφ/dt = vbl B E v E = vb E = v B 8

13 Changing magnetic flu through the area generates non-zero circulation of E-field Eds = 0 Recall, that for electrostatic field this was 0! V = dφ/dt = vbl B + - E v E = vb E = v B 9

14 Changing magnetic flu through the area generates non-zero circulation of E-field Eds = 0 Recall, that for electrostatic field this was 0! V = dφ/dt = vbl B + - E v E = vb I E = v B 9

15 Changing magnetic flu through the area generates non-zero circulation of E-field Eds = 0 Recall, that for electrostatic field this was 0! V = dφ/dt = vbl B + - Bind E v E = vb I E = v B 9

16 Changing magnetic flu through the area generates non-zero circulation of E-field Eds = 0 Recall, that for electrostatic field this was 0! V = dφ/dt = vbl B + - Bind E v E = vb I E = v B Lenz s Law! 9

17 Lenz s Law - stability 10

18 Lenz s Law - stability 10

19 Lenz s Law - stability Eternal force 10

20 Lenz s Law - stability Eternal force 10

21 Lenz s Law - stability Eternal force 10

22 Lenz s Law - stability System s response - opposes eternal effect Eternal force 10

23 Lenz s Law - stability System s response enhances eternal effect - instability System s response - opposes eternal effect Eternal force 10

24 Lenz s Law - Induced current - Induced magnetic field in the direction so to oppose the change 11

25 Lenz s Law - Induced current - Induced magnetic field in the direction so to oppose the change 11

26 Lenz s Law - Induced current - Induced magnetic field in the direction so to oppose the change 11

27 Lenz s Law - Induced current - Induced magnetic field in the direction so to oppose the change i 11

28 Lenz s Law - Induced current - Induced magnetic field in the direction so to oppose the change E i 11

29 Eddy currents E-field line is circular. If within a conductor, it will drive current: eddy current. That current will produce its own B-field, opposing the eternal change (Lenz s law) 12

30 Eddy currents E-field line is circular. If within a conductor, it will drive current: eddy current. That current will produce its own B-field, opposing the eternal change (Lenz s law) 12

31 Eddy currents E-field line is circular. If within a conductor, it will drive current: eddy current. That current will produce its own B-field, opposing the eternal change (Lenz s law) 12

32 Eddy currents E-field line is circular. If within a conductor, it will drive current: eddy current. That current will produce its own B-field, opposing the eternal change (Lenz s law) E 12

33 Eddy currents E-field line is circular. If within a conductor, it will drive current: eddy current. That current will produce its own B-field, opposing the eternal change (Lenz s law) E i 12

34 demo: magnet falling through pipe The B-field induced by eddy currents will oppose the effect that induced the current = motion of the magnet. There will be breaking force. B-field generated by eddy currents will cause a repulsive or drag force between the conductor and the eternal magnet: magnetic breaking. gravitational energy is dissipated by resistivity, can heat an object. 13

35 Inductance i The unit of inductance is the henry (H) given by 14

36 Inductance i B The unit of inductance is the henry (H) given by 14

37 Inductance Magnetic flu through the contour is proportional to current, Φ i i B The unit of inductance is the henry (H) given by 14

38 Inductance Magnetic flu through the contour is proportional to current, Φ i Φ=Li i B The unit of inductance is the henry (H) given by 14

39 Inductance Magnetic flu through the contour is proportional to current, Φ i Φ=Li L - inductance (self-inductance) i B The unit of inductance is the henry (H) given by 14

40 Inductance Magnetic flu through the contour is proportional to current, Φ i Φ=Li L - inductance (self-inductance) i compare with capacitance: q= C V The unit of inductance is the henry (H) given by B 14

41 Inductance of a current loop. 15

42 B-field of a ring current B = µ0 4π I dl r r 3 db z = db cos β = µ 0 4π idl r 2 cos β cos β = R/r r = z 2 + R 2 B = µ 0 4π 2πR 2 i (z 2 + R 2 ) 3/2 Recall: E-field of a dipole ~ 1/z 3, similarly, B ~ 1/z 3 16

43 Self-inductance of a current loop. In the plane of the loop Φ R 2 B µ 0 ir B µ 0 2 i R Φ=Li L µ 0 R Similar to capacitance, inductance is a geometrical property 17

44 Inductance (1) Consider a long solenoid with N turns carrying a current i Same current, flu adds: large inductance. This current creates a magnetic field in the center of the solenoid resulting in a magnetic flu of Φ B The quantity NΦ B, called the flu linkage, is always proportional to the current with a proportionality constant called the inductance L 41

45 Inductance (1) Consider a long solenoid with N turns carrying a current i Same current, flu adds: large inductance. This current creates a magnetic field in the center of the solenoid resulting in a magnetic flu of Φ B Φ=Li The quantity NΦ B, called the flu linkage, is always proportional to the current with a proportionality constant called the inductance L 41

46 Inductance of a Solenoid Consider a solenoid with cross sectional area A and length l The flu linkage is n is the number of turns per unit length and B = µ 0 in The inductance of a solenoid is then The inductance of a solenoid depends only on its geometry 43

47 Self Inductance and Mutual Induction Consider the situation in which two coils, or inductors, are close to each other A current in the first coil produces magnetic flu in the second coil Changing the current in the first coil will induce an emf in the second coil However, the changing current in the first coil also induces an emf in itself This phenomenon is called self-induction The resulting emf is termed the self-induced emf 44

48 Self Induction Faraday s Law of Induction tells us that the self-induced emf for any inductor is given by Thus in any inductor, a self-induced emf appears when the current changes with time This self-induced emf depends on the time rate change of the current and the inductance of the device Lenz s Law provides the direction of the self-induced emf The minus sign epresses that the induced emf always opposes any change in current 45

49 Self Inductance: Increasing Current In the figure below, the current flowing through an inductor is increasing with time Thus a self-induced emf arises to oppose the increase in current V = L di dt 46

50 Self Inductance: Decreasing Current In the figure below, the current flowing through an inductor is decreasing with time Thus a self-induced emf arises to oppose the decrease in current V = L di dt 47

51 Self-inductance V = ir + L di dt di dt + R L i = V L,i(0) = 0 1 e Rt/L i(t) = V R τ = L R time constant inductor prevents the current from rising (or falling) much faster than the time L/R 24

52 Mutual inductance Φ 2 = L 12 i 1 Φ 1 = L 21 i 2 L 21 = L 12 25

53 Demo: Faraday s law 26

54 Energy of a Magnetic Field The instantaneous power provided by the emf source is the product of the current and voltage in the circuit Integrating this power over the time it takes to reach a final current yields the energy stored in the magnetic field of the inductor 59

55 Energy of B-field L = µ 0 n 2 V W = Li2 2 = µ 0 2 n2 i 2 V ni = B/µ 0 W = B2 2µ 0 V w = B2 2µ 0 Energy density of B-field 28