UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 1 Prof. Steven Errede LECTURE NOTES 1

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1 UIUC Physics 435 EM Fields & Suces I Fall Semeste, 007 Lectue Ntes 1 Pf. Steen Eede Intductin: LECTURE NOTES 1 In this cuse, we will study/inestigate the natue f the ELECTROMAGNETIC INTERACTION (at {ey} lw enegies, i.e. E ~ 0 GeV, {1 GeV = 10 9 electn lts = Jules}). The electmagnetic inteactin is ONE f FOUR knwn FORCES ( INTERACTIONS) f Natue: 1) Electmagnetic Fce binds electns & nuclei tgethe t fm atms - binds atms tgethe t fm mlecules, liquids, slids.... gases ) Stng Fce binds ptns & neutns tgethe t fm nuclei 3) Weak Fce espnsible f adiactiity (e.g. β decay) (weak fce high enegies) 4) Gaity binds matte tgethe t fm stas, planets, sla systems, galaxies, etc. At the MICROSCOPIC (i.e. QUANTUM) LEVEL (elementay paticle physics) the fces f natue ae mediated by the exchange f a fce-caying paticle e.g. between tw chaged paticles: mediating fce caie chage B chage A Mass Range Intinsic Chage Quantum Fce Fce f fce f fce spin f assciated Field They Fce Mediat Type mediat mediat fce mediat w/ fce QED 1) EM single attactie & PHOTON epulsie ± e QCD ) STRONG ctet f attactie &, g, b GLUONs epulsie ~ 1fm 1, g, b QWD 3) WEAK W ±, Z attactie & M w 80.4 GeV/c epulsie M z 91. GeV/c ~ 1fm 1 ± gw QGD 4) GRAVITY single attactie GRAVITON nly MASS, m (unquantized) At high enegies, QED & QWD unify t becme a single fce, knwn as the ELECTROWEAK FORCE = Planck s cnstant diided by π = h/π = x Jule secnds m ptn = 0.93 GeV/c = kg 1 fm = 1 femt-mete = 1 Femi = metes Pfess Steen Eede, Depatment f Physics, Uniesity f Illinis at Una-Champaign, Illinis All ights eseed. 1

2 UIUC Physics 435 EM Fields & Suces I Fall Semeste, 007 Lectue Ntes 1 Pf. Steen Eede Patten f Masses f Fundamental, Spin-½ Matte Paticles - Femins dublets f quaks: u c t hae electic chage + /3 factinal electic chage d s b hae electic chage 1/3 chage!!! Each quak cmes in 3 stng ( cl ) chages: ed, geen, blue dublet f anti-quaks: u c t q = -/3 with 3 anti-cl chages: ed, geen, blue d s b q = +1/3 (i.e. anti-ed, anti-geen, anti-blue) Pfess Steen Eede, Depatment f Physics, Uniesity f Illinis at Una-Champaign, Illinis All ights eseed.

3 UIUC Physics 435 EM Fields & Suces I Fall Semeste, 007 Lectue Ntes 1 Pf. Steen Eede Questins: Why ae thee 3 geneatins f quaks & leptns? Intenal Quantum #? Why nt just ne? Ae thee me? (seemingly nt ) What physics is espnsible f the bseed patten f quak/leptn masses? Why ae thee fu fces f natue? Why nt just ne? Ae thee me fces? Nte that ALL 4 fundamental fces f natue hae bth electic & magnetic fields!!! Magnetic field aises fm mtin f electic chage in space elatiity & space-time inled hee! FORCE ELECTRIC FIELD MAGNETIC FIELD EM EM electic EM magnetic STRONG chm electic chm magnetic WEAK weak electic weak magnetic GRAVITY gait electic gait magnetic Ndedt Effect e.g. affects mtin f mn s bit aund eath (ey small) n mtin/mement Electic Field time-aeaged field (macscpic) pesent f static chages exchanging itual quanta assciated w/gien fce Magnetic Field time aeaged field (macscpic) aises/assciated w/ming chages mtinal effect Magnetic field aises fm mtin f chage Any/all/each f any/all/each f 4 fundamental 4 fundamental fces f natue fces f natue Magnetic field esults fm chage + space-time stuctue f u uniese!! At micscpic leel, EM fce mediated by (itual) phtns tw electically chaged paticles knw ut each the by exchanging itual phtns. Vitual phtn chage e chage e Pfess Steen Eede, Depatment f Physics, Uniesity f Illinis at Una-Champaign, Illinis All ights eseed. 3

4 UIUC Physics 435 EM Fields & Suces I Fall Semeste, 007 Lectue Ntes 1 Pf. Steen Eede Vitual phtns cay linea mmentum, but hae ze ttal enegy: p = h λ Planck s cnstant DeBglie waelength If If 4 E = p c m c 0 + = c = speed f light = m /sec Real Phtns (e.g. isible light): E = p 0 R c m R = R 4 E = 0, then p c m c = p = h, E = hf > 0 i.e. p =± im c i= -1 cmplex! λ R R R R E = 0 then: E 0 0 = hf f = = itual phtns hae ze fequency, but hae nn-ze DeBglie waelength, λ V > 0! (Newtn s nd Law) ( dp Δp d m ) FORCE: F = ma d F = = = = m = m a t t dt Δt dt dt electic chages emit & sb itual phtns (lts f them!!!) each such phtn caies with it mmentum, P depending n sign f mmentum (emitted/sbed), a net fce will esult, acting n each chaged paticle + + e1 e Like chages epulsie: F F + + e1 e Oppsite chages attact: + e1 e F F + e1 e n.b. Yu wn bdy can sense itual phtns!!! Get yu cmb ut, cmb yu hai seeal times - chages up cmb ia static electicity Bing cmb nea t e.g. hai n yu feam & feel the pull n feam hais fm electic chage n cmb (wks best in winte/dy cnditins). 4 Pfess Steen Eede, Depatment f Physics, Uniesity f Illinis at Una-Champaign, Illinis All ights eseed.

5 UIUC Physics 435 EM Fields & Suces I Fall Semeste, 007 Lectue Ntes 1 Pf. Steen Eede COULOMB S LAW ELECTROSTATIC FIELDS IN A VACUUM It has been expeimentally bseed (Chales Augustin Culmb, 1785) that the net, time-aeaged fce (i.e. summed e many, many itual phtns) between tw statinay pint chages Q a & Q b : 1) Acts alng the line jining the tw pint chages, Q a & Q b (i.e. adial fce!) ) Is linealy pptinal t the pduct f the tw pint chages, Q a * Q b (n.b. Fce is chage-signed!) Net fce is epulsie if Q a is same sign as Q b. Net fce is attactie if Q a is ppsite sign as Q b. 3) Is inesely pptinal t the squae f the sepaatin distance, b a = between the tw pint chages. The net fce exeted by pint chage Q a ON pint chaged Q b is gien by: QQ a b F = K ˆ (SI UNITS Newtns) cnstant f unit ect pptinality (pints fm Q a at a t Q b at b ) b a = ˆ = = unit ect pinting fm pint A t pint B. F is a adial fce, ne which pints fm (t) pint A t (fm) pint B, depending n sign f the chage pduct (Q a Q b ) Q a Q b < 0 is attactie fce (F < 0) Q a Q b > 0 is epulsie fce (F > 0) Pfess Steen Eede, Depatment f Physics, Uniesity f Illinis at Una-Champaign, Illinis All ights eseed. 5

6 UIUC Physics 435 EM Fields & Suces I Fall Semeste, 007 Lectue Ntes 1 Pf. Steen Eede The NET fce exeted by pint chage Q b ON pint chage Q a : QQ b a F = K ˆ F is adial fce, pint fm (t) pint B t (fm) pint A, depending n sign f chage pduct (Q a Q b ) Q a Q b < 0 is attactie fce (F < 0) Q b Q a > 0 is epulsie fce (F > 0) a b = ˆ ˆ = = F Q Q = K a b ˆ F Q Q = K b a ˆ Nw =, since b a = and = a b = but nte that = = = = = and/ since ( ) ( ) b a a b thus, we see that: F = F This is Newtn s 1 st Law: F eey actin, thee is equal and ppsite eactin. SI units f electic chage Q: Culmbs (C) Fundamental unit f electic chage, Q e = 1.60 x Culmbs Questin: What is the physics that dictates (specifies/detemines) the alue f e?? i.e. Why is e = 1.60 x Culmbs? What is K? 1 K = in SI units πε 4 6 Pfess Steen Eede, Depatment f Physics, Uniesity f Illinis at Una-Champaign, Illinis All ights eseed.

7 UIUC Physics 435 EM Fields & Suces I Fall Semeste, 007 Lectue Ntes 1 Pf. Steen Eede ε = electic pemittiity f fee space = x 10 1 Culmbs Faads = Newtn -m mete (Faad is SI unit f capacitance) Questin: If fee space is tuly empty, hw can it hae any measule physical ppeties assciated with it??? Answe: Fee space is NOT empty!!! It is filled with itual paticle-anti-paticle pais!! (e.g. e + -e, μ + -μ, q q, W + W, etc. pais) existing f sht time(s), as allwed by the Heisenbeg Uncetainty Pinciple can ilate enegy (mmentum) cnseatin nly f time inteal Δt / ΔE (and e a distance f Δx / Δpx ). ε 0 is the macscpic, time-aeaged (e many many such itual pais) electic pemittiity f (quantum) acuum - the physical acuum behaes like a dielectic medium!!! 1 QQ a Thus: F = b ˆ ẑ Q A Q B A B a b F ϑ ŷ = b a = ˆx Fact f 4π = flux fact f slid angle assciated with flux f itual phtns emitted by pint chage!!! Vitual phtns emitted fm Q A ae emitted int 4π pint A: Q A Fce deceases as 1 Just like/analgus t eal Phtns emitted fm e.g. 100 watt light bulb - Intensity deceases as 1 fm light suce. A Pfess Steen Eede, Depatment f Physics, Uniesity f Illinis at Una-Champaign, Illinis All ights eseed. 7

8 UIUC Physics 435 EM Fields & Suces I Fall Semeste, 007 Lectue Ntes 1 Pf. Steen Eede Nte the similaity between Culmb s Law and Newtn s Law f Gaity: 1 QQ a b MaMb FC = ˆ F ˆ G = GN Newtn s cnstant, G N = x 10 m kg s O can define: Define: 1 1 g 1 G E GN ε g 4ε 4πG F C = G E 0 QQ a b ˆ Culmb s Cnstant 1 MM a then FG = g 0 b ˆ N QQ a b Culmb s Law FC = ˆ nthing Nte that if dielectic ppeties f fee space (acuum) wee diffeent than they ae, then Culmb s Law, i.e. the fce between electically chaged paticles wuld be diffeent. Cnside a uniese in which we culd change the EM ppeties f the acuum at will: im( ε 0 ): FC!! stng electmagnetism im( ε ): F 0!! weak electmagnetism C (assuming this desn t als affect alue f fundamental electic chage, e) Nte futhe/we shall see that: c = speed f light = x 10 / sec εμ = m μ = magnetic pemeility f fee space = 4 π x 10 7 Newtns/Ampee 1 Ampee f electic cuent = 1 Culmb/sec (I = dq/dt) Thus: im ε im ( 0) ( ε ) c c 0 If μ is unchanged 8 Pfess Steen Eede, Depatment f Physics, Uniesity f Illinis at Una-Champaign, Illinis All ights eseed.

9 UIUC Physics 435 EM Fields & Suces I Fall Semeste, 007 Lectue Ntes 1 Pf. Steen Eede THE ELECTRIC FIELD E (Vect Quantity!!) (Als knwn as the Electic Field Intensity) We e intduced/discussed the net/time aeaged fce, F e.g. f Q a acting n Q b : 1 QQ a b F = ˆ We nw intduce the cncept f a net/time aeaged electstatic field, E a, due t Q a, at a (sepaatin) distance, b a fm Q a (i.e. at Q b ), which is defined in tems f the ati f the net/time aeaged fce F ( b ) t the stength f the test chage Qb used as a pbe: E F Q ( ) ( ) a b b b ẑ : F ( ) = Q E ( ) F b b a b Q b B Pint A is knwn as suce pint Q a is knwn as suce chage Q a A electstatic fce & electstatic field ealuated a ŷ O b at pint B = field pint ˆx Q b is knwn as test chage a pints fm the lcal igin, O t pint A whee the suce chage Q a is lcated. pints fm the lcal igin, O t pint B whee the test chage Q b b is lcated. b pints fm the lcal igin, O t pint B whee the electic field (net/time aeaged) due t Q a is t be ealuated (i.e. by expeimentally measuing F, and knwing (apii) Q a and Q b ). 1 QQ 1 QQ F Q E ( ) = ( ) = a b a b ˆ = ( ) b b a b 3 b a b a 1 Q 1 Q a a Then: E ( ) = ˆ = ( ) a b 3 b a b a Vey ften, we will be cnsideing situatins in electstatics whee we use ne chage, Q T t TEST f the pesence/existence f suce chage(s) q s. Pfess Steen Eede, Depatment f Physics, Uniesity f Illinis at Una-Champaign, Illinis All ights eseed. 9

10 UIUC Physics 435 EM Fields & Suces I Fall Semeste, 007 Lectue Ntes 1 Pf. Steen Eede We want t knw e.g. the electic field due t q s, a sepaatin distance, fm it: with magnitude: = ect ( ) ẑ Suce Pint, S ) Field Pint, P (@ ) q s Q T FT ( ) = Fce n test chage Q T (at field pint ), a sepaatin distance fm suce chage q s (lcated at suce pint, ): 1 Qq 1 Qq T 3 T s T s Oigin, O ŷ F ( ) = ˆ = ( ) ˆx n.b. pimed quantities (e.g. ) always efe t suce (chage) distibutin. unpimed quantities (e.g. ) efe t field/bseatin pint. E ( ) = Electstatic pint due t suce chage qs a distance = away fm q s : ( ) 1 q 1 s q ( s ) 1 qs ( ) 1 qs E( ) = ˆ = = = 3 ( ) F Q E T ( ) = ( ) T cumbesme ntatin, but ey explicit!!! Obiusly, SI Units f E ( ) ae Newtns / C (als lts m) Units f E = fce pe unit chage (N/C) fm dimensinal analysis 10 Pfess Steen Eede, Depatment f Physics, Uniesity f Illinis at Una-Champaign, Illinis All ights eseed.

11 UIUC Physics 435 EM Fields & Suces I Fall Semeste, 007 Lectue Ntes 1 Pf. Steen Eede A Detail: F A me igus definitin f electic field intensity, E( ) is gien by: ( ) E ( ) im Q T 0 Q T We eally d need this limiting pcess expeimentally/in eal life, the pesence f a finite-singed test chage Q T necessaily petubs the suce chage distibutin that ne is attempting t measue!! This is especially tue f spatially-extended suce chage distibutins. As the test chage is made smalle and smalle, the petubing effect n the iginal/unpetubed suce chage distibutin is made smalle and smalle. In the limit Q T 0, the tue suce chage distibutin is btained. THIS IS VERY IMPORTANT TO KEEP THIS IN MIND!!! IT IS NOT A TRIVIAL POINT!!! Usually, we might think f e.g. Q T = 1 e and e.g. q s = e, thus q s >> Q T, and thus petubing effects ae negligible (in this case). We hae shwn that: E( ) If then F ( ) E( ) F( ) 1 qs = ˆ QT is a adial fce f pint suce chage, q s is als adial Cnentin: diectin f electic field lines f q s = +e and qs 1 qq s T and thus: F( ) = ˆ = Q E( ) = e T q s = e q s = +e inwad utwad Pfess Steen Eede, Depatment f Physics, Uniesity f Illinis at Una-Champaign, Illinis All ights eseed. 11

12 UIUC Physics 435 EM Fields & Suces I Fall Semeste, 007 Lectue Ntes 1 Pf. Steen Eede ELECTRIC FIELD LINES Assciated with Tw Pint Chages Equal but ppsite chages Figue.13 Equal chages Figue.14 1 Pfess Steen Eede, Depatment f Physics, Uniesity f Illinis at Una-Champaign, Illinis All ights eseed.

13 UIUC Physics 435 EM Fields & Suces I Fall Semeste, 007 Lectue Ntes 1 Pf. Steen Eede THE PRINCIPLE f LINEAR SUPERPOSITION -VERY IMPORTANT- F( ) Assuming we ae always in im (i.e. Q Q T 0 Q T << q s ) egime, then suppse we hae N discete pint T suce chages: q1, q, q3, q4 qn What is the (ttal net) fce, F ( ) TT due t all f the N suce chages? N = = Vectially, we knw that F ( ) F ( ) F ( ) F ( ) F ( ) F ( ) TT 1 3 N i i= 1. Me explicitly: N F F F F E F ( ) = ( ) + ( ) + ( ) + ( ) = ( ) TOT 1 3 N i i= 1 N Q T q1 q q3 q N QT qi = ˆ ˆ ˆ ˆ ˆ N = i πε 1 3 N 4 πε i= 1 i ˆ i = i whee: ( ) What is (ttal net) electic field intensity, ETOT ( ) We knw that: F ( ) = Q E ( ) : ( ) ( ) TOT T TOT due t all f the N suce chages? E F Q TOT TOT T N E E E E E E ( ) = ( ) + ( ) + ( ) + ( ) = ( ) TOT 1 3 N i i= 1 N 1 q1 q q3 q N 1 qi = ˆ ˆ ˆ ˆ ˆ N = i πε 1 3 N 4 πε i= 1 i We can extend the use f the pinciple f linea supepsitin t mathematically descibe the net/ttal fce + net/ttal electic field intensity at the field pint, f itay cntinuus chage distibutins: QT F ( ) ˆ and ( ) ˆ TOT = dqs ETOT = dq s =, =, = whee: ( ) ˆ Pfess Steen Eede, Depatment f Physics, Uniesity f Illinis at Una-Champaign, Illinis All ights eseed. 13

14 UIUC Physics 435 EM Fields & Suces I Fall Semeste, 007 Lectue Ntes 1 Pf. Steen Eede Then f lume, suface & line chage suce distibutins: A.) VOLUME CHARGE DISTRIBUTIONS: Vlume Chage Density, ρ ( ): (e.g. inside cylindes, sphees, bxes, etc.) Q 1 ( ) : T dqs = ρ dτ FTOT ( ) = ˆ ρ( ) dτ Culmbs/m ETOT ( ) = ˆ ρ ( ) dτ B.) SURFACE CHARGE DISTRIBUTIONS: Suface Chage Density, σ ( ): (e.q. n sufaces f cylindes, sphees, bxes, etc.) Q 1 ( ) : T dqs = σ da FTOT ( ) = ˆ σ ( ) da S Culmbs/m 1 1 ETOT ( ) = ˆ σ ( ) da S = = = whee: ( ),, ˆ 14 Pfess Steen Eede, Depatment f Physics, Uniesity f Illinis at Una-Champaign, Illinis All ights eseed.

15 UIUC Physics 435 EM Fields & Suces I Fall Semeste, 007 Lectue Ntes 1 Pf. Steen Eede λ C). LINE CHARGE DISTRIBUTIONS: Linea Chage Density, ( ) : (e.q. wie) Q 1 ( ) : T dqs = λ d FTOT ( ) = ˆ λ( ) d C 1 1 E TOT ˆ d Culmbs/m ( ) λ ( ) = 0 C = = = whee: ( ),, ˆ Thus, a cmplete desciptin f all pssible chage distibutins, cnsisting f discete and cntinuus chage distibutins: N Q T q ρ( ) σ i ( ) λ( ) F ( ) ˆ ˆ ˆ ˆ TOT = i + dτ + da + d i= 1 i V S C 1 N q ρ( ) σ i ( ) λ( ) E ( ) ˆ ˆ ˆ ˆ TOT = i + dτ + da + d i= 1 i V S C Please Nte: F all integals (e), when integals e dτ, da, and/ d ae caied ut, FTOT ( ) and thus E hae NO (i.e. suce-psitin) dependence - it has been integated e/integated ut!!! F TOT TOT ( ) ( ) and ETOT ( ) FTOT ( ) QT ae functins f the field pint aile ONLY i.e. they ae nt functins f (suce pint{s})!!! PLEASE wk/gind thugh example.1 Giffiths p. 6-63) n yu wn t bette lean/undestand this! ACTIVE LEARNING BY DOING Pfess Steen Eede, Depatment f Physics, Uniesity f Illinis at Una-Champaign, Illinis All ights eseed. 15

16 UIUC Physics 435 EM Fields & Suces I Fall Semeste, 007 Lectue Ntes 1 Pf. Steen Eede EXAMPLE.1 p. 6 Giffiths - ey explicit detailed deiatin- Find the electic field intensity, E() a distance z e mid-pint f a staight line segment f length L, which caies a unifm line chageλ (Culmbs/mete) (n.b. QTOT = λl) 1 λ ( ) Hee, E( = zzˆ) = ˆd C Ntice the symmety f this pblem cntibutin t net E ( field pint, P fm infinitesimal line chages dλ assciated with infinitesimal line segments, dl lcated at cmpnents f net electic field pint, P cancel each the: z z csθ = = x + z x x sinθ = = x + z + denet ( = zzˆ) = de ( = zzˆ) + de ( = zzˆ) 1 λdl = sin cs 1 λdl + ( + sinθxˆ ) + ( cs θzˆ ) de = zzˆ 1 λdl = csθ zˆ + ( θxˆ ) + ( θzˆ) de ( = zzˆ) ( ) ± x such that ˆx 16 Pfess Steen Eede, Depatment f Physics, Uniesity f Illinis at Una-Champaign, Illinis All ights eseed.

17 UIUC Physics 435 EM Fields & Suces I Fall Semeste, 007 Lectue Ntes 1 Pf. Steen Eede Nw nly need t integate this expessin e x fm 0 x L : L L L L 1 λdl E ( ˆ) ( ˆ) { ( ˆ) ( ˆ) } cs ˆ NET = zz = de = zz = de + = zz + de = zz = θ z 0 1 L 1 z = λ dx zˆ 0 ( x + z ) x + z λz L 1 = dx zˆ 0 3/ x + z ( ) λz x L λ z L λl 1 = ˆ 0 z = zˆ = zˆ z x + z z L + z z z + L ε L If z >> L; then (Tayl Seies Expansin) 1+ ε 1+ 1 f ε = 1 z L QTOT ENET ( = zzˆ) λ zˆ zˆ z = z same E-field as that due t a pint chage, q! If L (i.e. infinite staight wie): use the same Tayl seies expansin, but f L z : ε z i.e. 1+ ε 1+ 1 f ε = 1 L 1 λ λ ENET = zzˆ zˆ = zˆ z πε z Then: ( ) The E-field is actually in the adial ( ˆρ ) diectin f an infinite staight wie in cylindical cdinates: E NET 1 λ λ ˆ ρ = ˆ ρ ρ πε ρ ( ) Pfess Steen Eede, Depatment f Physics, Uniesity f Illinis at Una-Champaign, Illinis All ights eseed. 17