8.022 (E&M) Lecture 19

Transcription

1 8. (E&M) Lecture 19 Topics: The missing term in Maxwell s equation Displacement current: what it is, why it s useful The complete Maxwell s equations An their solution in vacuum: EM waves Maxwell s equations so far ie = 4πρ Gauss s law: relates E an charge ensity (ρ) ib = Magnetic fiel lines are always close! 1 B E = Faraay s law: change in B flux creates e.m.f. (E) c t 4π B = J Ampere s law: relates B an its sources (J) c Is this set of equations completely consistent? Not quite G. Sciolla MIT 8. Lecture 19 1

2 Maxwell s equations so far () ie = 4πρ ib = 1 B E = c t 4π B = J c Is this set of equations consistent? Not quite Take the ivergence of Ampere s law 4π 4π 4π ρ i J = ij = (using continuity equation) c c c t i B = ( i v is ALWAYS!) Ampere s law works only when ρ/t= which works in most cases but not always: Ampere s law is incomplete! G. Sciolla MIT 8. Lecture 19 3 Fixing the inconsistency Since i v we nee to a some term to the right han sie to that its ivergence will be ientically 4π Generalize Ampere s law: B = J + F c What is F? We know that its ivergence must be =: 4 π ρ i J + F = i ( cf ) = 4 π i J = 4 π c t ρ i ( cf ) = 4π Similar to Gauss's law! t Take time erivative of Gauss s law: E i = 4 i = i = i ( cf ) time an space erivatives commute t t t t 1 E F = c t ρ ( E ) π ( E ) G. Sciolla MIT 8. Lecture 19 4

3 Displacement currents Generalize Ampere s equation What is the J? 4 π 1 E B = J + c c t 4 π This can also be written as: B = ( J + J ) c 1 E With J = isplacement current (ensity): J = 4π t Not a real current: oes not escribe charges flowing through some region But it acts like a real current: whenever we have changing E fiel, we can treat its effect as if ue to as a real current J G. Sciolla MIT 8. Lecture 19 5 What is a isplacement current? onsier a current flowing in a circuit an charging a capacitor S S 4 π 4 π Stanar integral Ampere s law: B i l = I encl = J a c c i S Let s choose the path an the surface S as in the rawing above: It all makes sense! Now choose the same path but the surface S (ok by Stokes ) No stanar current J through the surface (no charge crosses!) But there is a flux of isplacement current J through the plates! G. Sciolla MIT 8. Lecture

4 What is a isplacement current? () We can use the generalize Ampere s Law: 4 π B i l = ( I encl + I ) c 1 E 1 1 Φ E with I = J S ' i a = a = a = 4 π i S ' t 4 π t E i S ' 4 π t The isplacement current is relate to the change over time of the flux of the electric fiel. In the example above, the electric fiel E is the one prouce in between the plates of the capacitor S E G. Sciolla MIT 8. Lecture 19 7 What is a isplacement current? (3) The electric fiel E: Points in the same irection of the current (+x) 4 π Q At a given instant in time: E = ˆ x A The flux of E will then be: Φ = 4πQ (yes, Gauss's law!) E E The rate of the change if this flux is: = 4π t Where I is the current that is charging the capacitor Φ Q =4 πi t omparing this with results in the previous page: I = a J i = J i a = I S ' generalize Ampere s Law is vali no matter what surface we use G. Sciolla MIT 8. Lecture 19 S 8 4

5 The importance of isplacement currents When we examine the following circuit: we sai 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 valiity of Kirchoff s laws. G. Sciolla MIT 8. Lecture 19 9 Maxwell s equations (complete!) ρ i E = 4 πρ cgs i E = ε i B = i B = 1 B E = B c t E = t 4 π 1 E B = J + = µ c c t + µ ε E B J t Generalize Ampere s law NB: when Maxwell introuce the term E/t in the generalize Ampere s law, his arguments were base purely on symmetry Yes, he was a theorist! SI Typo in Purcell Eq 15 ch 9 G. Sciolla MIT 8. Lecture

6 Maxwell s equations: integral form Φ = Ei a = 4 π Q enc ( Gauss's law) E S Φ = ( Magnetic fiel line are clo se) B 1 Φ = i B emf E l = (Faraay's law) c t 4 π B i l = ( I + I ) ( Generalize Ampere's law) c where the curren ts I an I are efine as I= J i a an I= 4 1 Φ ( S ) E π t S G. Sciolla MIT 8. Lecture goo reasons to remember Maxwell s equations 1) They compactly an beautifully summarize all the E&M we learne so far! ) 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 ay of 8.3 next semester you will be aske to write them own on a piece of paper to check what you learne in your first semester at MIT: save your honor (an mine) G. Sciolla MIT 8. Lecture

7 Displacement current: application onsier the following R circuit: As charges up, I flows I inuces B insie the plates Assuming cylinrical plates of raius a alculate B insie the plates 4πQ ( t ) 1) Fin E(t): E ( t ) = 4 πσ = π a 1 E 1 E ( t ) 1 Q ( t ) I ( t ) ) Disp l. current ensity: J = = = = 4 π t 4 π t π a t π a t / R 3) Remember that I ( t ) = e R 4) Magnetic fiel insie the pl ate (Am pere's law ): V b G. Sciolla MIT 8. Lecture 19 4 π B i l = a c J i B ( r ) = rv b ca R e t / R 13 Maxwell equations in vacuum What happens when we write Maxwell s equations in vacuum? Vacuum: no sources, ρ= an J= ie = 4πρ ie = ib = ib = = 1 B E = 1 B E c t c t 4π 1 E 1 E B = J + B = c c t c t Except for a sign, these equations are exquisitely symmetric! onsequence: an electric file E varying in time will create a magnetic file B; a B varying in time creates a E: E an B are intimately relate! G. Sciolla MIT 8. Lecture

8 Maxwell equations in vacuum: solution i E = (1) i B = () How? 1 B E = (3) Take the curl of equations (3) an (4) c t Use other equations as neee 1 E B = (4) c t Start from (3): Left: ( E ) = ( i E ) E = E ( since i E = in vacuum) 1 B 1 1 E Right: = B = ( using (4)) c t c t c t 1 E E = G. Sciolla c MIT t 8. Lecture How to solve these equations? Uncouple them! Separate E an B in equations Maxwell equations in vacuum: solution 1 E Now repeat the proceure starting from B = (4) c t Left: ( B ) = ( i B ) B = B ( since i B = in vacuum) 1 E 1 1 B Right: = E = ( using (3)) c t c t c t 1 B B = c t This is a special case of a known equation: the wave equation: 1 f f = where f = f ( x ± vt ) v t where f is any function that has well-behave erivatives NB: we are restricting ourselves to the 1D case; extension to 3D next lecture G. Sciolla MIT 8. Lecture

9 Solution of wave equation: prove Prove that f = f ( x ± vt ) is a solution of the wave equation Just calculate time an space erivatives. Keep in min that = + + x y z Define u = x ± vt f ( x ± vt ) f f f f( x ± vt ) f = = ± v = v t u t u t u f ( x ± vt ) f f f f( x ± vt ) f = = = x u x u t u f 1 f Plug the above results into the equation = v ientity! u v u As we wante to prove! = f 1 f v t G. Sciolla MIT 8. Lecture Wave equation solution = f 1 f v t What is a function such as? Assume v=1 cm/s f = f ( x vt ) f(x) At time t=: Position of the max: x x At time t=1 s: The peak still occurs when the argument of f is x But since the time is not the function will be shifte in x by vt =1 cm Position of the max: x1 =x +1 f(x) x 1 = x +1 f = f ( x vt ) represents a wave traveling in the +x irection with velocity v G. Sciolla MIT 8. Lecture

10 Wave equation solution = f 1 f v t What is a function such as? Assume v=1 cm/s f = f ( x + vt ) f(x) t= At time t=: Position of the max: x x At time t=1 s: The peak still occurs when the argument of f is x But since the time is not the function will be shifte in x by vt =1 cm Position of the max: x1 =x -1 f = f ( x + vt ) f(x) x 1 = x -1 represents a wave traveling in the -x irection with velocity v t=1 G. Sciolla MIT 8. Lecture EM waves 1 f Wave equation: f = v t Solution: f = f ( x ± vt ) Any function of argument x ± vt These solution represent waves traveling with velocity v x vt represents a wave traveling in the +x irection x + vt represents a wave traveling in the x irection 1 f Maxwell s equation: f = c t Same equation! Only ifference: v=c Solution: EM waves traveling with spee of light The light IS an EM wave!!! G. Sciolla MIT 8. Lecture 19 1

11 EM waves in SI This same result looks much more interesting in SI. Maxwell s equations in SI: f f = µε t where ε is the permittivity of free space an µ is the permeability of free space Maxwell s equations tell us what the velocity of an EM wave is: v = 1/ µ ε ε an µ can be measure we can preict velocity of EM waves: v= m/s which is the spee of light! Maxwell was the first to realize that E&M equations were leaing to a wave equation that was propagating at the spee of light: light is an EM wave! G. Sciolla MIT 8. Lecture 19 1 How to measure c (emo A4) Experimental setup: a neon laser beam is sent into a beam splitter. Part of it is reflecte an part of it is refracte first an then reflecte by a mirror. Difference in path between the beams: laser ~ m x = 34.3 meters h h1 Measure the elay of channel wrt channel 1 on the scope: 116 ns v=34.3 m / 116 ns = m/s Beam splitter h 1: longer path h : shorter path G. Sciolla Mirror MIT 8. Lecture 19 11

12 Summary an outlook Toay: omplete Maxwell s equations The missing term leas to isplacement currents Solution of Maxwell s equations in vacuum Wave equation light is an EM raiation Next time: Properties of EM raiation Polarization an scattering of light G. Sciolla MIT 8. Lecture