MIT OpenCourseWare http ocw mit edu 6 013 ESD 013J Electromagnetics and Applications Fall 2005 Please use the following citation format Markus Zahn Erich Ippen and David Staelin 6 013 ESD 013J Electromagnetics and Applications Fall 2005 Massachusetts Institute of Technology MIT OpenCourseWare http ocw mit edu accessed MM DD YYYY License Creative Commons AttributionNoncommercial Share Alike Note Please use the actual date you accessed this material in your citation For more information about citing these materials or our Terms of Use visit http ocw mit edu terms 6 013 Electromagnetic Fields Forces and Motion Prof Markus Zahn Sept 15 2005 Lecture 3 Differential Form of Maxwell s Equations I Divergence Theorem 1 Divergence Operation Courtesy of Krieger Publishing Used with permission A i dS S div A dV V div A lim V 0 A i dS S V 6 013 Electromagnetic Fields Forces and Motion Prof Markus Zahn Lecture 3 Page 1 of 10 Courtesy of Krieger Publishing Used with permission Courtesy of Krieger Publishing Used with permission 6 013 Electromagnetic Fields Forces and Motion Prof Markus Zahn Lecture 3 Page 2 of 10 A x y z dydz A x x y z dydz x x 1 1 A y x y y z dxdz A y x y z dxdz 2 2 A z x y z z dxdy A z x y z dxdy 3 3 A x x y z A x x x y z A y x y y z A y x y z x y z x y A z x y z z A z x y z z A y A A V x z y z x div A lim A i dS S V 0 V Del Operator i x div A i A A y A x A z x y z iy iz x y z A y A x A z x y z 2 Gauss Integral Theorem Courtesy of Krieger Publishing Used with permission 6 013 Electromagnetic Fields Forces and Motion Prof Markus Zahn Lecture 3 Page 3 of 10 A i dS N i 1 N S A i dSi dSi lim N V 0 n i A V N i i 1 i A dV V V i A dV S A i da 3 Gauss Law in Differential Form 0 S V i E 0 S E i da i 0 E dV dV V 0 H i da i 0 H dV 0 V i 0 H 0 II Stokes Theorem 1 Curl Operation A i ds S Curl A Curl A i da C n lim dan 0 A i ds C dan 6 013 Electromagnetic Fields Forces and Motion Prof Markus Zahn Lecture 3 Page 4 of 10 S A i da C A i ds Courtesy of Krieger Publishing Used with permission 6 013 Electromagnetic Fields Forces and Motion Prof Markus Zahn Lecture 3 Page 5 of 10 x x A i ds C y y A x x y dx x 1 y 2 y A y x x y dy x A x x y y dx x x 3 A y x y dy y y 4 A x x y A x x y y A y x x y A y x y x y y x A y A x daz y x Curl A z A i ds daz A y x A x y By symmetry Curl A Curl A y x A i ds A x A z z x A i ds A z A y y z day dax A y A x A z A y A x A Curl A i x z iz iy z x y z y x ix det x A x iy y Ay iz z A z A 6 013 Electromagnetic Fields Forces and Motion Prof Markus Zahn Lecture 3 Page 6 of 10 2 Stokes Integral Theorem Courtesy of Krieger Publishing Used with permission N lim N i 1 dC i A i dsi A i ds C A i da N i i 1 A i da S Courtesy of Krieger Publishing Used with permission 6 013 Electromagnetic Fields Forces and Motion Prof Markus Zahn Lecture 3 Page 7 of 10 3 Faraday s Law in Differential Form d E i ds E i da dt C S E 0 0 H i da S H t 4 Amp re s Law in Differential Form d H i ds H i da J i da dt C S S H J 0 0 E i da S E t III Applications to Maxwell s Equations 1 Vector Identity A i ds 0 A i da i A dV lim C 0 S C V i A 0 2 Charge Conservation E i H J 0 t E 0 i J 0 t 0 i J t 3 Magnetic Field H i E 0 t 0 i 0 H i 0 H 0 t 6 013 Electromagnetic Fields Forces and Motion Prof Markus Zahn Lecture 3 Page 8 of 10 4 Vector Identity b E i dl a b a if a b E i dl a a 0 C E i dl 0 C f i da f i dl 0 f 0 S C IV 6 013 Electromagnetic Fields Forces and Motion Prof Markus Zahn Lecture 3 Page 9 of 10 Summary of Maxwell s Equations in Free Space Integral Form Differential Form Faraday s Law E i dl 0 C d H i da dt S E 0 H t Ampere s Law H i dl J i da C 0 S d E i da dt S H J 0 E t Gauss Law E i da dV i 0 E i 0 H 0 0 S V 0 H i da 0 S Conservation of charge 1 d J i da dt dV 0 C 2 V J S iJ 0 E i da 0 t E i J 0 0 t EQS Limit MQS Limit E 0 E 0 E i E i 2 x y z 0 t 0 Poisson s Eq H J x y z dx dy dz i 0 H 0 0 H A 1 2 x y z H t 2 2 2 4 0 x x y y z z 2 A 0 J i A 0 A x y z x y z 0 J x y z dx dy dz 2 2 2 4 x x y y z z 6 013 Electromagnetic Fields Forces and Motion Prof Markus Zahn 1 2 Lecture 3 Page 10 of 10
View Full Document
Unlocking...