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 afVf = -, + Oi, + i,ax ay OzaA, aA, aA,V .A= a+ +ax ay az(LAX, A AaA\ .a, OaA.ay z az Ox ax ayV2f a' +f + a'fOx jy azCylindricalCoordinates (r, 4, z)Of. I af af.Vf = r • 4 + 1ar r04 az1 a 1iA, aA,V *A=--(rAr)+M +Mrr rr a OzI aBA, aA. A, DAA. I(rAs) Ar rr) 1xzaz'L ar a4JV~ f l a0af\ 1 82f a2frr --r-Ora + r) 14 -2azSphericalCoordinates (r, 0, 4)a. af. 1 af.Vf= ar ,+ ae+I If-14r •O r sin 0 aOA 1 (r 1 a(sin OAo) 1 oA*V" -A= (rPA,)+ +r ar r sin 0 ae r sin 0 a4x 1 a(sin OAs) aA]r sin a80 04,aS 1 MA, a(rA,)) 1[ra(rAo) dA,1r sio arsin rOr O-V'f = a-"r-r r+ a+sin0 O+ I a•fCartesianCylindricalSphericalx= r cosc= r sin 0 cos 4y= r sinq= r sin 0 sin 4z= z= r os 0= cos i, -sin 0i= sin 0 cos i, + cos 0 cos 4ie-sin Ois1= sin 0 sin 4i, + cos 0 sin ,Y= sin 0i, + cos 0ik/ie+ cos 46 i= iz= cos Oi,-sin OieCylindricalCartesianSpherical=r sin 0= tan-1y/x-= zSr cos 0= cos kix,+sin i,= sin Oi, +cos ie= -sin 0ix +cos 4iy= i4= i= cos Oi, -sin 0iOSphericalCartesianCylindricalr/x 2+y2+zIf,-ý+z0 -1 z= cos= cos/x2'+y2+z'2= cot-x/yi, = sin 0 cos ,ix +sin 0 sin (i,= sin Oi,+cos Oi,+ cos Oi.is = cos 0 cos oi, +cos 0 sin 4i,= cos Oi, -sin Oi,-sin Oi.i, = -sin 46i, +cos di,= i4,Geometric relations between coordinates and unit vectors for Cartesian, cylirdrical, and spherical coordinate systems.VECTOR IDENTITIES(AxB). C= A. (B xC)= (CxA). BAx(BxC)=B(A C)-C(A -B)V* (VxA)=OVx(Vf)=oV(fg) = fVg + gVfV(A B) =(A * V)B + (B -V)A+Ax(VxB)+Bx(VxA)V. (fA)= fV. A+(A -V)fV *(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)BVx(fA)= VfxA+fVxA(V x A) x A = (A V)A -'V(A .A)Vx (Vx A) = V(V -A) -V AINTEGRAL THEOREMSLine Integral of a GradientVf dlI =f(b) -f(a)Divergence Theorem:f V-AdV= sA dSCorollariest VfdV=f dSV VxAdV=-s AxdSStokes' Theorem:fA dl= (Vx A) dSCorollaryffdl= -fVfxdSIMAXWELL'S EQUATIONSIntegral Differential Boundary ConditionsFaraday's LawE'*dl=-d B-dS VxE=-aB nx(E2'-E')=0.dtJI atAmpere's Law with Maxwell's Displacement Current CorrectionH.dI=s J,.dS VxH=Jjf+a-nx (H2-HI) =Kf+ D dSdtiJsGauss's LawsD-dS= pfdVB dS=0Conservation of ChargeV D=pV*B=0n *(D2-D1) = ofJdS+d pfdV= O V J,+f=0 n (J2-JI)+ = 0s dt at atUsual Linear Constitutive LawsD=eEB=LHJf =o(E+ vx B) =0E'[Ohm's law for moving media with velocity v]PHYSICAL CONSTANTSConstant SymbolSpeed of light in vacuum cElementary electron charge eElectron rest mass m,eElectron charge to mass ratio eProton rest mass mnBoltzmann constant kGravitation constant GAcceleration of gravity gPermittivity of free space 60Permeability of free space Al0Planck's constant hImpedance of free space 110=Avogadro's numberValue2.9979 x 108 =3 x 1081.602 x 10-'99.11 x 10- s3'1.76 x 10"1.67 x 10-271.38 x 10-236.67 x 10-"9.80710-8.854x 10-12= 3636?r4Tr x 10- 76.6256 x 10-34376.73 -120ir6.023 x
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