maxwells_equations

maxwells_equations - Va MAXWELL’S E UATIONS:...

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Unformatted text preview: Va MAXWELL’S E UATIONS: RADIATION-:5? LI‘ (5H “T To summarize,- the field Equations derived from Experiments are: GAUSS’ LAW FOR COULOMB E 1. 2C EoAA = —.-EQ,. ' (1) F’ a 130 . GAUSS’ LAW FOR B V 2 C BvAfl E 0 I ' ' (2) LENZ’S LAW M) B EC EWCQAE = ” At 1 V ' (3) AMPERE’S LAW _ , . 2C 39A] = #024. (4') When Maxwell began to study these equations, he realized that there was a serious problem. Scientists believe that at its most fundamental level nature must be symmetric. Maxwell noticed that Whereas a time varying flux of B gave rise to an E --field [ENC in Eq.(3 ] ‘ ~> —> _) I) there was no corresponding term in Eq. (4 ’). He immediately asserted that the above field equations could not be regarded as being complete. This was a FUNDAMENTAL PROBLEM - Maxwell also noted a “PRACTICAL PROBLEM” in using Eq. (4/. Imagine that we charge a capacitor to i q and then connect a Wire between the two plates as shown. 2f a A , . It is clear that a conduction current A—:] begins to flow through the mm and so [using Eq. (4 9] it must create a B -field encircling the wire as shown. However, as soon as you cross one of the .—> capacitor plates, both the current and B must be zero. Again, Maxwell asserted that such a -—) discontinuity cannot be physically meaningful. ‘ To resolve the fundamental problem Maxwell postulated that if the flux of E varies with .—) time it mUSt be equivalent to a current. He called this new type of current a displacement current 80 A Of course, Eq. (4’) implies that every current generates a B so Maxwell “completed” Eq. (4") by —> and introduced the definition z'D = A writing 203- ha]: r10210+yoao ME (4) —> Where IC explicitly signifies a conduction current = flow of charge in a conductor while the second term on the right comes from z'D [Eq. (5)] . Let us see if introduction of iD also solves the practical problem. If the capacitor plates have an area A the {)3 -field between them is E=—q 92,14: A)? a go —> .9 (D =— so E 80 ND Aq an 1D 80 At t 1C . . A . . [iD is from —1ve to +1ve because of A—i is —1ve] Since iD = iC we will have rm discontinuity in either the current or the B -field on crossing —> the capacitor plate. Maxwell has solved both the fundamental and the practical problem by proposing Eq. (5). war EM. wcfiéélé’Mfigi .V‘. ,_ , g . ~51"? c: ., . {12424450. ,. 1.: fizzgz, 727m, wax .673 . 5mg . ,Wmé” ,x 4: z ,. ¥. £932“ .flégéiéé 7 .. . V A w ‘ a V . ‘ , . A" Aé‘lhceg‘ 4% _ _. _. paw; _..T4>_Tf¥:4 , . _ ‘ ' Q axe/v0ch .. ., ‘ 2&3 5 f f f' f .} f G2); ' ” ' u . wa.‘ , Ii 74 .WWHA A Nam C’Qétffi-é‘f‘fl’gwiwfifféb WITLLMMPEJM IN .Evmv Cmsa. 3.?wa JiaMmWfi/Né, 77*?5W _ . . . Ream (WP‘E’QEHTHET waéée’i V $37 .VVMY’Wéizw , 7M , 35W? , Ora. riais.._.,l:<yWs“%pz@3493es, @WDVJE THE , Wham Ti‘wv IN “75%. ,,FW¥.P,F1_ THAT (Wasafs IT; HERE; CMCMM ’T’b’v ‘35 4% 5%” 4);; a 0mm?» . .4517" .. w .. . . , F ‘ 7 E :17. CAUTION: iD exists in vacuum. It never involves flow of charge. No conduction current can exist inside the capacitor!!! Maxwell’s Equations (1) through (4) have profound consequences. Let us recall his work using these in outer space, Where there is vacuum, q=0, Q. = 0. so the Equations become: ECE-AAst I —> —> ECB-A :0 I] —) —> Mt ZCEévaaL— At III V ME legal—#080 At IV and now indeed there is total symmetry with respect to E and B . This is what led Maxwell to —) —> propose that rather than think of E and B fields, one should think of a single entity: ~> ~> l Electromagnetic or EM field 1 And call Equations 1 through IV, EM field Equations. He next used these Equations to predict that in vacuum there must exist EM-waves! He was able to show that the structure of these Equations is such that both the E and B have the functional form (propagation along x for ~> —) example) f (x i ct) . That is, they propagate as an Electromagnetic wave with the enormous 1 speed c = = 3 x 108 ’% . This was a giant step forward: Maxwell had solved the problem 4/1080 , . of the nature of Radiation or Radiant energy. :> Radiation is an Electromagnetic wave. O_ur observable universe = Matter + Radiation ' Incidentally, Einstein demonstrated that matter and radiation convert into one another there by further simplifying our picture of the universe. —> Heat —> Light —> x — rays —> radiowaves are all cases of EM waves. They are distinguished only by their frequencies (or wavelengths). We will concentrate on M FM 3 LLY we CBME "773° 3 LIGHT LIGHT: is a transverse EM wave (E and B fields perpendicular to direction of propagation and —> —> also E i B) whose wavelength lies between 400 nm and 800 nm and whose speed in vacuum is —> —> 3 x 108 ’% . As always, light waves transport energy. Let us compare transport of Energy by: Wave on a string: Power I J A1,. ‘ zlfimzwzv [Emmetvrfifi sweat PM 2 , F Light} [mg/CK) v: — Wu.»pr deal, L93” l ' ‘ __ V A A F-vaV—W WWW—WWW :. Sound: Intensity 1 $2 2 Std-5%“?! “P44 = —po a) v 2 m 1» Val—e2 Mme, P . v : h 100 EM-wave Light: Intensity I Bm2 1 E 2 : —‘ : — E Z’uo c 2 0 m c ...
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maxwells_equations - Va MAXWELL’S E UATIONS:...

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