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MIT OpenCourseWarehttp://ocw.mit.edu6.013/ESD.013J Electromagnetics and Applications, Fall 2005Please use the following citation format:Markus Zahn, Erich Ippen, and David Staelin, 6.013/ESD.013JElectromagnetics and Applications, Fall 2005. (Massachusetts Instituteof Technology: MIT OpenCourseWare). http://ocw.mit.edu (accessedMM 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

Massachusetts Institute of TechnologyDepartment of Electrical Engineering and Computer Science6.013 Electromagnetics and ApplicationsLecture 1, Sept. 8, 2005I. Maxwell’s Equations in Integral Form in Free Space1. Faraday’s Lawd E i ds - dt µCH i da0SCirculationof EMagnetic Fluxµ0 4π 10-7 henries/meter[magnetic permeability offree space]EQS form: E i ds 0(Kirchoff’s Voltage Law, conservative electricCfield)MQS circuit form: v Ldi(Inductor)dt2. Ampère’s Law (with displacement current) H i ds C J i da SddtCirculation ConductionCurrentof HMQS form: H i dsC6.013 Electromagnetics and ApplicationsProf. Markus Zahn ε0E i daSDisplacementCurrent J i daSLecture 1Page 1 of 7

dv(capacitor)dt3. Gauss’ Law for Electric FieldEQS circuit form: i C ε E i da0 S ρ dVV-910 8.854 10-12 farads/meter36π1c 3 108 meters/second (Speed of electromagnetic waves inε0 µ 0ε0 free space)4. Gauss’ Law for Magnetic Field µ H i da0 0SIn free space:B µ0 (amperes/meter)5. Conservation of ChargeTake Ampère’s Law with displacement current and let contour C 0limC 0d H i ds 0 J i da dt ε E i da0CSS ρ dVV6.013 Electromagnetics and ApplicationsProf. Markus ZahnLecture 1Page 2 of 7

J i daS ddt ρ dV 0VTotal currentTotal chargeleaving volume inside volumethrough surface6. Lorentz Force Law(f q E v µ0 H)II. Electric Field from Point Charge ε E i da ε E 4π r00 r2 qSEr q4π ε0r2T sin θ fc q24π ε0r2T cos θ Mgtan θ 6.013 Electromagnetics and ApplicationsProf. Markus Zahnq2r 4π ε0r2Mg 2lLecture 1Page 3 of 7

2π ε0r3Mg q l 12III. Faraday Cagedddq J i da i - dt ρ dV - dt (-q) dtS idt qIV. Edgerton’s Boomer1. Magnetic Field, Current, and InductanceCourtesy of Hermann A. Haus and James R. Melcher. Used with permission.6.013 Electromagnetics and ApplicationsProf. Markus ZahnLecture 1Page 4 of 7

H i ds H 2 π a N1i H1 1 1Cb()λ N1 π a2 µ0 H1 L N12 π a 2 µ02π aN1 i12πai1 N12 a µ0i1221λ N a µ0 i12ω 1LC11L ip2 C vp2 ip vp CL22C 25 µ f, vp 4 k V, N1 50, a 7 c mL1 0.1 mHip 2000 A, ω 20 x 103 / s f ω 3k Hz2πHp 2.3 x 105 A / m Bp µ0 Hp 0.3 Teslas 3000 Gauss2. Electrical Breakdown in Single Turn Coil with Small GapR Bp 0E E0Inside Metal CoilSmall Gap 6.013 Electromagnetics and ApplicationsProf. Markus ZahnLecture 1Page 5 of 7

d Eids E dt ( B πR0p2)CBp Bm cos ωtE0 BmωπR 2sin ωt Take: Bm 0.3 Tesla, ω 20, 000 radians/second, R 0.07 m, 0.01 mmBm ωπR 2 0.3(20, 000)π(0.07)2 9 106 Volts/meter 10 5Breakdown strength of air 3 106 Volts/meter.Em Courtesy of Hermann A. Haus and James R. Melcher. Used with permission.3. Force on Metal Disk E i ds 2 π aEφ CbdBpdB i da π a2 πa2Bmω sin ωt dt SadtJφ σ Eφ F J x µ0 H ,f F dVσ a dBpσaBm ω sin ωt 2dt2 Force per unitvolume Jx µ0H dVVVtotal forceK φ Jφ Hr Hr Jφ F J µ0 H Jφ iφ µ0Hr ir µ0 JφHr iz µ0 Jφ2 iz2 σa Fz µ0 J2φ µ0 Bmω sin2 ωt2 fz Fz πa2 πµ0 2 σ2 a4 2 2Bmω sin2 ωt46.013 Electromagnetics and ApplicationsProf. Markus ZahnLecture 1Page 6 of 7

σaluminum 3.7 107 Siemens/meter, a 0.07 m, 2 mm, ω 20, 000radians/second, Bm 0.3 Tesla, M 0.08 kgfz µ0π σa2 ωBm4π()(2sin2 ωt)(2)2 10 7 π 2 10 3 3.7 107 (.07 ) 20, 000 ( 0.3) sin2 ωt 62 4.7 10 sin ωtMg (0.08)9.8 0.8 Newtonsfmax4.7 106 5.9 106Mg0.8Neglecting losses:11CV2 Mv2 (t 0 ) Mgh22CVMC 25µf , M .08 kg, Vp 4000 voltsv(t 0 ) v(t 0 ) 70.7 meters/secondh (Initial velocity)2v( t 0 ) 255 meters2g(Maximum height)Courtesy of Hermann A. Haus and James R. Melcher. Used with permission.6.013 Electromagnetics and ApplicationsProf. Markus ZahnLecture 1Page 7 of 7

Prof. Markus Zahn Page 7 of 7 Courtesy of Hermann A. Haus and James R. Melcher. Used with permission. Title: Microsoft Word - 6.013 Lecture 1 - Zahn 2005.doc Author: Mary Created Date: