Maxwell s Equations and Electromagnetic Waves
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1 Phsics 36: Waves Lecure 3 /9/8 Maxwell s quaions and lecromagneic Waves Four Laws of lecromagneism. Gauss Law qenc all da ρdv Inegral From From he vecor ideni da dv Therefore, we ma wrie Gauss Law as ρ Differenial From. Farada s Law MF da dl da Therefore, we ma wrie Farada s Law as 3. Gauss Law for Magneism (no magneic monopoles da 4. Ampére s Law (modified b Maxwell dl K m J da K m D da Inegral From Differenial From Inegral From Differenial From Inegral From Page of 8
2 Phsics 36: Waves Lecure 3 /9/8 Therefore, Km J KeKm Differenial From Now, if we ake he differenial form of Farada s Law and appl he curl operaor on boh sides, hen ( Le us aemp o derive equaions conaining onl elecric fields or magneic fields. To decouple he wo from he above equaion, we will subsiue oher equaions ino he above. The righ-hand side of he equaion is he negaive ime derivaive of Ampére s Law. Hence, Now we emplo he familiar vecor ideni J ( Km KeKm If we assume ρ or is a consan in space ( A ( A A ( So, K K e m K m J In emp space, J and K e K m. liminaing he negaive sign from boh sides we arrive a Page of 8
3 Phsics 36: Waves Lecure 3 /9/8 Page 3 of 8 i.e. z x So, we have raveling elecromagneic waves where we define he speed, c, of he wave as Similarl, Finall, we have equaions ha have isolaed elecric or magneic fields. I urns ou hese equaions are wave equaions because he saisf V x And he soluions are raveling waves! Suppose we r as soluions o hese wo differenial equaions ( kz i e ω ; ( kz i e ω which are wave funcions propagaing along he z-direcion. Since, in he absence of charges we ma wrie Gauss Law Subsiuing he rial soluion ino he above equaion and aking he divergence, we arrive a c
4 Phsics 36: Waves Lecure 3 /9/8 Therefore, we conclude i( kz ω ( ike z z ; z This resul means he elecromagneic waves are ransverse! The oscillaions of he elecric and magneic fields are perpendicular o he propagaion vecor. Wha is he direcion of wih respec o? Le us begin wih linearl polarized elecric field and î kˆ. Then, e i ˆ ( kz ω i The goal now is o find he magneic field vecor,, via he equaion relaing he elecric and magneic fields, viz. The lef-hand side becomes Wih all oher erms zero. Therefore, Therefore, x ˆj z ik e ik e iω i i ( kz ω ˆ j ˆ c ( kz ω i( kz ω j e j ω where c. Since he elecric field oscillaes in he x-direcion (î and he magneic k oscillaions are in he -direcion ( ĵ, we conclude ha he elecric and magneic fields are perpendicular o each oher and he direcion of propagaion. ˆ Page 4 of 8
5 Phsics 36: Waves Lecure 3 /9/8 To summarize, i. ii. c iii. and are in phase, since he have he same ime-dependen i( kz ω form: e, wih no phase shif. Now we will show he inensi average power / uni area is I P A c Consider he volume elemen L c The volume is V Ac area, A The energ densi, u, is u energ volume Solving for he energ we ge energ volume u Ac Therefore, u energ area ime c We alread concluded ha he magniudes for he elecric and magneic fields is: c. Hence, u c c Page 5 of 8
6 Phsics 36: Waves Lecure 3 /9/8 where we have used he relaion c c c c Since, c, we wrie u c c c If, cos ( kx ω Then he ime-average becomes So, we have he final resul ha he inensi of he elecromagneic wave is I u energ c area ime c Also, he Poning vecor is defined as S I can be riviall derived, ha I S To summarize, i. Irradiance wave inensi I avg. power/uni area, and dp ii. Radian inensi, where P power and Ω solid angle dω Page 6 of 8
7 Phsics 36: Waves Lecure 3 /9/8 Polarizaion of lecromagneic Waves A. Linear or Plane Polarizaion The ±-direcion of (or sas consan in ime. Propagaion direcion Propagaion direcion ½ period, T, laer Le Generall, e ( kz ω i iˆ x ˆj ( cos( kz ω iˆ cos( kz ω ˆj ( iˆ ˆj cos( kz ω Re x x No ime dependence cos ( kz ω x cos ( kz ω x. Unpolarized Page 7 of 8
8 Phsics 36: Waves Lecure 3 /9/8 The direcion of varies randoml wih ime. C. Circular Polarizaion The - (and - field roae wih ime. We wrie he elecric field as i( kz i( kz ( iˆ ω ω ij ˆ e ( xˆ i e ˆ From uler s equaion for complex variable θ e i cosθ isinθ We will subsiue in for he exponenial on he righ-hand side o ge i( kz ω ( xˆ iˆ e ( xˆ iˆ ( cos( kz ω isin( kz ω ( ( kz xˆ i ( kz xˆ i ( kz ˆ ( kz ˆ cos ω sin ω cos ω sin ω ( cos( kz ω xˆ sin( kz ω ˆ i( sin( kz ω xˆ cos( kz ω ˆ Hence, Re ( cos( kz ω xˆ sin( kz ˆ ω We see he x- and -componens are 9º ou of phase and he magniude,, is independen of ime. To see wha is happening in his complex siuaion, le us fix z and var ime. j a a laer ime sin( ω sin( ω k i a sars wih zero value and evolves increasing o be >. Looking ino an on-coming wave, roaes aniclockwise. This is lefhanded circular polarizaion Page 8 of 8
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