PC 1342 Electricity and Magnetism - Notes Semester 2

Transcription

1 PC 34 lecticity and Magnetism - Ntes Semeste ) lectic Chage and lectic Field Lectue: D. yan Andesn (Chapte, Yung and Feedman). Matte Simple. Cnducts, Semi-Cnducts and nsulats n slid state, the ute electn shells (the Valence electns) ae in bands f enegy levels. f a band is nt full, electns can jump acss enegy levels and mve aund. This gives cnductin. ands in mateials: Cnducts Semi-cnducts nsulats C C C V Ge 0.6eV Si.eV Seveal ev V Highe tempeatue gives bette cnductin V: Valence and C: Cnductin and V n nsulats, the Valence band is full. Thee is nwhee fee f an electn t mve t. As it takes a lt f enegy t get electns int the cnductin band (seveal ev) this desn t happen vey ften, and mst insulats simply melt when given the enegy needed f the tansitin. K T, m. temp ev 40.3 nduced Chage Take an islated atm: Nucleus lectn clud Placing a chage nea the islated atm causes the nucleus t be epelled, while the electn clud is attacted.

2 PC 34 lecticity and Magnetism - Ntes Semeste q The effect f chage sepaatin is called plaizatin. Placing a chage nea a plate (eithe an insulat metal): q n an insulat, the net chage mvement esults in suface chages, while in a metal the electn clud is affected. ae ins ae left n the back suface, and electns n the fnt suface. N: chage densities << pe suface lattice in. Mvement f chage is vey small i.e. 0 << 0 m. ample - nduced chage tansfe A chage is induced n a metal slab. What happens if this chaged slab is then cnnected t anthe piece f metal? q q Wie t wuld be pssible t beak the cnnectin, leaving bth pieces f metal ppsitely chaged.

3 PC 34 lecticity and Magnetism - Ntes Semeste.4 Culmb s Law The magnitude f the fce between tw pint chages, q and q, sepaated by is given by: qq F 4" 0 ε 0 is the pemittivity f fee space, and is a cnstant. The 4π appeas hee s it des nt in the equatins 9 " Nm C 4$# 0 # 0 " Fm Units ae Newtns, Faads, Culmbs and Metes. f q and q have the same sign, the fce between them is epulsive. f they have ppsite signs the fce is attactive. N: Culmb s law des nt apply t distibutive chages, just pint chages. ntduce the lectic field. Fce n q due t the pesence f q : & q # F $. q $ 4 0 % '( " q q whee 4" 0 Units f ae N.C - Vm -. pess as a vect (dp subscipts): q ˆ 4 " 0 F q Fce n test chage q. pints adially away fm q if q is psitive..5 lectic Field Calculatins Pinciple f Supepsitin F seveal pint chages, the net fce n a test chage is the vect sum f the fces epeienced when the chages ae taken ne at a time. F q and F n test chage Q F q and F n test chage Q tc 3

4 PC 34 lecticity and Magnetism - Ntes Semeste Then F F F Q( ) ample - Field f an electic diple: d d q -q a Pint chages y cmpnents f and will cancel. -diectin. y sin $ % y % sin $ cs $ ( ) iˆ a cs $ d 4"# a d 3 cs $ 0 y d q a d iˆ iˆ 4"# 0 q cs $ iˆ d ample -3 A ing-shaped cnduct f adius a has ttal chage Q unifmly distibuted aund the ing. What is the electic field n an ais a distance away fm the cente f the ing? N: thee is a much simple calculatin like this in Wkshp. Remembe t lk f symmety The appach t distibuted-chage pblem like this is t:. eak up the ing int tiny segments small enugh t be egaded as pint chages. Calculate the vect field fm a segment 3. f pssible fm any symmety, nly calculate the cmpnent which is ging t suvive in the final esult. 4. Sum integate ve all pssible cmpnents n this pblem, the field fm the segment is ging t be at an angle t the ais but the net field will be alng the ais. The cmpnents pependicula t the ais will integate ut. See the diagam. 4

5 PC 34 lecticity and Magnetism - Ntes Semeste y Radius a Small segment ds d d y α d Chage in element ds is dq. gne d because the segment n the ppsite side f the ing cancels it ut. y d d cs d a dq but d 4" 0 a dq Theefe d 4" 0 a i ˆ iˆ # 4" Q 0 d Q ˆ ( ) i 3 a At lage distances, this appeas t be a pint chage. n the middle f the ing, 0..6 lectic Field Lines An electic field line is the cuve dawn at a tangent t the diectin f. The line spacing tells yu abut the field stength. Since is unique at any pint, the field lines neve css. Cautin: a eleased test chage des nt mve alng an line due t the inetia f the paticle. ample: electic diple fm ample.9. q -q d 5

6 PC 34 lecticity and Magnetism - Ntes Semeste.7 lectic Diples A diple is tw equal and ppsite chages sepaated by a distance d. They ae imptant in physical situatins. amples: nduced plaizatin due t. Radiatin fm a diple antenna. nic substances ae natual diples e.g. NaCl. Aside: wate is a gd slvent f inic cmpunds (salts) because the H O mlecule has a diple mment. ample -4 Fce and Tque n a diple in an electic field. n a unifm field (the field stength and diectin ae the same eveywhee) thee is n tanslatinal fce n the diple but thee is a tque. q F q d P d sin F q -q Tque qd sin " Diple mment Theefe P qd (definitin) p sin " 30 e.g. H O mlecule has p 6.30 Cm. 30 P 6.30 Take q e d " 0 m. q 9.60 p sin " p Whee p is dawn fm the negative t the psitive chage. As dawn, is int the page. f the diple tates in the field, wk is dne. dw d". ecause the tque is in the diectin f deceasing φ we must wite it as " #p sin and dw " p sind. F change fm φ t φ, the wk dne is: W # (" p sin ) d ( cs " ) p cs Nw the wk dne s the negative f the change in P. S P f a diple in a unifm field is: p "p cs p " P ( ) 6

7 PC 34 lecticity and Magnetism - Ntes Semeste ample -5 Field f a diple n an ais at distance >>d We have seen fm ample - that, alng the bisect i ˆ. p 4" 3 0 On an ais with >>d: The chage due t psitive field q 4"# ( d) 4" q ' d $ % " 4 () & # n ut ( ) ( n ) n " n... n if n<<. q & d # Theefe ' $ ' 4() % " q d p # ( p qd) 4" 4 3 ".8 lectic Fields inside Cnducts Static is ze inside cnducts. Take a seies f metal plates: q All chages ae suface chages. Chage layes ae vey thin. 7

8 PC 34 lecticity and Magnetism - Ntes Semeste. Gauss s Law (Chapte 3, Yung and Feedman) Gauss s Law is useful f calculating f distibutive chages with sme st f symmety. The basic idea is t enclse the chage distibutin by a clsed suface. The ttal flu f ut f the suface depends nly n the ttal chage enclsed by the suface. Flu means the flw f smething. Analgy: cnside a velcity vect V athe than. The ttal flu f V velcity aea vlume pe secnd.. Chage and f the chage is nt inside the b, then nt all the flu lines will pass thugh the sufaces s the full chage will nt be shwn. Als, the flu enteing the b will als leave the b n the ppsite side, s a lt f the chage will cancel. Will equal 0 chage.. Calculating flu Assume that thee is a unifm electic field. φ th sufaces have the same aea. The flu f thugh the suface nmal t is A. F the the suface: " A cs i f yu epesent the suface by a vect A Anˆ whee nˆ is a unit vect nmal t the suface. n the geneal case, " A d " da. n geneal, the ttal flu ut f a clsed suface is ". da. The integal needs t be applied ve the whle f the suface..3 Gauss s Law Stat with a pint chage. q ˆ 4 " 0 Chse a suface f adius R abut the chage. n 8

9 PC 34 lecticity and Magnetism - Ntes Semeste ( 4" R ) ( q % q ). da & # * 4" ' R $ 0 whee( 4 R ) & q # is the suface aea f the sphee, and $ is the at the suface f the % 4'( R " sphee. is actually independent f the psitin f the chage in the sphee and the size f the sphee. Gauss s Law states that the ttal flu f ut f the whle f a suface is equal t the net chage enclsed by the suface divided by ε 0. " q $ #. da. A d q d A A A d d d A d A amples include a Paallel plate capacit. Qenclsed #. da whle. suface " ample - Take a spheical cnduct with chage Q and adius R. q R 9

10 PC 34 lecticity and Magnetism - Ntes Semeste Take a small aea f the suface f the ute sphee, da. # # ( ) ( ). da da 4" f R<, Qenclsed Q Q ( ). " 4 f <R, Q enclsed 0 ( ) 0. 4" QR " # $ % & ' R ample - Take a lng, cylindical cnduct f adius R and chage λ pe unit length (Cm - ), which has an infinite length. Take pat f the cylinde: R l Field lines must be nmal t the cylinde, thee wuld be an electic fce alng the cylinde causing electns t mve. This wuld then set up an electic fce in the ppsite diectin; these fces wuld then balance themselves ut, ensuing the field lines have n hizntal cmpnent. Thee is n ut f the ends f the suface, nly the sides. At adius the aea f the suface f the cylinde (nt cunting the ends) is l. ( > R)" l Q # $ l # enclsed 0

11 PC 34 lecticity and Magnetism - Ntes Semeste ( > R) "# ( ) R > 0 "# R R ample -3 Cnside an infinite plane sheet f chage with a chage density f σ cm -. A Thee is nly an flu ut f the ends f the cylinde. A " A # " # ample -4 Field at a suface f infinite plane cnduct: A Plane

12 PC 34 lecticity and Magnetism - Ntes Semeste The nly place whee flu cmes ut is at the tp end. " A A # σ is the suface chage density.. " ample -5 Unifmly chaged sphee with adius R and chage Q. Chage is distibuted thughut the sphee with cnstant density. R F >R Q ( > R) ( ) 4 " Q 4 " F R> Q ( < R) ( < R) 4 " Q # $ " % R & ' ( Q 3 4 " R 3 R

13 PC 34 lecticity and Magnetism - Ntes Semeste Cavities in a cnduct. Put a chage n the cnduct. Suface chage laye As thee can be n electic fields within the cnduct, thee is n electic field in the cavity. What happens if yu put a chage within the cavity? Field lines cme ut fm evey pat f the suface. 3. lectic Ptential (Chapte 4, Yung and Feedman) 3. lectic Ptential and negy Wk dne fce distance. dw F dl Fdl cs " f dw > 0 the fce des wk n the utside wld and the system ceating the fce lses enegy. The ptential enegy change du dw F " dl. n electstatics F q q theefe du q " dl Take a unifm field. y a q Y b 3

14 PC 34 lecticity and Magnetism - Ntes Semeste Wa b Fd qd. F gavity: F electstatics: F mg F q y y U mgy U qy When the test chage mves fm a b then: Wa b "# U " ( Ua " Ub ) q ( ya " yb ) Just like gavity ecept yu can t have a negative mass. 3. lectic ptential f tw pint chages: qq Thee is a fce between the chages F. q 4 " is a test chage which will mve fm psitin a t psitin b. a -q b b b % F d % a qq b d qq 4#$ a 4#$ Wa b " W a b depends nly n the end pints nt n the path between. Justificatin: dw qdl qdl cs qd The static -field is cnsevative. S the ptential enegy f u test chage q is infinity) F seveal chages q, q, q 3,... Test chage at fm q, fm q q q q " U 4 #... $ %& ' ( q qi " U # $ 4%& ) i ' i ( Using this, we can find U f test chage f any distibutin f chage. Ttal enegy f the system f chages: qi q j U 4 "# $ i j a ij q b a q U q (with espect t the enegy at 4 " 3.3 lectic Ptential Ptential V is the ptential enegy pe unit (test) chage. S U V q U qv Unit f V is the vlt. VJC -. 4

15 PC 34 lecticity and Magnetism - Ntes Semeste FU and U as defined peviusly V V a b a U q a U q b b U qq " q Ptential due t a single pint chage is V # $ q ' 4%& ( q 4%& This is the ptential at a distance fm chage q. V and U ae scala ppeties. F an assembly f chages qi V 4 " # i i F a chage distibutin: dq V 4 " # We can calculate V fm whee is knwn. b b a b # " " a # a W F dl q dl Thus b V V " dl cs # dl a b $ $ a b a Can see that V field distance, theefe the unit f can be epessed as Vm -. 3 TV signal " 0 Vm. 3.4 amples ample 3- Thee is a chage q in a cnducting sphee f adius R. We have shwed that, f field is the same as f a pint chage. Take V 0 at. > R, -R q V (f > R ) 4 " V R 5

16 PC 34 lecticity and Magnetism - Ntes Semeste V is cnstant inside the cnduct since static 0 inside a cnduct. q 6 suface < 0 Vm in ai. 4"# R V R 3MV f R m, 30KV f R cm, 300V f R 0.mm. suface suface ample 3- Take an infinite line f chage density " cm. Fund that adial "# b b d V ln b a $ Vb dl d " #, % a, & a ' (), () * a We culd define V t be ze at say. # " V ln $ %. &' ( ) ample 3-3 Take a lng cnducting cylinde f adius, with chage density " cm. " R # ( > R) ln &' $ % ( ). V ( R) 0 V. dq k.g. emembev 4 " #. A disk f adius R and chage density " " e Cm whee k is a cnstant. Find the field at distance h abve the disc n the ais pint. h R The inne cicle has a adius a, with the ute adius a da. The aea f the annulus is " ada if da << a. Theefe the chage n the annulus k is " a# e da. The distance t an ais pint is just Theefe V ( h) V 0 at h 0. h a. ka ka h " a# h e da # ae da 4"$ h a $ h a % %

17 PC 34 lecticity and Magnetism - Ntes Semeste 3.5 quiptential Sufaces These ae thee-dimensinal sufaces n which V is the same eveywhee. V V - Plates f a capacit. ) is always pependicula t the P suface. ) quiptential sufaces cannt tuch css (c.f. field lines) 3) The whle f a cnduct (static fields) is an equiptential suface. 4) Since static is always pependicula t the suface f a cnduct, equiptential sufaces ae lcally paallel t the sufaces f cnducts. 7

18 PC 34 lecticity and Magnetism - Ntes Semeste 3.6 Ptential Gadient a b b a b V V " dl dv dv a b a b # dv " dl a dv " dl,,,,, d dy dz b a y z $ V $ V $ V, y, z, $ $ y $ z % ˆ $ V ˆ $ V ˆ $ V & # ( i j k ) ' V * $ $ y $ z 3.7 Cathde Ray Tube L v gap d θ y Phsph V Sceen Wking ut the speed f the electns mv 7 v.650 ms. V ev. v e etween the plates the acceleatin upwads is a.. m d L The electns spend a time t within the plates. Theefe vy v Vy y eit the plates. tan V " D. Theefe ev. fv KV, m V DeV L LD " V " y # $. mdv % & % & # d $ V ev L at when they md v 4. Capacits and Capacitance (Chapte 5, Yung and Feedman) 4. Capacits A capacit is a device f sting chage. Diffeent capacits can have diffeent capacitance; the capacit s ability t ste chage. Diffeent aangements f pais f electdes metal sufaces give diffeent capacits. 8

19 PC 34 lecticity and Magnetism - Ntes Semeste We define capacitance as C (Chage sted n electdes) / (Ptential diffeence between electdes). The unit f capacitance is the Faad (F). F CV. ample: a paallel plate capacit. unifm d Assume d is small (cmpaable t A ) Assume is unifm, and that we can neglect edge effects. Assume thee is a chage Q n the plates. Q Chage density Cm ". A Gauss s Law:. " As is unifm, VA Q A Theefe C. V d ab 4 Aea A Qd d battey vltage. A 4 Take A 0 m, d 0 m. C F 8.85pF. ample 4- slated metal sphee The secnd electde in this case is the est f the univese. Q We peviusly fundv. 4 " R Q Theefe C 4" R. V Take R 0cm C 4" # 0pF. ample 4- Cncentic metal sphees 9

20 PC 34 lecticity and Magnetism - Ntes Semeste < < 4 " Q ( a b ) This is fm cnsideing a thid sphee midway between the tw abve, and applying Gauss s law. lectic field is 0 utside the capacit. Q Va 4#$ a " %" Altenatively Q Vb 4#$ " b & " Vab Q $ # % 4&' ( a b ) # " dl Q a b " C 4#$ % & V ab ( b ' a ) Take 9.5 cm, 0.5cm C 0pF. a b 4. Cmbinatins f Capacits in Seies and Paallel Seies: Q -Q C V Q C -Q V The Q s ae the same since same chaging cuent and q V Q C Q V C Theefe " Vab V V Q # $ % C C & V Q C C C eq idt. C eq is the equivalent capacitance f tw capacits in seies. This is cmpaable with esists in paallel. 0

21 PC 34 lecticity and Magnetism - Ntes Semeste Paallel: V V Q Q C V C V Qttal Ceq C C V This is cmpaable t esists in seies. 4.3 negy Stage in Capacits and lectic Fields Q CapacitV. f we mve chage dq fm the negative plate t the psitive plate then the C wk dne dw Vdq. dldq ( ) Theefe the ttal wk dne in chaging the capacit is Q W dw Qdq CV C f we define the ptential enegy f a capacit t be 0 when Q 0 then; Q U CV QV. C.g. camea flash C " µ F V " 300V U CV " 4.50 J f the flash lasts in the de f The enegy sted in fields (altenative view) F paallel plates. " V d Vlume f space ccupied by is Ad negy density in a capacit is U CV u Ad Ad A d d Ad ( Jm 3 ) 3 0 secnds, pwe in flash 45W. 5. Magnetic Fields and Fces (Chapte 8, Yung and Feedman) 5. Magnetism Magnetism has been knwn f a lng time. The ldest use is the magnetic cmpass used in navigatin. The ath is a big magnet:

22 PC 34 lecticity and Magnetism - Ntes Semeste S N N S The gemagnetic Nth Ple is the S side. The cmpass needle will pint NS while the magnet in the ath is SN as like ples epel, while unlike ples attact. All magnetism is assciated with cuents i.e. with mving chages. ven pemanent magnets can be eplained by this they have pemanent atmic cuents. Yu can induce magnetism int sft magnetic mateials, e.g. sft in. These mateials becme magnetized in the pesence f an etenal field. f yu emve the etenal field, this magnetism disappeas. Applicatins f magnets: Pemanent magnets: Cmpass needle Fidge d D seal lecticity metes Small mts Tl hlde nduced magnetism: Mts Tansfmes nducts e.g. the ignitin cil in a ca, tuned cicuits LC, filtes t pass eject cetain fequencies (50Hz in pwe supplies) 5. Magnetic Field A mving chage cuent ceates a magnetic field. A magnetic field eets a fce n mving chages cuents. Repesent a magnetic field by. Fce n a chage q mving at a velcity v in a field : v q F q v F qv

23 PC 34 lecticity and Magnetism - Ntes Semeste F q v v F qv sin F q ( v). ( q is the electic fce). The unit f is the Tesla T. T NsC m NA m (Newtn secnd) / (Culmb Mete) (Newtn) / (Ampee Mete) dq Cuent Ampees dt A Cs Centimete Gam Secnd (cgs) unit f is Gauss (G) 4 G 0 T. Typical sizes f magnetic fields: The ath s magnetic field G. A ba magnet 5000G 0.5T Atmic fields 0T. lectmagnets few T. Steady fields in the lab 50T. Pulsed field in the lab 0T 8 Neutn sta (suface) ~ 0 T. 5.3 Magnetic Field Lines a magnet: S N Slenid (Cylindical cil f wie): 3

24 PC 34 lecticity and Magnetism - Ntes Semeste lectmagnet: Lng staight wie: Wie lp caying a cuent magnetic diple: 5.4 Magnetic Flu, Gauss s Law f Define the magnetic flu: d" da cs # da $ da 4

25 PC 34 lecticity and Magnetism - Ntes Semeste da Ttal flu thugh a suface is: % % " # da cs $ da % # da S unit f flu is the Webe (Wb ). Wb Tm NmA Aside: Old bks smetimes define in tems f called the magnetic flu density. f the suface cmpletely enclses the vlume then: # " da 0. This is Gauss s law f. This eflects the fact that neve stats ends. 5.5 Mtin f Chaged Paticle in F q v i.e. d. Units fwbm. is da ( ) F q ( v) is static and unifm in all but ne f the fllwing eamples. Speed f the paticle is nt affected. v v F q Mtin f paticle is a cicle with adius. mv F q v R mv R q 5

26 PC 34 lecticity and Magnetism - Ntes Semeste The angula fequency f the mtin q m f q " " m v R f is called the cycltn fequency. F an electn e.g. micwave ven 9 f.450 Hz 0.09T f and v ae nt pependicula: v q v ~ v 0 f.70 N fce as v 0 v is nt affected by. v causes cicula mtin in that plane. Theefe the path is helical. n a nn-unifm field, the paticle can be tapped in the magnetic field. This is called a magnetic bttle. t is pssible f the paticle t bunce back fm the ight-hand-side, and then bunce backwad and fwad in the bttle. The cmpnent f the fce n the ight-hand-side is t the left. The cmpnent f the fce n the left-hand-side is t the ight. This can be bseved in the Van Allen adiatin belts aund the ath. 6

27 PC 34 lecticity and Magnetism - Ntes Semeste 5.6 Applicatins Velcity select Select mn-velcity paticles fm a beam. eam q Unifm field int the pape; unifm field. F n vaiatin, F q qv 0 Theefe v. This wks whethe q is psitive negative. Thmpsn s deteminatin f the e m f the electn used this. His appaatus cnsisted f an electn gun acceleating the electns thugh a ptential diffeence V. He used the velcity acceleat and a flescent sceen t view the beam f electns. We can equate the kinetic enegy f the electn t ev. mv ev. ev Theefe v. m F n deflectin, e " Theefe m V # $ % &. ev m He fund nly a single value f e m independent f the suce f the electns. This shws that thee is nly ne st f electns. This pves that they ae basic cnstituents f matte. e (53) 0 Ckg. m Mass Spectmete (Thid yea lab) 7

28 PC 34 lecticity and Magnetism - Ntes Semeste n suce eam Velcity select Thin, mnvelcity beam Unifm field R R Radius depends n mass f paticle Paticle detect Velcity v. and ae the fields in the velcity select. mv m R q q Discvey f istpes (Same chemical species, diffeent masses) pat f the evidence f the eistence f a neutn. F helium, masses f 0 and cme ut. 5.7 Magnetic Fce n a cnduct caying a cuent Cnduct: fied psitive ins, mbile chage caies (e.g. electns) v j A unifm F q L J is the cuent density. Let the caie dift velcity be v d upwads. F qvd Let n numbe f caies pe unit vlume. Vlume f cylinde AL Theefe the numbe f caies in the cylinde nal. Theefe the ttal fce n all the caies is F nalqv d ( ) nqv A L d ut J nqv d J 8

29 PC 34 lecticity and Magnetism - Ntes Semeste Theefe: JA F L F L F a nn-staight cnduct: Cnside a little segment. df dl. What if the chage is negative? Upwads cuent is a dwnwad dift f negative chage caies, and q is negative. These tw cancel each the ut. Theefe this deivatin is nt sensitive t the chage plaity. 5.8 Fce and Tque n a cuent lp (Unifm ) F F (z-diectin) z b -F a φ -F y Net fce F y 0 N tanslatinal fce n lp. F a ( ) F ' b sin 90 " b cs " b Tque abut y diectin F sin " A sin " whee A is the aea f the lp A ab. A is called the magnetic diple mment, dented by µ A. The sign f µ is psitive if, lking alng µ the cuent ciculates clckwise. Thus µ Tque n a magnetic diple µ in a field. Cmpae with the electic diple, P and P U P " y analgy, the ptential enegy f a magnetic diple in a magnetic field is U ample 5- A cicula lp f adius 0.05m, n 30, 5A,.T. µ µ ". unifm The aea f the lp A 9

30 PC 34 lecticity and Magnetism - Ntes Semeste F ne tun µ A µ ttal nµ n Tque n ne tun A sin " 0.047Nm Tque n n tuns n.4nm. ample 5- Change the angle in the pevius eample. nitial, 90 p U µ ttal cs " 0 Final, 0, p U µ ttal cs ".4J This is negative as wk is dne n the utside wld. Fce n a magnetic diple in nn-unifm F N µ Net tanslatinal fce F Magnetic mateials Many elements have n pemanent magnetic diple mment. n has seveal pemanent magnetic mments aligned. t can eist in many diffeent states. n the unmagnetised state, thee is andm ientatin f the diples. n the magnetized state, thee is alignment f the diples. 30

31 PC 34 lecticity and Magnetism - Ntes Semeste 5.9 Diect Cuent Mt Mst mts >50W, e.g. vacuum cleane, dill, mie, These all wk ff AC because the magnet is an electmagnet pweed ff the same AC. N Rtates Rtating cntacts S Fied cabn backets 5.0 Hall ffect J v d Fz qvd Net tansvese fce: F q qv 0 z d y " v z d y q is psitive, is negative. F z The cuent density z J nqv Jy Theefe nq. z z Negative chage caie z d J y unifm F a metal, n is lage and q e. vd is vey small, and the Hall effect is small. F semicnducts, n is elatively small and v d is lage, and the Hall ffect is elatively lage. z F z Psitive chage caie v V 3

32 PC 34 lecticity and Magnetism - Ntes Semeste 6. Suces f (Chapte 9, Yung and Feedman) Mving chages a cuent-caying cnduct epeience a fce in a magnetic field. f the magnetic field is static, it altes the velcity but nt the speed enegy f the mving chage. Due t the cnsevatin f mmentum, thee must be a eactin back n the system pducing the magnetic field. The magnetic field is ceated by mving chage cuent. Theefe deflectin f a mving chage in a magnetic field is eally mving chages inteacting with each the. Thee is symmety just as f static chage t static chage inteactins. 6. Magnetic field f a mving chage P φ q,,sin ", v ntduce a cnstant f pptinality: " µ q v sin # $ % 4 ' & ( Making this int a vect equatin: µ " vˆ # $ q 4% & ' This is the magnetic field f a chage q mving with velcity v. ˆ is the psitin f the pint whee is detemined with espect t the psitin f the chage. Of cuse, the chage als pduces an electic field. This is neglected hee. Nte the symmety: the field fms cicles abut the diectin f mtin. Cnstant f pptinality: µ 4 T NsC m NA m Units f µ ae Ns C NA WbA m TmA Hm H is the Heny unit f inductin. 7 µ is defined t be 4" 0 Hm Remembe " µ q v " Fm (definitin) 8 30 ms c the velcity f light in a vacuum. 3

33 PC 34 lecticity and Magnetism - Ntes Semeste ample 6- Fce between tw phtns a distance apat, mving in ppsite diectins with velcity v. y F v v z F is in the y diectin n the uppe chage/. F is als upwads. q F epulsive. 4 " Lwe chage: v viˆ and ˆ ˆj µ ˆ ˆ " vi j µ qv At uppe chage: q kˆ # $. 4 4 & % ' % Fce n the uppe chage due t ceated by the mtin f the lwe chage: ( ˆ µ q v F ) ˆ q vi j 4 " This is als epulsive and is the invese squae. F v Nte µ v F c 6. Magnetic field f a cuent element P φ A d Vlume f element is Ad 33

34 PC 34 lecticity and Magnetism - Ntes Semeste Theefe the mving chage is: dq naqd which mves with a dift velcity v d. n is the chage caie density q is the chage f the caie. At pint P: " µ # vd sin d $ % dq 4& ' ( " µ # vd Ad sin $ % n q 4& ' ( ut n q v A the cuent in the element. d µ " d sin # $ d % & 4 ( ' ) µ ˆ " d d % & 4' ( ) This is called the it-savat Law. µ " d ˆ F the ttal field, # $ (. 4% & ' 6.3 Magnetic field due t cuent in a staight cnduct a d y y -a int pape -a d is pependicula t dl and. Theefe is int the pape. dl dy sin ( " #) sin # y µ dl sin " µ dy µ a 4# 4# 4# When a ( ) a a $ a $ a 3 y a ( y ) ( a ) >> (infinite wie appimatin) µ " # ut because f aial symmety: µ whee is the distance f the pint ff the ais. 34

35 PC 34 lecticity and Magnetism - Ntes Semeste ample 6- lng paallel wie, cuents in ppsite diectins. -3d -d d d T. 3 T P P P 3 Wie 3 cuent ut Wie f pape cuent int pape T3 ij i wie, j pint (T ttal) At pint P, µ " d T µ " 4 ( ) ( d) µ 8 " d At pint P, µ d µ d µ T d At pint P 3, µ 3 3d 3 T 3 ( ) µ " d µ " 3 d. 6.4 Fce between paallel cnducts caying cuent (lng wies) 35

36 PC 34 lecticity and Magnetism - Ntes Semeste F is the field at the uppe wie due t the cuent in the lwe wie. µ Fce n length L f the uppe wie is: µ F ' L ' L Theefe the fce pe unit length is: F µ ' L qual and ppsite fces act n the lwe wie. Cuents in same sense attact pinch effect. Cuents in ppsite senses epel. An eample: ' m F 0 Nm L Fce n ais field f a cicula lp dl y a θ y d d Neglect the cnnecting wies. µ dl d 4 a d d cs " d y d sin " # a $ % cs " & % a & % & sin " % & ' a ( 36

37 PC 34 lecticity and Magnetism - Ntes Semeste y symmety, f the whle lp thee is n net. Theefe µ µ µ a µ µ adl a dl " ( a 4 " ) ( a ) ( a ) ( a ) y whee µ a the magnetic diple mment f the lp. N: dl 3 a On ais field: " µ µ µ At the cente, 0. 3 a a µ µ µ F >> a, a i.e. (n ais) Cmpae with the electic diple. 6.6 Ampee s Law Using the it-savat law, we fund f an infinite staight wie caying cuent : µ at adius. # Take dl " µ Lp f adius abut wie µ µ " # $ % dl ' & ( 0 ) * Fist path enclsed the cuent µ Secnd path did nt enclse the cuent 0. The actual path des nt matte. 37

38 PC 34 lecticity and Magnetism - Ntes Semeste " dl µ enclsed Ampee s Law lp is the cuent enclsed by the lp. ample 6-3 Cnside the field inside and utside a vey lng, cylindical cnduct caying a chage. R # Aund caial cicles dl " F lage cicles ( R) Theefe ( R) utside enclsed µ " F the small path ( < R) enclsed assuming unifm cuent density R µ enclsed µ Theefe inside R µ R R ample

39 PC 34 lecticity and Magnetism - Ntes Semeste Field f a Slenid (lng): A Slenid is a helical cil f cylindical fm, with n tuns pe unit length. Regad as having n adjacent cicula cuent lps pe unit length. Cnside cental pat f the vey lng slenid. c L d Appealing t symmety; is in the diectin. inside utside might be in the diectin. f lp abcd is utside the slenid, thee is n cuent enclsed theefe utside 0. F the lp staddling the slenid; ". b dl L µ nl inside inside µ n and it is the same eveywhee inside. F a slenid f finite length: a..... Culd use it-savat law. Slutin n ais. Numeical slutin ff ais. ample

40 PC 34 lecticity and Magnetism - Ntes Semeste Field f a Tidal (dughnut) slenid. b a Ttal f N tuns. # " > b dl " 0 < a dl 0 This is because thee is n cuent enclsed. # a < < b dl " µ n µ Theefe n eveywhee inside the tid. We can use it-savat geneally, but yu have t integate which can be cmplicated. Yu can deive Ampee s law fm it-savat. Ampee s law can be used in pblems whee thee is sme st f symmety this eases the calculatins. Have t knw that is unifm und the lp ( pependicula t dl ). Ampee s Law is imptant because Mawell fund an e eta displacement cuent which needs t be added.. Cected Ampee s law analgus t Faaday s law M waves. ample 6-6 nfinite cuent sheet k Am Sheet d a h e c b g f Lp : 40

41 PC 34 lecticity and Magnetism - Ntes Semeste # ( )( ) dl l 0 " " L 0 L µ kl µ k Lp : $ ( ) ( ) dl L 0 " L 0 " L µ 0 0 # lectic Magnetic dqˆ Culmb s Law d 4 " it-savat Gauss s Law nfinite line chage with Q " " # " $ % 4&' $ % ( ) ( ) nfinite chaged sheet % " & #$ ( 4#' ) ( ) µ " dlˆ d # $ 4 & % ' " enclsed da Ampee s Law dl µ enclsed $ # " Cm " Cm F fields diveging in dimensins Fields diveging in dimensins Fields nt diveging at all n dependence n. 9.0 Displacement Cuent nfinite cuent Cs µ " 4% # $ & ' nfinite cuent sheet µ " $ % # & 4# ' ( K) f the cuent is spead ve an aea as a cuent density J ( Am ): " enclsed " dl µ µ J da The last integal is any suface which has the lp as a peimete. K Am A Mawell pinted ut a pblem with Ampee s law. 4

42 PC 34 lecticity and Magnetism - Ntes Semeste Vey distant capacit Suface f integal N cuent thugh the suface but thee is a time vaying electic field..g. paallel plate capacit. " A # q CV % & d A $ ' d ( dq d Chaging cuent c " dt dt d Mawell defined this t be a displacement cuent D " and mdified Ampee s Law dt " t be dl µ ( ) c D The fist cuent is the cnductin cuent (the cuent in the wie which can be measued), while the secnd is the displacement cuent (which is fictitius). D d Nte als that the displacement cuent density JD A dt With a dielectic in the capacit, pat f the displacement cuent is due t the chage mvement as atms plaise. ut D is in a vacuum. Faaday s law d dt Mawell d dt This symmety leads t M waves. ample 6-7 Resistivity 8 " 0 # Aea A 4mm a) What is? L L R, V R A A V " Theefe 0.5Vm L A 4

43 PC 34 lecticity and Magnetism - Ntes Semeste d b) f 6000As what is d dt dt? d " d 30Vm s dt A dt c) What is D? d" d d 5 D # # A # $ ~ 0 A dt dt dt 7. Magnetic nductin (Chapte 30, Yung and Feedman) We knw that if we mve a magnet nea a cil, we get an induced MF. f yu change a cuent n ne cil, yu induce an MF in a neaby cil (mutual inductance) and an MF in the fist cil (self inductance) 7. Faaday s Law dl φ Magnetic flu: d" # da da da cs $ % d % " " # da Faaday s law states that the MF induced in a clsed lp is equal t the minus f the ate d f change f magnetic flu thugh the lp i.e. " # whee ε is the MF. Altenatively, dt d " # N if the cil has N tems. dt Diectin f induced MF: inceasing A deceasing A ε ε A A ε inceasing ε deceasing 43

44 PC 34 lecticity and Magnetism - Ntes Semeste 7. amples ample 7- Seach cil t measue. A seach cil is a cil f wie with N tuns, aea A in. Resistance f the whle system is R. Galv R Step : align with A (Maimise thugh the lp) A Step : Quickly tate the cil thugh 90 degees. " 0 d MF " NA dt Cuent R Theefe a chage Q flws thugh the galvanmete. NA d NA Q " dt dt R " dt R The special galvanmetes calibated in Q. ample 7- A simple altenat geneates AC cuent. φ A Slipings ushes " A A cs # d e $ da % A sin % & % t V ( t) 44

45 PC 34 lecticity and Magnetism - Ntes Semeste ε t f the slip ings ae eplaced by a cmmulat, the path in dtted lines is fllwed. ample 7-3 AM Radi e.g. C 5 live, 909kHz. c 330m f N tuns Feite d, nsulat, in-like Aea A Vaiable capacit L induct An LC esnant cicuit is fmed with Q 00 d d " N n ( AF) dt dt F is the value by which the feite cncentates in the d. cs ft (at fied psitin) and f ( ) 4 " 3.30 T, F " 0, N 30, A 0 m 6 4 " # Q..0 V.0 V ample 7-4 Faaday s Disc Dynam ε Rtates at ω R θ ushes V 45

46 PC 34 lecticity and Magnetism - Ntes Semeste R Aea A d" # $ dt " A d " dt da dt da R d R % dt dt R % & # # " $ T, R 0. m, " 300 s % f ~ 50Hz e.g. & ( ' ) *.5V 7.3 Lenz s Law The diectin f any magnetic inductin effect is such as t ppse the cause f the effect.g. baking tque n Faaday geneat and eamples n the sense f. 7.4 Mtinal MF All chages in any mateial mving in a magnetic field epeience magnetic fces. Only the fee chage caies in a cnduct ae fee t mve. Psitive F m F e q Negative mving cnducting ba v unifm. Chage builds up n the ends f the cnduct t pduce an intenal electic field. n equilibium, qv q, theefe the mtinal MG between the ends f the ba L whee L is the length f the ba. Theefe ample 7-5 An aiplane. v 300ms 4 vl 0 T (ath s field) L 33m (using span). The mtinal MF between the tips f the wings vl vlt. ven thugh appeas acss the ends f the ba it cannt be used f anything since mving cnnectins t it wuld have the same mtinal MF induced in it, meaning that thee is n easn f chage t flw aund the lp. Nw cnside: 46

47 PC 34 lecticity and Magnetism - Ntes Semeste Fied ails Sliding cntact unifm int pape Velcity v ample 7-6 Cnside a d antenna (length L << wavelength) in the field f an M wave, paallel t the d. The MF induced in the d L. We culd equally well detemine the mtinal MF induced in the d as the cmpnent f the wave as it passes at velcity c. Velcity c Mtinal MF cl We want Theefe C 3 0 ms µ " 8 This is tue in a vacuum. 7.5 nduced lectic Fields Lng slenid. inside µ n n tuns pe unit length. Galv G 47

48 PC 34 lecticity and Magnetism - Ntes Semeste G T fce a cuent thugh the lp thee must be a tangential field. d d " # #µ na dt dt % $ dl " Theefe we can state Faaday s Law as: d $ " dl # dt 7.6 ddy Cuents Any cnduct mving in epeiencing changing has MFs induced in it. n the latte case, cuents can flw. Whee the induced cuents ae unwanted, they ae called eddy cuents, e.g. in electic mts and tansfmes. These have femagnetic ces (in steel) and ae made f metal. ddy cuents cause heating, meaning that pwe is lst. ample 7-7 Aipt secuity ~ AC Geneat V MF geneated by magnetic inductin 7.7 Mawell s quatins These ae a cllectin f the fundamental equatins f &M. Qenclsed Gauss s Law f : # da " " Gauss s Law f : da 0 Ampee s Law Faaday s Law " d # $ dl µ & c % dt ' ( ) * d " dl # dt $ n empty space, thee ae n chages cuents s: " Gauss s Law f da 0 48

49 PC 34 lecticity and Magnetism - Ntes Semeste " Gauss s Law f da 0 Ampee s Law Faaday s Law $ " dl µ # d " dl # dt $ d dt Thee is an bvius symmety between the tw Gauss s Laws and Ampee s and Faaday s laws, ecept f the sign and sme cnstant. 8. nductance (Chapte 3, Yung and Feedman) A cuent in a cnduct pduces a magnetic field, and as a esult thee is a magnetic flu linking any neaby cnducts. Changing will esult in MFs being induced in all cnducts neaby, including the iginal cnduct. 8. Mutual nductin A cuent change in ne cnduct induces an MF in a secnd cnduct. Cil (N tuns) Cil (N tuns) d The MF induced in the secnd cil is " # N whee is the magnetic flu dt thugh ne tun f the secnd cil due t cuent in the fist cil, as the cils ae linked magnetically. N " M whee M is the mutual inductance. N is the ttal magnetic flu thugh the secnd cil. N M ttal flu thugh the secnd cil due t. d d Theefe " # N # M dt dt Similaly, a changing cuent n cil induces an MF in cil. d d " # N # M dt dt This is nt self-evident but M M M. d d N N Theefe " M and " M whee M. dt dt Unit f inductance is the Heny. H WbA VsA " s 49

50 PC 34 lecticity and Magnetism - Ntes Semeste ample 8- Aea A Length L N tuns (lng) N Teat this as an infinitely lng slenid. 0 utside inside N " & L ' # µ $ % Magnetic flu thugh tun f the secnd cil is N " N # Theefe M µ $ % N A. & L ' e.g. L 0.5m A 0 N N m 6 " M 50 H 5µ H N " # µ $ % & L ' A A Applicatins: Tansfme maimizes flu linkage as fa as pssible. V N V Lad V N V N N N N N V V V. N N 8. Self nductin f yu ty t change the cuent thugh a cil, yu will change the magnetic flu giving an induced MF. d d The induced MF " # # L. L is the self-inductance f the cil. dt dt ttal flu N L. cuent 50

51 PC 34 lecticity and Magnetism - Ntes Semeste Devices designed t ehibit inductance ae called inducts, which ae epesented by (a schematic f a cil). Real inducts have winding esistances i.e. the wie fm which they ae wund has finite esistance equivalent t, with finite Q. What is the sense f MFs in L and M? F L it tends t ppse the change in s it helps t keep flwing. F M it tends t pduce in secnday t pduce a magnetic field which ppses the change in. ample 8- Switch V R V - R L V L Clse switch at t 0. 0 f t < 0. V V V R L d R L dt Slutin: t " # " $ % $ # e& ' % $ % & ' whee V L 0 g R R V R V V pnential t 0 t t ample 8-3 Self-nductance f a tidal cil. Mean adius R Css-sectinal aea A N tuns Apply Ampee s Law: 5

52 PC 34 lecticity and Magnetism - Ntes Semeste " dl µ µ enclsed N Assume inside is unifm ( Theefe: # dl " µ N " thugh each tun is N µ N A Theefe L. " e.g.: N 00 4 A 50 m 0.m 6 " L 40µ H 400 H >> A ) µ NA " A. # ample 8-4 Self inductance pe unit length f a lng slenid, with length L, N tuns and aea A. N " F an infinite slenid, utside 0 and inside µ # L $ % &. Flu linking each tun inside A. Theefe f unit length (i.e. N L tuns) the ttal flu linkage is N N " # µ A L $ L %. & ' Theefe the inductance pe unit length is 8.3 Magnetic Field negy P V d V L f an induct. dt Duing time dt du Pdt d L dt dt Ld dlinduc tance N " A dl µ # L $ % & U L (negy sted in an induct) Cmpae this with U CV f a capacit. ample 8-5 The ignitin cil f a ca. 00 spaks / secnd, 6000pm, 4 cylindes. ( Hm ). 5

53 PC 34 lecticity and Magnetism - Ntes Semeste.4 Vlts V 5A L Time cnstant < sec nds R 00.4 L ".0 H 00 L negy pe spak U 0.5J c.f. camea flash U 0.05J. 8.4 Magnetic negy Density Cnside the tidal slenid. Vlume ccupied by A" Theefe the enegy density U µ u L N A A µ ut N " Theefe u µ This is a geneal esult but it is mdified in mateials. 9. Cuse Summay See PC 34 Teaching Web. lectstatics Chages nsulats and Cnducts Chages n mateials Cnductin Supepsitin Pinciple Culmb s Law lectic field lectic diples, mments, tque, enegy lectic flu Spak plug Gauss s Law Ptential negy lectic Ptential Capacitance negy in a chaged capacit negy in the field inside a capacit ( ) 53

54 PC 34 lecticity and Magnetism - Ntes Semeste Magnetism Magnetic field Fce n a chage mving in a magnetic field Magnetic flu Gauss s Law f Fce n a cuent element in a field Magnetic diple mment f cuent in lp f aea A Tque f magnetic diple, enegy. Magnetic field due t a mving chage Magnetic field due t a cuent element (it-savat) Ampee s Law Mawell intduced displacement cuent mdified Ampee s Law lectmagnetic nductin Faaday s Law Lenz s Law Mtinal MF Mawell s (Summey f) equatins Gauss s Law f Gauss s Law f Ampee s Law Faaday s Law Self and Mutual nductance negy in an induct negy in a field 0. Techniques Lk f symmeties, e.g. chse the suface t use while applying Gauss s Law - the integatins ae tivial if yu use the ight suface. eak the pblem dwn t a simple, me manageable pblem e.g. a distibuted chage. Use a segment small enugh t be egaded as a pint chage. dq d Yu can attack pblems fm abve belw. q V ut given V, V. Smetimes yu can als find q. 54