8.022 (E&M) Lecture 15

Transcription

1 8.0 (E&M) Lecture 5 Topics: More on Electromagnetic Inductance Mutual and self inductance Practical applications Last time Electromagnetic inductance Faraday s (and Lentz s) law: Φ Integral form: emf... Differential form: E Let s elaborate a bit more on this important law G. ciolla MIT 8.0 Lecture 5

2 Cu pendulum in field (H3) A copper pendulum is oscillating Application of Lentz s law Turn on the magnetic field for the following 3 different situations: Pendulum #: crosses area with cuts No effect crosses area above cuts tops slowly: Lentz s law Pendulum #: No cuts in Cu tops abruptly: Lentz s law Pendulum # Pendulum # G. ciolla MIT 8.0 Lecture 5 3 Three ways of creating e.m.f. Faraday s law can be used to build generators: emf... da i 3 ways of creating e.m.f.: Vary the area: (t) Vary the angle between and da Vary magnitude of : (t) G. ciolla MIT 8.0 Lecture 5 4

3 Changing the area emf... da i liding rod on rails: -- v L R I ++ x vl As derived last week: emf... c ecause of Lentz s law, direction of current is counterclockwise to oppose the change of flux of Demo H4: Loop + light bulb moving in created by electromagnet G. ciolla MIT 8.0 Lecture 5 5 Changing angle between and Constant and loop rotating around its axis with angular velocity ω ω θ emf... da i If is the area of the loop: da i cosθ cosωt ω emf... ( cosωt ) sinωt c This is an easy way to build an AC power generator G. ciolla MIT 8.0 Lecture 5 6 3

4 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: I(t) 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 G. ciolla MIT 8.0 Lecture 5 7 Changing magnitude of emf... da i uppose you have a way to vary over time the magnitude of : (t) Flux of : Φ () t ida () t cosθ Generated e.m.f.: How to created (t)? Loop of wire: I If II(t) (t) ( t) emf... Φ AC in a enoid will do the trick! G. ciolla MIT 8.0 Lecture 5 8 4

5 Induced e.m.f. Consider a loop of wire with radius r inside a long enoid olenoid: N# of loops, ltotal length nn/l I I (t) What is the e.m.f. generated in the loop? Find inside enoid: E.m.f. generated in loop: 4 πni ( t ) c ( t) 4π nr I () t... Φ ( πr ) emf Q: can you derive this in 60 sec? The e.m.f. will depend by the geometry of the setup and on the rate of change of the I over time G. ciolla MIT 8.0 Lecture 5 9 I Induced e.m.f. on enoid itself What if the loop is the enoid itself? Will any e.m.f. be created? Remember Faraday s law: 4 πni ( t ) inside enoid: c loop 4 πni ( t ) Flux of through each loop: Φ loop πr c Flux of through N loops: emf... da i Tot loop 4π RN Φ NΦ I () t cl I 4π RN I () t Induced e.m.f. on enoid: emf... c l t G. ciolla MIT 8.0 Lecture 5 0 5

6 ack e.m.f. Magnitude of induced e.m.f. on enoid: 4π RN I () t emf... c l t How about the direction? And the effect? I Use Lentz s law to predict direction of induced current If I increases increases flux increases I loop will fight change opposite direction as I If I decreases decreases flux decreases I loop will fight change same direction as I 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 G. ciolla MIT 8.0 Lecture 5 Example of back e.m.f. (H7) Fe R 5 V Close switch: wire jumps I flows (30 A) Open switch: big spark due by back emf G. ciolla MIT 8.0 Lecture 5 6

7 elf Inductance L elf-induced e.m.f. in the enoid: 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 enoid: Units: cgs: I: 4 π R N I ( t ) emf... c l t 4π RN L cl I ( t ) emf... L t [ emf...] esu/ cm sec [ L] [ current ]/[ time ] ( esu / s )/ s cm [ emf...] V [ L ] Henry ( H ) [ current ]/[ time ] A / s G. ciolla MIT 8.0 Lecture 5 3 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: I P I e. m. f. IL t Calculate work to increase the current from 0 I when t: 0 t t t I I W Pdt LI dt L IdI LI t 0 t 0 t I 0 Energy stored in the inductor: W LI G. ciolla MIT 8.0 Lecture 5 4 7

8 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 itself ame as energy stored in electric field of a capacitor Not surprising: special relativity! Energy density of magnetic field (enoid example) Energy stored in enoid: U L LI / elf inductance of a enoid: L4π R N /lc created by enoid: 4πN/lc Energy density of : π N πn 4 4 U L LI I ( π R l ) I Volume c l 8π cl 8π 8π imilar to energy density of the electric field: u E 8π G. ciolla MIT 8.0 Lecture 5 5 u E How do we calculate L in psets? Just some examples trategy : L is the proportionality constant between induced emf and variation over time of current: trategy : ( ) emf... L I t t Exploit the fact that energy stored in the magnetic field is the energy stored in the inductor: dv LI 8π V Energy stored in G. ciolla MIT 8.0 Lecture 5 6 8

9 Mutual inductance ack to the loop inside the enoid Label enoid with and loop with e.m.f. induced on loop (ε ) depends on di /dt and constant M I ε M t where M is the coefficient of mutual inductance 4π r N For this particular configuration we already calculated that M c l Now do the opposite: run a current I (t) in the loop and calculate e.m.f. induced on enoid (ε ): I ε M t How to calculate M??? No need to calculate it! Reciprocity theorem: M M G. ciolla MIT 8.0 Lecture 5 7 Reciprocity theorem Consider loops of wire: Loop Loop Current I runs through loop. What is Φ through loop due to? Φ ida Now rewrite this result in terms of vector potential and use tokes: Φ ida A ida A idl I dl ince A we obtain c C r ( ) C Φ I dl idl Φ c r C C ame fluxes if currents are the same: M M G. ciolla MIT 8.0 Lecture 5 8 9

10 Transformers Devices to step up (or down) AC currents Practical application of mutual inductance implest implementation: Primary enoid (black): N turns econdary enoid (red): N turns N N I(t) in the primary will induce a varying Φ through itself: N d Φ ε c dt where Φ magnetic flux through single turn Flux is the same in second enoid induced e.m.f. is: ε Comparing: ε N G. ciolla MIT 8.0 Lecture 5 9 N d Φ c dt ε Depending on number of turns we can N increase voltage (N >N ) reduce the voltage (N <N ) Demos on mutual inductance ingle turn around primary coil (H0) Emf: 08 V AC Primary coil: N 0 turns econdary coil: N turn Effect: V goes down, but current goes up and melts the nail! Explanation: Power VI is conserved between the coils Variable turns around primary coil (H9) ame primary; show how current in secondary goes as we add loops High turn secondary (H) Emf: 08 V AC Primary coil: N 0 turns econdary coil: N 0,000 turn Effect: mall currents, but very large V will cause big sparks! G. ciolla MIT 8.0 Lecture 5 0 0

11 ummary and outlook Today: elf inductance Energy stored in inductor Mutual inductance And its applications: transformers Next time: Inductors in circuits Quiz II-preparation supplies available here! G. ciolla MIT 8.0 Lecture 5