Electricity and Magnetism - Physics 121 Lecture 10 - Sources of Magnetic Fields (Currents) Y&F Chapter 28, Sec. 1-7

Transcription

1 Electrcty and Magnetsm - Physcs 11 Lecture 10 - Sources of Magnetc Felds (Currents) Y&F Chapter 8, Sec. 1-7 Magnetc felds are due to currents The Bot-Savart Law Calculatng feld at the centers of current loops Feld due to a long straght wre Force between two parallel wres carryng currents Ampere s Law Solenods and torods Feld on the axs of a current loop (dpole) Magnetc dpole moment Summary Copyrght R. Janow Fall 013

2 Prevously: movng charges and currents feel a force n a magnetc feld Magnets come only as dpole pars of N and S poles (no monopoles). Magnetc feld exerts a force on movng charges (.e. on currents). N S N S N S The force s perpendcular to both the feld and the velocty (.e. t uses the cross product). The magnetc force can not change a partcle s speed or KE F B q v B A charged partcle movng n a unform magnetc feld moves n a crcle or a spral. mv qb R c qb c m Because currents are movng charges, a wre carryng current n a magnetc feld feels a force also usng cross product. Ths force s responsble for the motor effect. For a current loop, the Magnetc dpole moment, torque, and potental energy are gven by: F B LB N A nˆ m U m B B Copyrght R. Janow Fall 013

3 Magnetc felds are due to currents Oersted - 180: A magnetc compass s deflected by current Magnetc felds are due to currents (free charges & n wres) In fact, currents are the only way to create magnetc felds. The magntude of the feld created s proportonal to s (current - length) Copyrght R. Janow Fall 013

4 The Bot-Savart Law (180) Same basc role as Coulomb s Law: magnetc feld due to a source Source strength measured by current-length ds Falls off as nverse-square of dstance New constant 0 measures permeablty Drecton of B feld depends on a cross-product (Rght Hand Rule) Dfferental addton to feld at P due to dstant source ds db 0 ds rˆ 4 r Unt vector along r - from source to P P db (out ofpage) ds db x r 10-7 exactly vacuum permeablty 0 = 4x10-7 T.m/A. Fnd total feld B by ntegratng over the whole current regon (need lots of symmetry) For a straght wre the magnetc feld lnes are crcles wrapped around t. Another Rght Hand Rule shows the drecton: db 0 4 ds sn( ) r Copyrght R. Janow Fall 013

5 Drecton of Magnetc Feld 10 1: Whch sketch below shows the correct drecton of the magnetc feld, B, near the pont P? B nto page B nto B nto page B B page P P P P P A B C D E Use RH rule for current segments: thumb along ds - curled fngers show B Copyrght R. Janow Fall 013

6 Example: Magnetc feld at the center of a current arc 0 ds rˆ db 4 r Crcular arc carryng current, constant radus R Fnd B at center, pont C f s ncluded arc angle, not the cross product angle Angle for the cross product s always 90 0 db at center C s up out of the paper ds = Rdf db 0 ds 4 R ntegrate on arc angle f from 0 to f f B 0 df' f R R 0 0 d' 4 R For a crcular loop of current - f = radans: B B B 0 R f nradans (loop) Another Rght Hand Rule (for loops): Curl fngers along current, thumb shows drecton of B at center?? What would formula be for f = 45 o, 180 o, 4 radans?? B R ds R Rght hand rule for wre segments Thumb ponts along the current. Curled fngers show drecton of B Copyrght R. Janow Fall 013

7 Examples: FIND B FOR A POINT LINED UP WITH A SHORT STRAIGHT WIRE ds P 0 dsrˆ 0 db rˆ Fnd B AT CENTER OF A HALF LOOP, RADIUS = r 4r ds sn( ) 0 Fnd B AT CENTER OF TWO HALF LOOPS B 4r 4r 0 0 nto page OPPOSITE CURRENTS B x 4r r 0 0 same as closed loop PARALLEL CURRENTS B 0 4r nto page 0 4r Copyrght R. Janow Fall out of page 0

8 Magnetc Feld from Loops 10 : The three loops below have the same current. The smaller radus s half of the large one. Rank the loops by the magntude of magnetc feld at the center, greatest frst. A. I, II, III. B. III, I, II. C. II, I, III. D. III, II, I. E. II, III, I. I. II. III. B 0 4 R f f nradans Hnt: consder radus, drecton, arc angle Copyrght R. Janow Fall 013

9 Magnetc feld due to current n a thn, straght wre 0 ds r 3 4 r a Current flows to the rght along x axs Wre subtends angles 1 and db Fnd B at pont P, a dstance a from wre. db s out of page at P for ds anywhere along wre Evaluate db along ds usng Bot Savart Law Magntude of.ds X r =.r.dx.cos(). 0 dx cos( ) db kˆ 4 r x negatve as shown, postve, 1 postve, negatve d r a / cos( ) x atan( ) [tan( )] sec ( ) 1/ cos ( ) d dx a d /cos ( ) 0 db cos( ) d 4a Integrate on from 1 to : 0 0 B db cos( )d [sn( 1) sn( )] 1 4a 1 4a General result applcatons follow Copyrght R. Janow Fall 013

10 Magnetc feld due to current n thn, straght wres B 0 [sn( 1) sn( )] 4a Example: Infntely long, thn wre: Set /, / 1 [ drecton was CW n sketch] 0 B a RIGHT HAND RULE FOR A WIRE a s dstance perpendcular to wre through P FIELD LINES ARE CIRCLES THEY DO NOT BEGIN OR END Example: Feld at P due to Sem-Infnte wres: Zero contrbuton Set /, 1 0 B 4a 0 Into slde at pont P Half the magntude for a fully nfnte wre Copyrght R. Janow Fall 013

11 Magnetc Feld lnes near a straght wre carryng current out of slde When two parallel wres are carryng current, the magnetc feld from one causes a force on the other. F a,b b L b B a B a 0a R. The force s attractve when the currents are parallel.. The force s repulsve when the currents are ant-parallel. Copyrght R. Janow Fall 013

12 Magntude of the force between two long parallel wres d x x x x x x x x x x x x x x x x x x x x L 1 x x x x x End Vew 1 Thrd Law says: F 1 = - F 1 Use result for B due to nfntely long wre B 1 01 d Due to 1 at wre Into page va RH rule Evaluate F 1 = force on due to feld of 1 L B F 1 1 F 1, L B 1 L s normal to B Force s toward wre1 0 1 F 1, L F 1 = - F 1 d Attractve force for parallel currents Repulsve force for opposed currents Example: Two parallel wres are 1 cm apart 1 = 1 = 100 A. F/L force per unt length F 0. N for L 1 m x10-7 x 100 x N / m.01 Copyrght R. Janow Fall 013

13 Forces on parallel wres carryng currents 10 3: Whch of the four stuatons below results n the greatest force to the rght on the central conductor? The currents n all the wres have the same magntude. B 0 R F tot LB tot greatest F? A B. C. D. Hnts: Whch parngs wth center wre are attractve and repulsve? or What s the feld mdway between wres wth parallel currents? What s the net feld drectons and relaatve magntudes at Copyrght center wre R. Janow Fall 013

14 Ampere s Law Dervable from Bot-Savart Law Sometmes a way to fnd B, gven the current that creates t But B s nsde an ntegral usable only for hgh symmetry (lke Gauss Law) An Amperan loop s a closed path of any shape Add up (ntegrate) components of B along the loop path. B ds 0 enc enc = net current passng through the loop To fnd B, you have to be able to do the ntegral, then solve for B Pcture for applcatons: Only the tangental component of B along ds contrbutes to the dot product Current outsde the loop ( 3 ) creates feld but doesn t contrbute to the path ntegral Another verson of RH rule: - curl fngers along Amperan loop - thumb shows + drecton for net current Copyrght R. Janow Fall 013

15 Example: Fnd magnetc feld outsde a long, straght, possbly fat, cylndrcal wre carryng current We used the Bot-Savart Law to show that 0 B for a thn wre r Now use Ampere s Law to show t agan more smply and for a fat wre. B ds 0enc Amperan loop outsde R can have any shape Choose a crcular loop (of radus r>r) because feld lnes are crcular about a wre. B and ds are then parallel, and B s constant everywhere on the Amperan path B ds Bxr 0 enc R The ntegraton was smple. enc s the total current. Solve for B to get our earler expresson: B 0 r outsde wre R has no effect on the result. Copyrght R. Janow Fall 013

16 Magnetc feld nsde a long straght wre carryng current, va Ampere s Law B ds 0 enc Assume current densty J = /A s unform across the wre cross-secton and s cylndrcally symmetrc. Feld lnes are agan concentrc crcles B s axally symmetrc agan Agan draw a crcular Amperan loop around the axs, of radus r < R. The enclosed current s less than the total current, because some s outsde the Amperan loop. The amount enclosed s enc r R Apply Ampere s Law: B ds Br enc 0 r B r R R R 0 0 nsde r R wre B ~r R ~1/r Outsde (r>r), the wre looks lke an nfntely thn wre (prevous expresson) Insde: B grows Copyrght lnearly R. up Janow to R Fall 013 r

17 Countng the current enclosed by an Amperan Loop 10 4: Rank the Amperan paths shown by the value of along each path, takng drecton nto account and puttng the most postve ahead of less postve values. All of the wres are carryng the same current.. B ds A. I, II, III, IV, V. B. II, III, IV, I, V. C. III, V, IV, II, I. D. IV, V, III, I, II. E. I, II, III, V, IV. B ds 0 enc I. II. III. IV. V. Copyrght R. Janow Fall 013

18 Another Ampere s Law example Why use COAXIAL CABLE for CATV and other applcatons? Fnd B nsde and outsde the cable Cross secton: Amperan loop Amperan loop 1 sheld wre current nto sketch center wre current out of sketch Insde use Amperan loop 1: B ds 0 Bxr Outsde use Amperan loop : B ds 0 0 enc B 0 r Outer sheld does not affect feld nsde Remnscent of Gauss s Law Zero feld outsde due to opposed currents + radal symmetry Losses and nterference Copyrght R. Janow suppressed Fall 013

19 Solenods strengthen felds by usng many loops L cancellaton d n # cols / unt length N/L Approxmaton: feld s constant nsde and zero outsde (just lke capactor) Long solenod d << L strong unform feld n center FIND FIELD INSIDE IDEAL SOLENOID USING AMPERIAN LOOP abcda only secton that has nonzero contrbuton B ds Outsde B = 0, no contrbuton from path c-d B s perpendcular to ds on paths a-d and b-c Insde B s unform and parallel to ds on path a-b B ds 0 B enc h nsde 0 enc 0 B 0n nsde deal solenod Copyrght R. Janow Fall 013 nh

20 Torod: A long solenod bent nto a crcle LINES OF CONSTANT B ARE CIRCLES outsde flows up AMPERIAN LOOP IS A CIRCLE ALONG B Fnd the magntude of B feld nsde Draw an Amperan loop parallel to the feld, wth radus r (nsde the torod) The torod has a total of N turns The Amperan loop encloses current N. B s constant on the Amperan path. B ds B r 0enc 0N 0N B r nsdetorod N tmes the result for a long thn wre Depends on r Also same result as for long solenod n N r (turns/un t length) Fnd B feld outsde Answer B 0 outsde B Copyrght R. Janow Fall n

21 Fnd B at pont P on z-axs of a dpole (current loop) We use the Bot-Savart Law drectly db db 0 ds rˆ 4 r r R z cancels by symmetry (normalto z - axs) cos R r db db z db db cos( ) 0 R 4 (R z z 3 / ) ds 0 ds cos( ) 4 R z ds R df Integrate around the current loop on f the angle at the center of the loop. The feld s perpendcular to r but by symmetry the part of B normal to z-axs cancels around the loop - only the part parallel to the z-axs survves. B z db z 0 4 (R R z ) 3 / ds 0 4 (R R z ) 3 / 0 d f s nto page B(z) R 0 (R z ) 3 / as before B(z 0) 0 R recall defnton of Dpole moment Copyrght NA R. Janow R Fall 013

22 B feld on the axs of a dpole (current loop), contnued Far, far away: suppose z >> R B(z) 0R 0R 3 / 3 (R z ) z Same 1/z 3 dependence as for electrostatc dpole Dpole moment vector s normal to loop (RH Rule). NA ˆ N number of turns 1 R above A area of loop R For any current loop, along z axs wth z >> R B(z) 0 Dpole-dpole nteracton: z 3 1 For charge dpole E(z) Current loops are the elementary sources of magnetc feld: Creates dpole felds wth source strength Dpole feels torque to another n external B feld B Torque depends on 1 r 3 1 p z Copyrght R. Janow Fall

23 Try ths at home 10-5: The three loops below have the same current. Rank them n terms of the magntude of magnetc feld at the pont shown, greatest frst. A. I, II, III. B. III, I, II. C. II, I, III. D. III, II, I. E. II, III, I. I. II. III. B 0 4 R f f nradans Hnt: consder radus, drecton, arc angle Answer: B Copyrght R. Janow Fall 013

24 Summary: Lecture 10 Chapter 9 Magnetc Felds from Currents Thn wre, asymmetrc pont I B 0 [sn( 1) sn( )] 4a Copyrght R. Janow Fall 013