Electricity and Magnetism- Maxwell's Equation

Topics: 􀂄 􀂄 􀂄 􀂄 8.022 (E&M) – Lecture 19 The missing term in Maxwell’s equation Displacement current: what it is, why it’s useful The complete Maxwell’s equations And their solution in vacuum: EM waves Maxwell’s equations so far 􀁇􀁇⎧∇iE = 4πρ 􀃅 Gauss’s law: relates E and charge density (ρ)⎪􀁇􀁇 ⎪∇iB = 0 􀃅 Magnetic field lines are always closed! ⎪ 􀁇 􀁇 1 ∂B⎨∇× E = − 􀃅 Faraday’s law: change in B flux creates e.m.f. (E) ⎪ ct∂ ⎪ 􀁇 􀁇 4π 􀁇 ⎪∇× B = J 􀃅 Ampere’s law: relates B and its sources (J) ⎩ c Is this set of equations completely consistent? Not quite… G. Sciolla –MIT 8.022 – Lecture 19 2 1 Maxwell’s equations so far (2) 􀁇􀁇⎧∇iE = 4πρ⎪􀁇􀁇⎪∇iB = 0⎪􀁇 􀁇 1 ∂B⎨∇×E =−⎪ ct∂⎪􀁇 􀁇 4π 􀁇⎪∇× B = J⎩ c 􀂄 Is this set of equations consistent? Not quite… 􀂄 Take the divergence of Ampere’s law 􀁇⎛4π􀁇⎞ 4π􀁇􀁇 4π ρ∂∇i⎜ J ⎟= ∇iJ =− (using continuity equation) 􀂄 ⎝ c ⎠ cc ∂t 􀁇􀁇 􀁇 􀁇􀁇􀁇 􀂄 ∇∇× B = 0(∇∇× v is ALWAYS 0!)ii Ampere’s law works only when dρ/dt=0 which works in most cases but not always: Ampere’s law is incomplete! G. Sciolla –MIT 8.022 – Lecture 19 3 4 Fixing the inconsistency 􀂄 Since 􀂄 􀂄 􀂄 0v∇ ∇× ≡􀁇i 4π∇× = + 􀁇 􀁇 B J F c 4 0 ( ) 4 4 ( ) 4π ρπ π ρπ ∂⎛ ⎞∇ + = = − ∇ =⎜ ⎟ ∂⎝ ⎠ ∂⇒∇ = ∂ 􀁇 􀁇 􀁇 i i i 􀁇 􀁇 i J F J c t cF t ( ) ( )4 ( )1 ρπ ⎛ ⎞∂ ∂ ∂ ∂∇ = ⇒ ∇ =∇ =∇⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ ∂ ⇒ = ∂ 􀁇􀁇􀁇 􀁇􀁇 􀁇 􀁇 􀁇 i i i i 􀁇􀁇 EE E t t t t EF ct G. Sciolla – MIT 8.022 – Lecture 19 we need to add some term to the right hand side to that its divergence will be identically 0 Generalized Ampere’s law: What is F? We know that its divergence must be =0: Take time derivative of Gauss’s law: 􀁇 􀁇 Similar to Gauss's law! ⇒ ∇ 􀁇 􀁇 􀁇􀁇 cF time and space derivatives commute cF 2 S G. Sciolla – MIT 8.022 – Lecture 19 5 Displacement currents 􀂄 Generalized Ampere’s equation 􀂄 This can also be written as: 􀂄 With Jd = displacement current (density): 􀂄 What is the Jd? 􀂄 Not a real current: does not describe charges flowing through some region 􀂄 But it acts like a real current: whenever we have changing E field, we can treat its effect as if due to as a real current Jd 4 1π ∂∇× = + ∂ 􀁇􀁇􀁇 􀁇 EB J c c t 4 ( )π∇× = + 􀁇􀁇 􀁇􀁇 dB JJ c 1 4π ∂ = ∂ 􀁇􀁇 d EJ t G. Sciolla – MIT 8.022 – Lecture 19 6 What is a displacement current? 􀂄 Consider a current flowing in a circuit and charging a capacitor C 􀂄 Standard integral Ampere’s law: 􀂄 Let’s choose the path C and the surface S as in the drawing above: 􀂄 It all makes sense! 􀂄 Now choose the same path C but the surface S’ (ok by Stokes…) 􀂄 No standard current J through the surface (no charge crosses C!) 􀂄 But there is a flux of displacement current Jd through the plates! 4 4 encl S C Bdl I J da c c π π = =∫ ∫ 􀁇􀁇 􀁇 􀁇i i􀁶 CCS’ 37 What is a displacement current? (2) 􀂄 􀂄 􀂄 the plates of the capacitor C C S’ ( ) ' ' ' 4 1 1 1 4 4 4 encl d C E d dS S S I I c EI da a t t t π π π π = + ∂Φ∂ ∂ = = = = ∂ ∂ ∂ ∫ ∫ ∫ ∫ 􀁇 􀁇􀁇 i 􀁇􀁇 􀁇􀁇 􀁇 􀁇i i i 􀁶 E G. Sciolla – MIT 8.022 – Lecture 19 We can use the generalized Ampere’s Law: The displacement current is related to the change over time of the flux of the electric field. In the example above, the electric field E is the one produced in between with Bdl J da E d8 What is a displacement current? (3) 􀂄 􀂄 􀂄 􀂄 􀂄 􀂄 􀂄 􀃆 4 ˆQE xA π = 􀁇 4E QπΦ =􀁇 4E Q I t t π π∂Φ ∂ = ∂ ∂ 􀁇 'd dS S I a a I= = =∫ ∫ 􀁇 􀁇􀁇 􀁇i i G. Sciolla – MIT 8.022 – Lecture 19 The electric field E: Points in the same direction of the current (+x) At a given instant in time: The flux of E will then be: The rate of the change if this flux is: Where I is the current that is charging the capacitor Comparing this with results in the previous page: generalized Ampere’s Law is valid no matter what surface we use (yes, Gauss's law!) =4 J dJ d 49 The importance of displacement currents 􀂄 􀂄 􀂄 􀂄 laws. G. Sciolla – MIT 8.022 – Lecture 19 When we examined the following circuit: we said the same current I was flowing in each circuit element. How is it possible? No current flows through the plates of a capacitor! Displacement currents fix this inconsistency! Displacement current “continues” the “real” current across the capacitors ensuring the validity of Kirchoff’s 10 Maxwell’s equations (complete!) 􀂄 Yes, he was a theorist! 4 0 4 1 1 πρ π ⎧∇= ⎪∇=⎪ ⎪∂⎨∇×⎪∂ ⎪ ⎪∂∇×=+ ⎩∂ 􀁇􀁇 i 􀁇􀁇 i 􀁇􀁇 􀁇 􀁇 􀁇 􀁇􀁇 E B BE ct EB J c c t 0 0 0 0 0 µµ ρ ε ε∂∇ ⎧∇=⎪ ⎪ ⎪∇= ⎪⎨∂∇×⎪∂⎪ ×=+ ∂ ⎪ ⎪⎩ i 􀁇 􀁇􀁇 􀁇 i 􀁇 􀁇 􀁇 E B BE t EB J t cgs SI G. Sciolla – MIT 8.022 – Lecture 19 NB: when Maxwell introduced the term dE/dt in the generalized Ampere’s law, his arguments were based purely on symmetry = − Generalized Ampere’s law = − 􀁇􀁇 􀁇 􀁇 Typo in Purcell Eq 15 ch 9 511 Maxwell’s equations: integral form 4 0 () (1 sed) ) πΦ== Φ= ∂Φ = ∂ ∫ ∫ ∫ 􀁇 􀁇 􀁇 􀁇 􀁇 i 􀁇􀁇 i 􀁇 i 􀁶 􀁶 encE S B B C C Q c t Bd d d 4 ( ) () ( )1I= 4 π π ⎧⎪⎪⎪⎪⎨⎪⎪⎪=+ ∂ ⎪ Φ ⎩ ∂ ∫ 􀁇 􀁇 􀁇 􀁇 􀁇 􀁇 i 􀁇 􀁇 􀁇 􀁇 d S E J l I a I S c d t G. Sciolla – MIT 8.022 – Lecture 19 Gauss's lawMagnetic field line are clo (Faraday's law=− Eda emf E dl where the curren ts I and I are defined as I= and Generalized Ampere's law3 good reasons to remember Maxwell’s equations 1) They compactly and beautifully summarize all the E&M we learned so far! 2) You will see them on T shirts for the rest of your life at MIT: better to get familiar with them ASAP! 3) On the first day of 8.03 next semester you will be asked to write them down on a piece of paper to check what you learned in your first semester at MIT: save your honor (and mine) G. Sciolla –MIT 8.022 – Lecture 19 12 6 13 Displacement current: application 􀂄 􀂄 d 􀂄 Id induces B inside the plates 􀂄 Assuming cylindrical pl􀂄 Calculate B inside the plates /iiliield iil lbVIt eR −= 􀁇 2 d 2 2 4() 4 1 1 1J 4 4 4 dC C QtEt a E t t a t a a c ππσ π π π π π π = = ∂ ∂ ∂ = = = = ∂ ∂ ∂ =∫ ∫ 􀁇 􀁇􀁇 􀁇􀁇 􀁇 􀁇i i􀁶 /2 2 brVBr e−⇒ = G. Sciolla – MIT 8.022 – Lecture 19 Consider the following RC circuit: As C charges up, Iflows ates of radius a 1) Fnd E(t): 2) Dsp. current density: 3) Remember that ( ) 4) Magnetc fnsde the pate (Am pere'saw ): tRC ( ) ( ) ( ) ( ) E t Q t I t Bdl J d( ) tRC ca R Maxwell equations in vacuum 􀂄 What happens when we write Maxwell’s equations in vacuum? 􀂄 Vacuum: no sources, ρ=0 and J=0 􀁇􀁇 􀁇􀁇 ⎧∇iE = 4πρ ⎧∇iE = 0 ⎪􀁇􀁇 ⎪ 􀁇􀁇⎪∇iB = 0 ⎪∇iB = 0⎪ 􀁇 􀁇⎪ 􀁇 􀁇 = − 1 ∂B 􀁇 􀃆 ⎨∇× E = − 1 ∂B 􀁇⎨∇× E⎪ ct⎪ ct ⎪ 􀁇 􀁇 4π 􀁇 ∂ 1 ∂E 􀁇 ⎪ 􀁇 􀁇 1 ∂E 􀁇∂ ⎪∇× B = J + ⎪∇× B =∂⎩ cc ∂t ⎩ ct􀂄 Except for a – sign, these equations are exquisitely symmetric! 􀂄 Consequence: an electric filed E varying in time will create a magnetic filed B; a B varying in time creates a E: E and B are intimately related! G. Sciolla –MIT 8.022 – Lecture 19 14 7 G. Sciolla – MIT 8.022 – Lecture 19 15 Maxwell equations in vacuum: solution 􀂄 􀂄 􀂄 􀂄 How? 􀂄 􀂄 􀂄 0 (1) 0 (2) 1 (3) 1 (4) ⎧∇ = ⎪∇ =⎪ ⎪ ∂⎨∇× = −⎪ ∂ ⎪ ∂⎪∇× = ∂⎩ 􀁇􀁇 i 􀁇􀁇 i 􀁇 􀁇 E B BE ct EB ct ( ) ( ) 2 2 2 2 2 2 2 2 2 () 1 1 1 (1 (4)) = ∇ = ⎛ ⎞∂ ∂ ∂ = − ∇× = − ∂⇒∇ = ⎟∂ ∂⎝ ⎠ ∂ ⎜ ∂ 􀁇 􀁇􀁇 􀁇􀁇 i i 􀁇 􀁇􀁇 􀁇􀁇 􀁇 􀁇 􀁇 E E E E E B EB ct c t EE c t t c How to solve these equations? Uncouple them! Separate E and B in equations Take the curl of equations (3) and (4) Use other equations as needed Start from (3): 􀁇 􀁇 􀁇 􀁇 Left: since 0 in vacuumRight: using ∇× ∇× =∇ ∇ −∇ −∇ ∇× − 􀁇 􀁇 􀁇􀁇􀁇 􀁇􀁇 16 Maxwell equations in vacuum: solution 􀂄 Now repeat the procedure starting from 􀂄 1 (4)∂∇× = ∂ 􀁇􀁇 􀁇 EB ct ( ) ( ) 2 2 2 2 2 2 2 2 2 () 1 1 1 ((3)) 1 = ∇ = ⎛ ⎞∂ ∂ ∂∇× = ∇× = ∂⇒∇ = ∂ −⎜ ⎟∂ ∂ ∂⎝ ⎠ 􀁇 􀁇􀁇 􀁇􀁇 i i 􀁇 􀁇􀁇 􀁇􀁇􀁇 B B B B B E BE ct c t BB c t t c 2 2 2 2 1 ( )∂∇ = = ± ∂ 􀁇 ff f f x v t G. Sciolla – MIT 8.022 – Lecture 19 This is a special case of a known equation: the wave equation: where f is any function that has well-behaved derivatives NB: we are restricting ourselves to the 1D case; extension to 3D next lecture Left: since 0 in vacuumRight: using ∇× ∇× =∇ ∇ −∇ −∇ 􀁇 􀁇 􀁇􀁇􀁇 􀁇􀁇 􀁇 􀁇 wherevt 817 Solution of wave equation: prove 􀂄 Prove that 􀂄 􀂄 􀂄 Define 􀂄 As we wanted to prove! 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ( ) ( ) ( ) ( ) 1 ∂ ± ∂ ∂ ± ∂ = =± ⇒ = ∂ ∂ ∂ ∂ ∂ ± ∂ ∂ ± ∂ = = ⇒ = ∂ ∂ ∂ ∂ ∂ ∂ ⇒ = ⇒ ∂ ∂ fx f fx f v v t u t u t u fx f fx f x u x u t u f f v u v u ( )= ±f fx =±u x 2 2 2 2 1 ∂∇ = ∂ 􀁇 ff v t 2 2 2 2 2 2 2 ∂ ∂ ∂∇= + + ∂ ∂ ∂ 􀁇 x y z G. Sciolla – MIT 8.022 – Lecture 19 is a solution of the wave equation Just calculate time and space derivatives. Keep in mind that Plug the above results into the equation identity! ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ vt f f vt vt f f vt vt vt 18 Wave equation solution 􀂄 ? 􀂄 􀂄 At time t=0: 􀂄 0 􀂄 At time t=1 s: 􀂄 0 􀂄 􀃆 􀂄 1=x0+1 represents a wave travel( )= −f fx x0 f(x) f(x) x1= x0+1 ( )= −f fx 2 2 2 2 1 ∂∇ = ∂ 􀁇 ff v t G. Sciolla – MIT 8.022 – Lecture 19 What is a function such as Assume v=1 cm/s Position of the max: xThe peak still occurs when the argument of f is xBut since the time is not 0 the function will be shifted in x by “vt”=1 cm Position of the max: xing in the +x direction with velocity v vt vt 919 Wave equation solution 􀂄 ? 􀂄 􀂄 At time t=0: 􀂄 0 􀂄 At time t=1 s: 􀂄 0 􀂄 􀃆 􀂄 1=x0-1 ( )= +f fx x0 f(x) f(x) x1= x0-1 ( )= +f fx 2 2 2 2 1 ∂∇ = ∂ 􀁇 ff v t t=1 t=0 G. Sciolla – MIT 8.022 – Lecture 19 What is a function such as Assume v=1 cm/s Position of the max: xThe peak still occurs when the argument of f is xBut since the time is not 0 the function will be shifted in x by “vt”=1 cm Position of the max: xrepresents a wave traveling in the -x direction with velocity v vt vt 20 EM waves 􀂄 􀂄 Solution: 􀂄 􀂄 􀂄 +x 􀂄 –x 􀂄 􀂄 􀂄 􀃆 ( )= ±f fx 2 2 2 2 1 ∂∇ = ∂ 􀁇 ff v t ±x vt −x vt +x vt 2 2 2 2 1 ∂∇ = ∂ 􀁇 ff c t G. Sciolla – MIT 8.022 – Lecture 19 Wave equation: Any function of argument These solution represent waves traveling with velocity v represents a wave traveling in the direction represents a wave traveling in the direction Maxwell’s equation: Same equation! Only difference: v=c Solution: EM waves traveling with speed of light The light IS an EM wave!!! vt 10G. Sciolla – MIT 8.022 – Lecture 19 21 EM waves in SI 􀂄 􀂄 where ε0 iand µ0 􀂄 􀂄 ε0 and µ0 􀃆 􀃆 8 m/s2 􀂄 2 2 2µε00 ∂∇ = ∂ 􀁇 ff t 1/µε0 0 =v G. Sciolla – MIT 8.022 – Lecture 19 This same result looks much more interesting in SI. Maxwell’s equations in SI: is the permittivty of free space is the permeability of free space Maxwell’s equations tell us what the velocity of an EM wave is: can be measured we can predict velocity of EM waves: v=2.998 10which is the speed of light! Maxwell was the first to realize that E&M equations were leading to a wave equation that was propagating at the speed of light: light is an EM wave! 22 􀂄 by a mirror. How to measure c (demo A4) laser Ch2 Ch1 Mirror 􀃆 8 m/s Ch 1: longer path Experimental setup: a neon laser beam is sent into a beam splitter. Part of it is reflected and part of it is refracted first and then reflected Beam splitter • Difference in path between the 2 beams: ~ 17.15 m x 2 = 34.3 meters • Measure the delay of channel 2 wrt channel 1 on the scope: 116 ns v=34.3 m /116 ns = 2.96 10Ch 2: shorter path 1123 Summary and outlook 􀂄 Today: 􀂄 􀂄 􀂄 􀂄 Wave equation 􀃆 􀂄 Next time: 􀂄 Properties of EM radiation 􀂄 G. Sciolla – MIT 8.022 – Lecture 19 Complete Maxwell’s equations The missing term leads to displacement currents Solution of Maxwell’s equations in vacuum light is an EM radiation Polarization and scattering of light 12