Induced e.m.f. on solenoid itself

Transcription

1 Induced e.m.f. Consider a loop of wire with radius r inside a long enoid Solenoid: 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 c ( t) nr I c c c... 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. Sciolla 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 inside enoid: c loop 4 πni Flux of through each loop: Φ S loop πr c Flux of through N loops: emf... da c i Tot loop RN Φ NΦ I cl S I RN I Induced e.m.f. on enoid: emf... c l G. Sciolla MIT 8.0 Lecture 5 0 5

2 ack e.m.f. Magnitude of induced e.m.f. on enoid: RN I emf... c l 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. Sciolla 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. Sciolla MIT 8.0 Lecture 5 6

3 Self Inductance L Self-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: SI: 4 π R N I emf... c l RN L cl I emf... L [ emf...] esu/ cm sec [ L] [ current ]/[ time ] ( esu / s )/ s cm [ emf...] V [ L ] Henry ( H ) [ current ]/[ time ] A / s G. Sciolla 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 I 0 Energy stored in the inductor: W LI G. Sciolla MIT 8.0 Lecture 5 4 7

4 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 Same 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 / Self inductance of a enoid: L R N /lc created by enoid: N/lc Energy density of : π N πn 4 4 U L LI I ( π R l ) I Volume c l cl Similar to energy density of the electric field: u E G. Sciolla MIT 8.0 Lecture 5 5 u E How do we calculate L in psets? Just some examples Strategy : L is the proportionality constant between induced emf and variation over time of current: Strategy : ( ) emf... L I t Exploit the fact that energy stored in the magnetic field is the energy stored in the inductor: dv LI V Energy stored in G. Sciolla MIT 8.0 Lecture 5 6 8

5 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 where M is the coefficient of mutual inductance 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 How to calculate M??? No need to calculate it! Reciprocity theorem: M M G. Sciolla 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 S Now rewrite this result in terms of vector potential and use Stokes: Φ ida A ida A idl I dl Since A we obtain c C r ( ) S S C Φ I dl idl Φ c r C C Same fluxes if currents are the same: M M G. Sciolla MIT 8.0 Lecture 5 8 9

6 Transformers Devices to step up (or down) AC currents Practical application of mutual inductance Simplest implementation: Primary enoid (black): N turns Secondary 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. Sciolla 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 Single turn around primary coil (H0) Emf: 08 V AC Primary coil: N 0 turns Secondary 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) Same primary; show how current in secondary goes as we add loops High turn secondary (H) Emf: 08 V AC Primary coil: N 0 turns Secondary coil: N 0,000 turn Effect: Small currents, but very large V will cause big sparks! G. Sciolla MIT 8.0 Lecture 5 0 0

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