Potential Formulation Lunch with UCR Engr 12:20 1:00

Transcription

1 Wed. Fri., Mon., Tues. Wed Emf & Induction Inductnce nd Energy of Mxwell s Equtions Potentil Formultion Lunch with UCR Engr 12:20 1:00 HW10

2 Generliztion of Flux Rule S(t) dl v d S(t+) d v dl Using vector identity (1) A C A C d v dl v dl v dl Thus chnge in mgnetic flux through the loop d v dl rte of chnge in mgnetic flux through the loop F v mg dl dl t q t Emf Wrning: our derivtion used tht the chnging, d/, corresponded to moving chrge, vdl. Not pplicble when tht s not the cse. thr be prdoxes (We will lter extend this resoning to discuss sttionry chrges but chnging fields) mg

3 q q Electric Genertor cosq Spin loop clockwise t w (perhps stem turbines mke it spin) d Emf d d cosq sin sin wt Genertes voltge between terminls V Emf sinwt w w wt w R Drives current I through resistive lod IR V sinwt Demo! - crnk genertor Electric Motor Sme process run in reverse Demo! - home-mde motor w

4 Frdy s Lw F mg q dl Emf t d vl v Which drives chrges round the loop, vi mgnetic force From the perspective of someone riding the loop t t From this perspective too we must see chrges move round the loop, there must be force ut mgnetic is defined s chrge force proportionl to chrge s velocity; from this perspective, there is no v, so we cn t cll it mgnetic, hve to cll it electric. F elect q dl Emf t

5 Frdy s Lw F mg q dl Emf t d vl v Which drives chrges round the loop, vi mgnetic force From the perspective of someone riding the loop t t In most generl cse Emf t From this perspective too we must see chrges move round the loop, there must be force t d d Full time derivtive

6 Frdy s Lw F elect dl q t qe dl d q t E dl d t E d d t E t circulting electric field is ccompnied by time vrying mgnetic field From the perspective of someone riding the loop oth re produced by time vrying current nd chrge distributions

7 Oh, Induction, let me count the wys induced emf in the coil 2 on the right I 1 1 Chnge the current in coil 1 v 1 1 Move coil 1 (with current through it) Come up with some more

8 Copper pipe Not mgnet Induction of the flling mgnet Copper pipe mgnet Why does the mgnet fll so slowly?

9 Copper pipe Not mgnet Induction of the flling mgnet Copper pipe mgnet Why does the mgnet fll so slowly? E t or d Emf Mens Emf s direction is by left hnd rule round re contining flux 23_Frdy_Mgnet.py To the right s downwrd flux increses To the left s downwrd flux decreses Thus drives chrges round pipe nd so trnsfers energy. These chrges in motion produce field which exerts force on moving chrges in mgnets.

10 induced emf in the coil 2 on the right Lenz s Lw Copper pipe mgnet nture bhors chnge in flux Chnge I 1 1 the current in coil v 1 1 Move coil 1 (with current through it) v 2 1 Move coil 2 (with current through coil 1) 1 Rotte coil 1 (with current) Induced Emf (or curled Electric field) drives current tht produce mgnetic field which prtly counters the chnge in mgnetic flux. To the left s downwrd flux decreses To the right s downwrd flux increses S N Demo! v 1 1 Thus drives chrges round pipe nd so trnsfers energy. These chrges in motion produce field which exerts force on moving chrges in mgnets.

11 Using Frdy s Lw d E or Emf or E dl t t re Exmple: very long solenoid of rdius with sinusoidlly vrying current such tht o coswt zˆ. A circulr loop of rdius /2 nd esistnce R is inserted. Wht is the current induced round the loop? ẑ I Demo!

12 Using Frdy s Lw d E or Emf or E dl t t re Exercise: very long solenoid, with rdius nd n turns per unit length, crries time vrying current, I(t). Wht s n expression for the electric field ẑ distnce s from xis? Recll tht inside solenoid oinzˆ. I

13 Using Frdy s Lw d E or Emf or E dl t t re Exmple: A slowly vrying lternting current, It I 0 coswt, flows down long, stright, thin wire nd returns long coxil conducting tube of rdius. In wht direction must the electric field point? ẑ I I E E Lenz lw sys in the direction to drive current tht would oppose chnging flux, so down nd up s the current vries up nd down. ẑ Wht s the electric field? E dl t d Clls for n Amperin loop E dl in E dl E d t E dl top E dl where out s z Es z in t s dsz out t s E dl 0 I ˆ s, 2s 0 s. nd It I 0 coswt bottom E dl oi dsz oi 2s ln z t 2 sin

14 Using Frdy s Lw d E or Emf or E dl t t re Exmple: A slowly vrying lternting current, It I 0 coswt, flows down long, stright, thin wire nd returns long coxil conducting tube of rdius. In wht direction must the electric field point? I I ẑ E E Wht s the electric field? s E in E Lenz lw sys in the direction to drive current tht would oppose chnging flux, so down nd up s the current vries up nd down. ẑ Right-hnd-side is independent of how fr E dl d out of loop s t out is, so E is constnt outside. ut it sout z oi ln z should be 0 quite fr wy, t 2 sin so must be 0 everywhere outside. E E s in s in oi ln t 2 s in t oiow sin wt ln 2 sin I cos wt o o 2 ln s in

15 Wed. Fri., Mon., Tues. Wed Emf & Induction Inductnce nd Energy of Mxwell s Equtions Potentil Formultion Lunch with UCR Engr 12:20 1:00 HW10