Physics 169 Luis anchordoqui Kitt Peak National Observatory Tuesday, April 24, 18 1 Maxwell’s Equations and Electromagnetic Waves 13.1 The Displacement Current In Chapter 9, we learned that if a current-carrying wire possesses certain symmetry, the magnetic field could be obtained by using Ampere’s law: ! ! B ! d s = µ I . (13.1.1) "" 0 enc The equationChapter states that the line integral of 11 a magnetic field around an arbitrary closed loop is equal to µ0 Ienc , where Ienc is the flux of the charge carries passing through the surface bound by the closed path. In addition, we also learned in Chapter 10 that, as a consequenceChapter of Faraday’s law of induction 11, an electric field is associated with a changing magnetic field, according to ! ! d ! ! E ! d s = # B ! dA . (13.1.2) Displacement"" dt "" Current and S Displacement Current and One mightMaxwell’s then wonder whether or not the converse Equations could be true, namely, a magnetic field is associated with a changing electric field. If so, then the right-hand side of Eq. (13.1.1) will11.1 have Displacementto be modified to reflect Currentsuch “symmetry” between electric and ! ! magneticMaxwell’s fields, E and B . Equations We can use Ampere’ s law to calculate magnetic fields due to currents To see how magnetic fields are associated with a time-varying electric field, consider the process of 11.1chargingIntegral a capacitor. Displacement DuringB~ the d ~s charging around process, any Currentthe electric close field loop strength C is equal to µ0iincl increases with time as more chargeC is accumulated· on the plates. The conduction current that carries the charges alsoI produces a magnetic field. In order to apply Ampere’s law to calculate11.1 thisWewhere field, saw let us i incl Displacementchoose in Chap.7 curve= currentC1 as thatthe Amperian passing we loop, can Current shown an use inarea Figure 13.1.1.bounded by closed curve C Amp`ere’s law to calculate magnetic Wefields saw due in to Chap.7 currents. that we can use Amp`ere’s law to calculate magnetic We know that the integral C B ds fieldsaround due any to currents. close loop C is¸ equal· to Weµ i know, where thati the= integralcurrent passingB d ans 0 incl incl C · aroundarea bounded any close by the loop closedC curveis¸ equal C. to e.g. µ0iincl, where iincl = current passing an e.g. area bounded by the closed curve C. Figure 13.1.1 Surfaces S and S bound by curve C . = Flat surface1 2 bounded by1 loop C e.g. = Flat surface bounded by loop C == Curved Flat surface surface bounded bounded by loopby loop C 13-C3 = Curved surface bounded by loop C If Ampere’ s law is true all time If Amp`ere’s= Curved law is true surface all the bounded time, then by loop the Ciincl determined should be inde- pendentthen of the i incl surface determined chosen. should be independent of surface chosen Tuesday, April 24, 18 2 If Amp`ere’s law is true all the time, then the iincl determined should be inde- pendent of the surface chosen. 11.1. DISPLACEMENT CURRENT 126 Let’s consider a simple case: charging a 11.1. DISPLACEMENT CURRENT 126 capacitorConsider. a simple case: charging a capacitor 11.1. DISPLACEMENT CURRENT 126 From Chap.5,Let’s consider we know a simple there case: is acharging current a t/RC We11.1. know DISPLACEMENT there is a current CURRENT flowing i(t)= E e− 126 capacitor. t/RC R flowing i(t)=Let’s considerER e− a simple, which case: charging leads a whichFrom leads Chap.5, to a magnetic we know therefield isobserved a currentB~ Let’s considercapacitor a simple. case: t/RCcharging⇤ a to a magneticflowing fieldi(t)= observedE e , whichB. With leads capacitor. From Chap.5,R we− know there is a current From Chap.5, we know⇤ there is a~ currentt/RC ⇤ Amp`ere’sWith toAmpere’ law, aflowing magnetic sB lawi(t field)=d ⇤s = observedEµBe−0iincld~s,B.= which.µ With0iinc leads C t/RC R flowing i(t)= E e−· ⇤, which leads· Amp`ere’sto a law, magneticR C B fieldCd⇤s = observedµ0iincl. B⇤ . With BUT WHAT IS¸ iincl? I· ⇤ to aBUT magnetic WHAT fieldi IS observed¸ iincl? B⇤. With BUT WHATAmp`ere’s IS ⇤ inc law, ? C B d⇤s = µ0iincl. Amp`ere’s law, C B d⇤s = µ0iincl.· BUT WHAT· IS¸ iincl? BUT WHAT IS¸ iincl? If we look at iinc = i(t) If we look at , iincl = i(t) If we look at , iincl = i(t) If we look atIf we look, i atincl = i(,t) iincl = i(t) If we lookIf we at look at iinc, =0iincl =0 If we look at , i =0 If we look at If we look, atincli =0, iincl =0 (⇥ There is no chargeincl flow between the (⇥ There is( noThere charge is flow no between charge flow the between the capacitorcapacitor plates.)⇥ plates.) Amp`ere’scapacitor law plates.) is either WRONG or (⇥ There* ∴ThereAmp`ere’s∴ is nois no law charge charge is either flowflow WRONG between between or capacitor the plates INCOMPLETE.INCOMPLETE.∴ Amp`ere’s law is either WRONG or capacitor plates.)INCOMPLETE. )TwoAmpere’Two observations: observations: s law is either WRONG or INCOMPLETE ∴ Amp`ere’s lawTwo observations: is either WRONG or 1. While1. While there is there no current is no between current the between capacitor’s the plates, capacitor’s there is plates, a time- there is a time- INCOMPLETE.Tuesday, Aprilvarying 24, 18 electric field between the plates of the capacitor. 3 varying1. While electric there field is no between current the between plates the of the capacitor’s capacitor plates,. there is a time- 2. We know Amp`ere’svarying electric law is mostly field between correct from the plates measurements of the capacitor of B-field. Two observations:around2. We circuits know. Amp`ere’s law is mostly correct from measurements of B-field 2. We know Amp`ere’s law is mostly correct from measurements of B-field around circuits. 1. While therearound is no circuits current. between the capacitor’s plates, there is a time- ⇓ varying electricCan we field revise between Amp`ere’s the law⇓ plates to fix it? of the capacitor. ⇓ Can we revise Amp`ere’sσ Q law to fix it? Electric field between capacitor’sCan plates: we reviseE = Amp`ere’s= , where lawQ = to charge fix it? on 2. We know Amp`ere’s law is mostly⇥0 ⇥0A correct from measurements of B-field capacitor’s plates, A = Area of capacitor’s plates. σ σ Q Q aroundElectricElectric circuits field fieldbetween. between capacitor’s capacitor’s plates: plates:E =E = = = , where, whereQ =Q charge = charge on on ⇥0 ⇥0A Q = ⇥0 E A = ⇥0ΦE ⇥0 ⇥0A capacitor’s plates, A∴ = Area of capacitor’s· plates. capacitor’s plates, A = AreaElectric of flux capacitor’s plates. We can define ⇤ ⇥Q⌅ = ⇥ E A = ⇥ Φ ∴ ∴ ∴ Q0= ⇥0 E A 0= ⇥E0ΦE dQ dΦ ⇓· · = ⇥ E = i ElectricElectric flux flux dt 0 dt disp ⇤ ⇥ ⌅ ∴ We canWe defineCan can define we revise Amp`ere’s⇤ ⇥ ⌅ law to fix it? where idisp∴is called Displacement Current (first proposed by Maxwell). dQ dQ dΦEdΦE Maxwell first proposed that this is the missing= ⇥ term=0 ⇥ for= theidisp= Amp`ere’si law: dt dt dt0 dt dispσ Q Electric field between⇤ capacitor’sdΦE plates: E = = , where Q = charge on wherewhereidispB isiddisp⇤s called=isµ0 called(iDisplacementincl +Displacement⇥0 ) Amp`ere-Maxwell Current Current(first(first proposedlaw proposed by byMaxwell). Maxwell). ˛C · dt ⇥0 ⇥0A capacitor’sMaxwell plates,Maxwell first A proposed first = proposed Area that of that this capacitor’s this is the is missingthe missing plates term term for. forthe the Amp`ere’s Amp`ere’s law: law: ⇤ dΦEdΦE B⇤ d⇤sB=dµ⇤s0=(iinclµ0(+iincl⇥0+ ⇥0 ) Amp`ere-Maxwell) Amp`ere-Maxwell law law ˛ ˛·C · Q =dt⇥ dtE A = ⇥ Φ C ∴ 0 · 0 E Electric flux ⇤ ⇥ ⌅ ∴ We can define dQ dΦ = ⇥ E = i dt 0 dt disp where idisp is called Displacement Current (first proposed by Maxwell). Maxwell first proposed that this is the missing term for the Amp`ere’s law: dΦE B⇤ d⇤s = µ0(iincl + ⇥0 ) Amp`ere-Maxwell law ˛C · dt Two obser vations ① While there is no current between capacitor’s plates there is a time-varying electric field between plates of capacitor ② We know Ampere’s law is mostly correct from measurements of B-field around circuits ⇓ Can we revise Ampere’s law to fix it? σ Q Electric field between capacitor’s plates E = = ✏0 ✏0A Q = charge on capacitor’s plates A = area of capacitor’s plates Q = ✏ E A = ✏ Φ ) 0 · 0 E Electric flux Tuesday, April 24, 18 | {z } 4 ) We can define dQ dΦ = ✏ E = i dt 0 dt dis where i dis is called Displacement Current (first proposed by Maxwell) Maxwell first proposed that this is missing term for Ampere’s law dΦE B~ d~s = µ (i + ✏ ) Ampere-Maxwell law · 0 inc 0 dt IC i inc = current through any surface bounded by C ΦE = electric flux through that same surface bounded by curve C Φ = E~ d~a E · ZS Tuesday, April 24, 18 5 11.2. INDUCED MAGNETIC FIELD 127 11.2 InducedWhere iincl Magnetic= current through Field any surface bounded by C, Φ = electric flux through that same surface☛ boundedcharges by curve C, Φ = E⇤ d⇤a. E E S · Electric field can be generated by ´ 11.2 Induced Magnetic☛ Fieldchanging magnetic flux Ampere-MaxwellWe learn law earlier that a that magnetic electric field field can can be begenerated generated by by charges .