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 Attribution-Noncommercial-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/terms6.013, Electromagnetic Fields, Forces, and Motion Prof. Markus Zahn September 20, 2005 Lecture 4: The Scalar Electric Potential and the Coulomb Superposition Integral I. Quasistatics Electroquasistatics (EQS) Magnetquasistatics (MQS) 0 ()0EH0∇× = ≈t∂− µ∂ ()0EHt∂∇× =− µ∂ 0 ()E0∇=εi ρ ()HJ Et0∂∇× = +∂ε ()HJ Et0∂∇× = +∂ε ()H00∇=µi J0t∂ρ∇+ =∂i J0∇=i ()E0∇=ρεi II. Irrotational EQS Electric Field 1. Conservative Electric Field 0 CSdHdt0µ∫∫Eds da 0=− ≈ii 6.013, Electromagnetic Fields, Forces, and Motion Lecture 4 Prof. Markus Zahn Page 1 of 6ba b bCab a aElectromotive Force(EMF)Eds Eds Eds 0 Eds EdsIII I II=+ =⇒ =∫∫∫ ∫ ∫ii i i i EMF between 2 points (a, b) independent of path E field is conservative () ()refrrefrScalarelectric potentialrrEΦ−Φ=∫ids ()() ()() () ()refref refrefrbbaarEds Eds Eds a r r b a b= + =Φ −Φ +Φ −Φ =Φ −Φ∫∫∫iii 2. The Electric Scalar Potential ___xyrxi yi zi=++z ___xyrxiyizi∆=∆+∆+∆z nrcos∆=∆ θ 6.013, Electromagnetic Fields, Forces, and Motion Lecture 4 Prof. Markus Zahn Page 2 of 6()()x x,y y,z z x,y,z∆Φ=Φ +∆ +∆ +∆ −Φ () ()x,y,z x y z x,y,zxyz∂Φ ∂Φ ∂Φ=Φ +∆+∆+∆−Φ∂∂∂ xyxyz∂Φ ∂Φ ∂Φ=∆+∆+∆∂∂∂z ___xyziiixyzgrad⎡⎤∂Φ ∂Φ ∂Φ=++⎢⎥∂∂∂⎣⎦Φ=∇Φir∆ ___xyziiixy∂∂∇= + +∂∂ z∂∂ ___xyzgrad iiixy∂Φ ∂Φ ∂ΦΦ=∇Φ= + +∂∂ z∂ () ()rrrEds r r r r E r+∆=Φ −Φ +∆ =−∆Φ=−∇Φ ∆ = ∆∫iii E =−∇Φ rcos n r rnn∆Φ ∆Φ∆Φ = ∆ θ = ∆ = ∇Φ ∆∆∆ii nnnn∆Φ ∂Φ∇Φ = =∆∂ The gradient is in the direction perpendicular to the equipotential surfaces. III. Vector Identity E0∇× = E =−∇Φ ()0∇× ∇Φ = 6.013, Electromagnetic Fields, Forces, and Motion Lecture 4 Prof. Markus Zahn Page 3 of 6IV. Sample Problem ()02Vxyx,yaΦ= (Equipotential lines hyperbolas: xy=constant) __xyEixy⎡⎤∂Φ ∂Φ=−∇Φ=− +⎢⎥∂∂⎣⎦i __0xy2Vyi xia⎛⎞−=+⎜⎟⎝⎠ Electric Field Lines [lines tangent to electric field] yxEdy xydy xdxdx E y==⇒ = 22yxC22=+ 22200yxyx−=−2 [lines pass through point ()00x,y] (hyperbolas orthogonal to xy) Courtesy of Hermann A. Haus and James R. Melcher. Used with permission. 6.013, Electromagnetic Fields, Forces, and Motion Lecture 4 Prof. Markus Zahn Page 4 of 6V. Poisson’s Equation ()200E∇=∇−∇Φ=ρ ⇒∇Φ=−ρεεii ()___ ___2xyz xyiii iiixyzxyz⎡⎤⎡∂ ∂ ∂∂Φ∂Φ∂Φ∇Φ=∇ ∇Φ = + + + +⎢⎥⎢∂∂∂∂∂∂⎣⎦⎣iiz⎤⎥⎦ 22222xyz∂Φ ∂Φ ∂Φ=++∂∂∂2 VI. Coulomb Superposition Integral 1. Point Charge r200qqECr44r∂Φ=− = ⇒Φ= +∂ππεεr Take reference ()r0CΦ→∞=⇒=0 0q4rΦ=πε 2. Superposition of Charges 6.013, Electromagnetic Fields, Forces, and Motion Lecture 4 Prof. Markus Zahn Page 5 of 6()12 12T01212q q dq dq1d P ... ...4rr rr''rr rr⎡⎤⎢⎥⎢⎥Φ= + + + +⎢⎥π−−−−⎢⎥⎣⎦ε ()NnT0n1nall line,surface, andvolume chargesq1dP4rr'rr=⎡⎤⎢⎥⎢⎥Φ= +⎢⎥π−⎢⎥−⎢⎥⎣⎦ε∑∫q Nsn0n1nLS V'''''rdl rda rdVq14rr''rr rr rr=⎡⎤⎛⎞ ⎛⎞ ⎛⎞λσ ρ⎜⎟ ⎜⎟ ⎜⎟⎢⎥⎝⎠ ⎝⎠ ⎝⎠⎢⎥=+++π⎢⎥−−− −⎢⎥⎣⎦ε∑∫∫ ∫'' Short-hand notation ()V0''rdVr'4rr⎛⎞ρ⎜⎟⎝⎠Φ=π−ε∫ 6.013, Electromagnetic Fields, Forces, and Motion Lecture 4 Prof. Markus Zahn Page 6 of
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