MIT OpenCourseWare http ocw mit edu Electromagnetic Field Theory A Problem Solving Approach For any use or distribution of this textbook please cite as follows Markus Zahn Electromagnetic Field Theory A Problem Solving Approach Massachusetts Institute of Technology MIT OpenCourseWare http ocw mit edu accessed MM DD YYYY License Creative Commons Attribution NonCommercial Share Alike For more information about citing these materials or our Terms of Use visit http ocw mit edu terms CartesianCoordinates x y z af af Vf af O i i ax Oz ay aA aA aA V A a ax az ay LAX A AaA ay z V2f a f Ox jy az a O aA ax Ox ay a f az CylindricalCoordinates r 4 z Of I af4 af 1 r04 az Vf arr 1a 1iA aA rr rr a Oz V A rAr M M I aBA aA rrr V f la0 af r rr 1 82f I rAs ar L a2f 1 ae I r O af If 14 r sin 0 aO A 1 r 1 a sin OAo 1 oA V A rPA r ar r sin 0 ae r sin 0 a4 x 1 r sin a sin OAs aA a80 a 04 S 1 r sio V f r r a r MA arsin a rA 1 ra rAo dA 1 rOr O a sin 0 O I A a4J 4 af a ar DA A Ora r 14 2az SphericalCoordinates r 0 Vf A 1xzaz a f Spherical Cylindrical Cartesian x r cosc r sin 0 cos 4 y z r sinq r sin 0 sin 4 z cos i sin 0i os 0 r sin 0 cos i cos 0 cos 4ie sin Ois 1 Y sin 0 sin 4i cos 0 sin sin 0i cos 0ik iz cos Oi sin Oie Spherical Cartesian Cylindrical r tan 1y x z cos kix sin i sin Oi cos i4 cos Oi sin 0iO If z x 2 y2 z ie Cylindrical Cartesian r cos 0 i Spherical sin 0 Sr sin 0ix cos 4iy 0 ie cos 46 i 1 cos x2 y2 z z cos 2 cot x y i sin 0 cos ix sin 0 sin i cos Oi is cos 0 cos oi cos 0 sin sin Oi i sin 46i cos di 4i sin Oi cos Oi cos Oi sin Oi i4 Geometric relations between coordinates and unit vectors for Cartesian cylir drical and spherical coordinate systems VECTOR IDENTITIES AxB C A B xC CxA B Ax BxC B A C C A B V VxA O Vx Vf o V fg fVg gVf V A B A V B B V A Ax VxB Bx VxA V fA fV A A V f V A x B B V x A A V x B v x A x B A V B B V A B V A A V B Vx fA VfxA fVxA V x A x A A V A V A A Vx Vx A V V A V A INTEGRAL THEOREMS Line Integral of a Gradient f b f a Vf dlI Divergence Theorem sA dS f V AdV Corollaries t VfdV f dS VVxAdV s AxdS Stokes Theorem fA dl Vx A dS Corollary ffdl fVfxdS I MAXWELL S EQUATIONS Differential Integral Boundary Conditions Faraday s Law E dl d B dS VxE aB nx E2 E 0 dtJI at Ampere s Law with Maxwell s Displacement Current Correction H dI s J dS dtiJs VxH Jjf a nx H 2 HI Kf D dS Gauss s Law V D p pfdV sD dS B dS 0 n D 2 D 1 of V B 0 Conservation of Charge JdS d pf dV O V J f 0 s dt Usual Linear Constitutive Laws D eE n J2 JI at at 0 B LH Jf o E vx B 0E Ohm s law for moving media with velocity v PHYSICAL CONSTANTS Constant Symbol Speed of light in vacuum Elementary electron charge Electron rest mass Electron charge to mass ratio c e m ee Value 2 9979 x 108 3 x 108 1 602 x 10 9 9 11 x 10 3s m sec coul kg 1 76 x 10 coul kg Proton rest mass Boltzmann constant Gravitation constant Acceleration of gravity mn k G g 1 38 x 10 23 6 67 x 10 9 807 Permittivity of free space 60 8 854x 10 Permeability of free space Planck s constant Impedance of free space Avogadro s number Al0 h 110 units 1 67 x 10 27 12 7 1036 r 36 kg joule OK nt m2 kg 2 m sec 2 farad m 4Tr x 10 6 6256 x 10 3 4 henry m joule sec 376 73 120ir ohms 6 023 x 1023 atoms mole
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