Electricity and Magnetism-Electromagnetic Induction

1 8.022 (E&M) – Lecture 14 Topics: 􀂄 Electromagnetic Inductance 􀂄 Faraday’s and Lentz’s laws G. Sciolla – MIT 8.022 – Lecture 14 2 Last time 􀂄 Parallel between Electric and Magnetic Fields 􀂄 Toward Maxwell’s equations: 􀂄 Vector Potential: 􀂄 Biot-Savart: E B A φ = −∇ ⇔ ≡ ∇ × 􀁇 􀁇 􀁇 􀁇 􀁇 4 04 0 πρ π ⎧∇ = ⇔ ∇ = ⎪⎨∇ × = ⇔ ∇ × = ⎪⎩ 􀁇 􀁇 􀁇 􀁇 i i 􀁇 􀁇 􀁇 􀁇 􀁇 E B E B J c 2 ˆ B wire I r dl c r = × ∫ 􀁇 􀁇2 G. Sciolla – MIT 8.022 – Lecture 14 3 Moving rod in uniform B 􀂄 Let’s move a conducting rod in a uniform B 􀂄 Charges move with velocity v//x axis 􀂄 B//y axis 􀂄 What happens? 1) Lorentz force: 2) Electric field E1 causes separation of charges on the wire 3) Separation of charges creates an opposite electric field E2 that exactly compensates E1 and equilibrium is established: 1 = × = 􀁇 􀁇 􀁇 􀁇 Lorentz v F q B qE c B v F 2 = − × 􀁇 􀁇 􀁇 v E B c B v E1 ++ ++ ----E2 G. Sciolla – MIT 8.022 – Lecture 14 4 Moving a loop in uniform B 􀂄 Now move a rectangular loop of wire in B 􀂄 Same velocity 􀂄 Same B 􀂄 What happens? 􀂄 Lorentz force 􀃆 E1 􀂄 E1 􀃆 separation of charges on the wire 􀂄 Separation of charges creates opposite electric field E2= -E1: v E2 E1 ++++ ----3 G. Sciolla – MIT 8.022 – Lecture 14 5 What if B is non uniform? 􀂄 Now move the rectangular loop of wire in non uniform B 􀂄 Velocity v 􀂄 B = B0 above ---􀂄 B = 0 below ---􀂄 What happens? 􀂄 Lorentz force 􀃆 E1 􀂄 E1 􀃆 separation of charges on the wire 􀂄 Separation of charges creates charges to flow in the loop (no opposing force in the bottom part!) 􀂄 This phenomenon is called electromagnetic induction v E1 ++ --I G. Sciolla – MIT 8.022 – Lecture 14 6 Comments on induction Please notice the following: 􀂄 End of electrostatics! 􀂄 The current flowing in top leg of the loop will feel a force FB from B pointing up 􀂄 Lentz’s law v E1 ++ --I 0 0 loop E dl or E ≠ ∇× ≠ ∫ 􀁇 􀁇 􀁇 i 􀁶 FB4 G. Sciolla – MIT 8.022 – Lecture 14 7 Induced emf 􀂄 Consider a sliding conducting bar on rails closed on a resistor R in a region of constant magnetic field B 􀂄 Charge separation in the bar will induce current 􀃆 e.m.f. 􀂄 Current flowing in the loop: B 􀁇 ( ) 1 1 1 . . . ( ) vBL e m f W F ds v B ds q q c c + + − − = −→+ = = × = ∫ ∫ 􀁇 􀁇 􀁇 􀁇 􀁇 i i L x R ++ --vBL I cR = v I G. Sciolla – MIT 8.022 – Lecture 14 8 Faraday’s law 􀂄 EMF in the loop: 􀂄 Magnetic flux in the rectangle is defined as: 􀂄 Combine the two keeping in mind that given the direction of v, flux decreases with time: 􀃆 Faraday’s law: 􀂄 The minus sign is important: Lentz’s law 􀂄 It indicates that the direction of the current is such to oppose the changes in flux of B: ~”electromagnetic inertia” . . . vBL BL dx e m f c c dt = = B Blx Φ = 1 . . . B e m f c t ∂Φ = − ∂5 G. Sciolla – MIT 8.022 – Lecture 14 9 Thoughts on Lentz’s law Lentz’s law: The current generated in wire opposes changes in flux of B 􀂄 v is L􀃆R: 􀂄 Flux of B decreases over time 􀃆 e.m.f. is created with direction that compensates this change: counterclockwise 􀂄 v is R􀃆L: 􀂄 Flux of B increases over time 􀃆 e.m.f. is created with direction that compensates this change: clockwise B 􀁇L x ++ --v I 1 . . . B e m f c t ∂Φ = − ∂ G. Sciolla – MIT 8.022 – Lecture 14 10 Another way of looking at Lentz When current flows in magnetic field it feels a force Lentz’s law: the force will be will try to slow down the bar 􀂄 If I clockwise: 􀂄 It creates a B pointing into the board 􀃆 I x B points to the left 􀂄 If I counterclockwise: 􀂄 It creates a B pointing out of the board 􀃆 I x B points to the right B 􀁇L ++ --v I 1 . . . B e m f c t ∂Φ = − ∂ NB: the – sign in Lentz’s law is what allows conservation of energy6 G. Sciolla – MIT 8.022 – Lecture 14 11 General proof of Faraday’s law 􀂄 Consider a loop of arbitrary shape moving with velocity v through a static magnetic field B 􀂄 At time t, the flux through the loop is: 􀂄 How does it change when t 􀃆 t+∆t? 􀂄 On the ribbon: Loop at time t Loop at time t+∆t v+∆t B S B da Φ = ∫ 􀁇 􀁇 i dl ( ) () B B B ribbon ribbon t t t Bda ∆Φ = Φ + ∆ − Φ = Φ = ∫ 􀁇 􀁇 i B ( ) da v t dl = ∆ × 􀁇 􀁇 􀁇 G. Sciolla – MIT 8.022 – Lecture 14 12 Proof of Faraday’s law(2) 􀂄 This means that: 􀂄 Using the identity we obtain: Since v/c x B is the magnetic force for unit charge 􀃆 its line integral on the loop is the work necessary to move a unit charge around the wire: e.m.f! 􀃆 ( ) ( ) B ribbon ribbon ribbon B da B v t dl t B v dl ∆Φ = = ∆ × = ∆ × ∫ ∫ ∫ 􀁇 􀁇 􀁇 􀁇 􀁇 􀁇 􀁇 􀁇 i i i ( ) ( ) a b c a b c × = × 􀁇 􀁇 􀁇 􀁇 􀁇 􀁇 i i ( ) For ∆t 0: B loop loop ribbon B loop t B v dl t B v dl tloop v B dl v c B dl t c ∆Φ = ∆ × = ∆ × = −∆ × ∂Φ ⎛ ⎞ → =− × ⎜ ⎟ ∂ ⎝ ⎠ ∫ ∫ ∫ ∫ 􀁇 􀁇 􀁇 􀁇 􀁇 􀁇 􀁇 􀁇 􀁇 i i i 􀁇 􀁇 􀁇 i 1 . . . B e m f c t ∂Φ = − ∂7 G. Sciolla – MIT 8.022 – Lecture 14 13 Work from B??? 􀂄 Faraday’s law: 􀂄 This means that integrated over the loop is the work that we have to do to move a unit charge around the loop 􀂄 But last time we proved that B cannot do work 􀂄 Are these 2 statements inconsistent??? 􀂄 No, the work done to move the charges is not done by B 􀂄 It’s done by whoever is moving the loop in B /v c B × 􀁇 􀁇 1 . . . B e m f c t ∂Φ = − ∂ G. Sciolla – MIT 8.022 – Lecture 14 14 Verification of Faradya’s law 􀂄 Faraday’s law: 􀂄 What does it mean? 􀂄 E.m.f. Is produced when the flux of B changes over time 􀂄 􀃆 area of the loop cannot be null! 􀂄 Demo H1: 1 . . . B e m f c t ∂Φ = − ∂ B -move loop in B -current flows in wire -if we used instead a wire with 0 area: no I vs. A8 G. Sciolla – MIT 8.022 – Lecture 14 15 “Relativity” 􀂄 What if loop is static and B changes? 􀂄 Relativity tells me that we should get the same result 􀂄 Same problem from another reference frame 􀂄 Does this make sense? 􀂄 Charges do not move in the other reference frame 􀂄 What causes the force? The induced electric field 􀂄 Since e.m.f. is the work necessary to move a unit charge around the loop: 􀂄 Demo H3: magnet bar moving in the loop . . . C e m f E dl = ∫ 􀁇 􀁇i 􀁶 G. Sciolla – MIT 8.022 – Lecture 14 16 􀂄 H5: disk falling in a magnetic field B 􀂄 Create B with e electromagnets (solenoid on Fe core) 􀂄 What happen if we drop a disk of conductor? 􀂄 With and without B 􀂄 What if we drop a full disk 􀂄 What if we drop a disk with a cut? Application of Lentz’s law B9 G. Sciolla – MIT 8.022 – Lecture 14 17 Explanation 􀂄 Falling Loop: 􀂄 B perpendicular to loop is limited in space 􀃆 flux changes during fall 􀃆 induced I 􀃆 loop will levitate (Eddie currents) 􀂄 Falling Disk 􀂄 Will it slow down? 􀂄 Falling open ring 􀂄 Will it levitate? Ifront Iback G. Sciolla – MIT 8.022 – Lecture 14 18 More demos on Faraday’s law 􀂄 H8: current generated by a solenoid 􀂄 Where to put the loop of wire to have current? 􀂄 Remember: B of solenoid is 0 outside 􀂄 Switch I on and off 􀂄 H22 Levitating rings A A B I F10 G. Sciolla – MIT 8.022 – Lecture 14 19 More demos on Faraday’s law 􀂄 H15a: current generated by a solenoid 􀂄 Spinning disk of conductor 􀂄 Magnet sitting on top separated by a plastic sheet 􀂄 When disk starts spinning, magnet levitates 􀂄 Why? +++ N v ---G. Sciolla – MIT 8.022 – Lecture 14 20 Faraday’s law in differential form 􀂄 Faraday’s law in integral form: 􀂄 Right term (apply Stokes): 􀂄 Left term: 􀃆 􀂄 Since this is valid for any surface: 􀂄 curl E is not longer zero: bye bye electrostatics! 􀂄 Explicit link between E and B, as in relativity! 1 . . . B e m f c t ∂Φ = − ∂ . . . C S e m f E dl E da = = ∇ × ∫ ∫ 􀁇 􀁇 􀁇 􀁇 􀁇 i i 􀁶 1 1 B S B da c t c t ∂Φ ∂ − =− ∂ ∂ ∫ 􀁇 􀁇 i 1 0 S B E da c t ⎛ ⎞ ∂ ∇× + = ⎜ ⎟ ⎜ ⎟ ∂ ⎝ ⎠ ∫ 􀁇 􀁇 􀁇 􀁇 i 1 B E c t ∂ ∇ × = − ∂ 􀁇 􀁇 􀁇11 G. Sciolla – MIT 8.022 – Lecture 14 21 Another step toward Maxwell’s equations… 􀂄 All the equations in differential form that we found so far: 􀂄 Another step toward Maxwell’s equations: one last missing ingredient… Can you guess what? 􀂄 Symmetry will guide you… Hint: 􀂄 Or vector calculus… Hint: take the divergence of Faraday’s law… 40 1 4πρπ ⎧∇ = ⎪∇ = ⎪⎪∂ ⎨∇ × = − ⎪ ∂ ⎪⎪∇ × = ⎩ 􀁇 􀁇 i 􀁇 􀁇 i 􀁇 􀁇 􀁇 􀁇 􀁇 EB B E c t B J c 􀃅 Relates E and charge density (ρ) -Gauss 􀃅 Magnetic field lines are closed 􀃅 Change in B creates E -Faraday 􀃅 Relates B and its sources (J) -Ampere G. Sciolla – MIT 8.022 – Lecture 14 22 Summary and outlook 􀂄 Today: 􀂄 Faraday’s (and Lentz’s) law: 􀂄 Integral form: 􀂄 Differential form: 􀂄 Next time: 􀂄 Mutual and self inductance 1 B E c t ∂ ∇ × = − ∂ 􀁇 􀁇 􀁇 1 . . . B e m f c t ∂Φ = − ∂