Electricity and Magnetism-Mutual and Self Induction

1 8.022 (E&M) – Lecture 15 Topics: 􀂄 More on Electromagnetic Inductance 􀂄 Mutual and self inductance 􀂄 Practical applications G. Sciolla – MIT 8.022 – Lecture 15 2 Last time 􀂄 Electromagnetic inductance 􀂄 Faraday’s (and Lentz’s) law: 􀂄 Integral form: 􀂄 Differential form: 􀂄 Let’s elaborate a bit more on this important law… 1 B E c t ∂ ∇ × = − ∂ 􀁇 􀁇 􀁇 1 . . . B e m f c t ∂Φ = − ∂2 G. Sciolla – MIT 8.022 – Lecture 15 3 Cu pendulum in B field (H13) 􀂄 A copper pendulum is oscillating 􀂄 Application of Lentz’s law 􀂄 Turn on the magnetic field for the following 3 different situations: 􀂄 Pendulum #1: 􀂄 B crosses area with cuts 􀂄 B crosses area above cuts 􀂄 Pendulum #2: 􀂄 No cuts in Cu • No effect • Stops slowly: Lentz’s law • Stops abruptly: Lentz’s law Pendulum #1 Pendulum #2 G. Sciolla – MIT 8.022 – Lecture 15 4 Three ways of creating e.m.f. 􀂄 Faraday’s law can be used to build generators: 􀂄 3 ways of creating e.m.f.: 􀂄 Vary the area: S=S(t) 􀂄 Vary the angle between B and da 􀂄 Vary magnitude of B: B=B(t) 1 . . . S e m f B da c t ∂ = − ∂ ∫ 􀁇 􀁇 i3 G. Sciolla – MIT 8.022 – Lecture 15 5 Changing the area 􀂄 Sliding rod on rails: 􀂄 As derived last week: 􀂄 Because of Lentz’s law, direction of current is counterclockwise to oppose the change of flux of B 􀂄 Demo H4: 􀂄 Loop + light bulb moving in B created by electromagnet x B 􀁇 L R ++ --v I . . . vBL e m f c = 1 . . . S e m f B da c t ∂ = − ∂ ∫ 􀁇 􀁇 i G. Sciolla – MIT 8.022 – Lecture 15 6 Changing angle between B and S 􀂄 Constant B and loop rotating around its axis with angular velocity ω 􀂄 If S is the area of the loop: 􀂄 This is an easy way to build an AC power generator ( ) 1 . . . cos sin e m f BS t BS t c t cω ω ω ∂ ⇒ = = ∂ ω θ cos cos S B da BS BS t θ ω = = ∫ 􀁇 􀁇 i B 􀁇 1 . . . S e m f B da c t ∂ = − ∂ ∫ 􀁇 􀁇 i4 G. Sciolla – MIT 8.022 – Lecture 15 7 DC vs. AC current 􀂄 DC current 􀂄 Electrons flow all in the same direction at the same rate 􀂄 AC current 􀂄 The flow of electron varies with time in amplitude and direction: 􀂄 DC/AC generator 􀂄 Uses DC to power electromagnet and induce AC on rotating loop 􀂄 Why AC? Easier to step up and down for efficient transportation I(t) G. Sciolla – MIT 8.022 – Lecture 15 8 Changing magnitude of B 􀂄 Suppose you have a way to vary over time the magnitude of B: B=B(t) 􀂄 Flux of B: 􀂄 Generated e.m.f.: 􀂄 How to created B=B(t)? 􀂄 Loop of wire: 􀂄 If I=I(t) 􀃆 B=B(t) 􀃆 AC in a solenoid will do the trick! 1 1 ( ) . . . B B t e m f S c t c t ∂ ∂ = Φ = ∂ ∂ ( ) ( ) cos B S B t da B t S θ Φ = = ∫ 􀁇 􀁇 i B I ∝ 1 . . . S e m f B da c t ∂ = − ∂ ∫ 􀁇 􀁇 i5 G. Sciolla – MIT 8.022 – Lecture 15 9 Induced e.m.f. 􀂄 Consider a loop of wire with radius r inside a long solenoid 􀂄 Solenoid: 􀂄 N=# of loops, l=total length 􀃆 n=N/l 􀂄 Isol = Isol(t) 􀂄 What is the e.m.f. generated in the loop? 􀂄 Find B inside solenoid: 􀂄 E.m.f. generated in loop: 􀃆 The e.m.f. will depend by the geometry of the setup and on the rate of change of the I over time Isol 4 () sol sol nI t B c π = 2 2 2 2 ( ) 1 1 ( ) 4 . . . ( ) sol B I t B t nr e m f r c t c t c t π π ∂ ∂ ∂ = Φ = = ∂ ∂ ∂ Q: can you derive this in 60 sec? G. Sciolla – MIT 8.022 – Lecture 15 10 Induced e.m.f. on solenoid itself 􀂄 What if the “loop” is the solenoid itself? 􀂄 Will any e.m.f. be created? 􀂄 Remember Faraday’s law: 􀂄 B inside solenoid: 􀂄 Flux of B through each loop: 􀂄 Flux of B through N loops: 􀃆 Induced e.m.f. on solenoid: Isol 4 () sol sol nI t B c π = 1 . . . S e m f B da c t ∂ = − ∂ ∫ 􀁇 􀁇 i 1 loop 2 B 1 loop 4 () sol nI t BS R c π π Φ = = 2 2 2 Tot 1 loop B B 4 ( ) sol R N N I t c l π Φ = Φ = 2 2 2 2 ( ) 4 . . . sol I t R N e m f c l t π ∂ = ∂6 G. Sciolla – MIT 8.022 – Lecture 15 11 Back e.m.f. 􀂄 Magnitude of induced e.m.f. on solenoid: 􀂄 How about the direction? And the effect? 􀂄 Use Lentz’s law to predict direction of induced current 􀂄 If Isol increases 􀃆 B increases 􀃆 flux increases Iloop will fight change 􀃆 opposite direction as Isol 􀂄 If Isol decreases 􀃆 B decreases 􀃆 flux decreases Iloop will fight change 􀃆 same direction as Isol 􀂄 Conclusion: The inductance always opposes the change in the current The e.m.f. created is called back e.m.f. as it acts back on the circuit trying to oppose changes Isol 2 2 2 2 ( ) 4 . . . sol I t R N e m f c l t π ∂ = ∂ G. Sciolla – MIT 8.022 – Lecture 15 12 Example of back e.m.f. (H17) 􀂄 Close switch: wire jumps 􀃆 I flows (30 A) 􀂄 Open switch: big spark due by back emf 125 V Fe R7 G. Sciolla – MIT 8.022 – Lecture 15 13 Self Inductance L 􀂄 Self-induced e.m.f. in the solenoid: 􀂄 Let’s examine this in detail: 􀂄 e.m.f. depends on change over time of current: dI/dt 􀂄 A bunch of constants depending on geometry called self inductance L 􀂄 For a solenoid: 􀂄 Units: 􀂄 cgs: 􀂄 SI: 2 2 2 2 4 ( ) . . . sol R N c I t e m f t l π ∂ = ⇒ ∂2 2 2 2 4 sol R N L c l π = [ . . .] [ ] [ ]/( ) [ ] /e m f V L current time Hen H A s ry = = ≡ 2 [ . . .] /sec [ ] [ ]/[ ] ( /)/e m f esu cm L current time esu s s cm = = = ( ) . . . sol I t e m f t L∂ = ∂ G. Sciolla – MIT 8.022 – Lecture 15 14 Energy stored in inductors 􀂄 Consider an inductor L in which we start flowing a current I 􀂄 As soon as the current starts flowing, a back-emf tries to fight this current back 􀂄 Power needed to fight the back-emf: 􀂄 Calculate work to increase the current from 0 􀃆 I when t: 0 􀃆 t 􀂄 Energy stored in the inductor: . . . I P I emf ILt ∂ = × = ∂ 2 0 0 0 12 t t I t t I I W Pdt LI dt L IdI LI t = = = ∂ = = = = ∂ ∫ ∫ ∫ 2 12 W LI =8 G. Sciolla – MIT 8.022 – Lecture 15 15 How is energy stored in inductors? 􀂄 We created a magnetic field where there was none: work necessary to create the magnetic field is the energy stored in the B itself 􀂄 Same as energy stored in electric field of a capacitor 􀂄 Not surprising: special relativity! 􀂄 Energy density of magnetic field (solenoid example) 􀂄 Energy stored in solenoid: UL=LI2/2 􀂄 Self inductance of a solenoid: L=4π2R2N2 /lc2 􀂄 B created by solenoid: B=4πN /lc 􀃆􀂄 Energy density of B: 􀂄 Similar to energy density of the electric field: ( ) 2 2 2 2 2 2 2 2 1 14 1 4 2 2 8 8 L N N B U LI I Rl I Volume c l cl π π π π π ⎛ ⎞ = = = = ⎜ ⎟ ⎝ ⎠ 2 8 B B u π = 2 8 E E u π = G. Sciolla – MIT 8.022 – Lecture 15 16 How do we calculate L in psets? Just some examples… 􀂄 Strategy 1: 􀂄 L is the proportionality constant between induced emf and variation over time of current: 􀂄 Strategy 2: 􀂄 Exploit the fact that energy stored in the magnetic field is the energy stored in the inductor: 2 2 1 Energy stored in B 8 2 V B dV I L π = = ∫ ( ) . . . L I t e m f t ∂ = ∂9 G. Sciolla – MIT 8.022 – Lecture 15 17 Mutual inductance 􀂄 Back to the loop inside the solenoid 􀂄 Label solenoid with 1 and loop with 2 􀂄 e.m.f. induced on loop (ε2) depends on dI1/dt and constant M21 where M21 is the coefficient of mutual inductance 􀂄 For this particular configuration we already calculated that 􀂄 Now do the opposite: run a current I2(t) in the loop and calculate e.m.f. induced on solenoid (ε1): 􀂄 How to calculate M12??? 􀂄 No need to calculate it! Reciprocity theorem: M12=M21 21 1 2 M It ε ∂ = ∂ 2 2 21 2 4 r N M c l π = 12 2 1 M It ε ∂ = ∂ 1 2 G. Sciolla – MIT 8.022 – Lecture 15 18 Reciprocity theorem 􀂄 Consider 2 loops of wire: 􀂄 Current I runs through loop 1. What is ΦB through loop 2 due to 1? 􀂄 Now rewrite this result in terms of vector potential and use Stokes: 􀂄 Since we obtain 􀂄 Same fluxes􀃆 if currents are the same: M12=M21 1 1 1 C dl I A c r = ∫ 􀁇 􀁇 􀁶 Loop 1 Loop 2 2 21 1 2 B S B da Φ = ∫ 􀁇 􀁇 i ( ) 2 2 2 21 1 2 1 2 1 2 S S C B da A da A dl Φ = = ∇ × = ∫ ∫ ∫ 􀁇 􀁇 􀁇 􀁇 􀁇 􀁇 i i i 2 1 1 21 2 12 C C dl I dl c r Φ = =Φ ∫ ∫ 􀁇 􀁇 i10 G. Sciolla – MIT 8.022 – Lecture 15 19 Transformers 􀂄 Devices to step up (or down) AC currents 􀂄 Practical application of mutual inductance 􀂄 Simplest implementation: 􀂄 Primary solenoid (black): N1 turns 􀂄 Secondary solenoid (red): N2 turns 􀂄 I(t) in the primary will induce a varying ΦB through itself: 􀂄 where ΦB=magnetic flux through single turn 􀂄 Flux is the same in second solenoid 􀃆 induced e.m.f. is: 􀂄 Comparing: 1 1 B N d c dt ε Φ = N1 N2 2 2 B N d c dt ε Φ = 2 2 1 1 NN ε ε = Depending on number of turns we can • increase voltage (N2>N1) • reduce the voltage (N2