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Maxwell's Equations Explained for Electromagnetic Fields and Waves

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AIMST University
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EEEC6430310 ELECTROMAGNETIC FIELDS AND WAVES Maxwell’s Equation FACULTY OF ENGINEERING AND COMPUTER TECHNOLOGY BENG (HONS) IN ELECTRICALAND ELECTRONIC ENGINEERING Ravandran Muttiah BEng (Hons) MSc MIET
Maxwell’s Equation I. Maxwell’s Equation For Linear Media 𝛻 Γ— 𝑬 = βˆ’πœ‡ πœ•π‘― πœ•π‘‘ Faradayβ€² s Law 𝛻 Γ— 𝑯 = 𝑱 + πœ€ πœ•π‘¬ πœ•π‘‘ Ampereβ€² s Law 𝛻 βˆ™ 𝑬 = 𝜌f πœ€ Gaussβ€² s Law 𝛻 βˆ™ 𝑯 = 0 Gaussβ€² s Law where, 𝛻 = vector differential (del) operator 𝑬 = electric field intensity 𝑯 = magnetic field intensity 𝑱 = electric current density 𝜌f = free electric charge density 1
2 II. Pointing’s Theorem A. Power Flow And Electromagnetic Energy 𝛻 βˆ™ 𝑬 Γ— 𝑯 = 𝑯 βˆ™ 𝛻 Γ— 𝑬 βˆ’ 𝑬 βˆ™ 𝛻 Γ— 𝑯 = βˆ’πœ‡π‘― βˆ™ πœ•π‘― πœ•π‘‘ βˆ’ 𝑬 βˆ™ 𝑱 + πœ€ πœ•π‘¬ πœ•π‘‘ = βˆ’ πœ‡ 2 πœ• πœ•π‘‘ 𝑯 2 βˆ’ πœ€ 2 πœ• πœ•π‘‘ 𝑬 2 βˆ’ 𝑬 βˆ™ 𝑱 𝛻 βˆ™ 𝑬 Γ— 𝑯 + πœ• πœ•π‘‘ 1 2 πœ€ 𝑬 2 + 1 2 πœ‡ 𝑯 2 = βˆ’π‘¬ βˆ™ 𝑱
3 V 𝛻 βˆ™ 𝑬 Γ— 𝑯 d𝑉 = S 𝑬 Γ— 𝑯 βˆ™ d𝑺 S 𝑬 Γ— 𝑯 βˆ™ d𝒂 + d d𝑑 V 1 2 πœ€ 𝑬 2 + 1 2 πœ‡ 𝑯 2 d𝑉 = βˆ’ 𝑉 𝑬 βˆ™ 𝑱 d𝑉 𝑺 = 𝑬 Γ— 𝑯 Poynting Vector Watt m2 π‘Š = V 1 2 πœ€ 𝑬 2 + 1 2 πœ‡ 𝑯 2 d𝑉 Electromagnetic Stored Energy 𝑃d = V 𝑬 βˆ™ 𝑱 d𝑉 Power dissipated if 𝐽 βˆ™ 𝑬 > 0 e.g., 𝑱 = πœŽπ‘¬ β‡’ 𝐽 βˆ™ 𝑬 = 𝜎 𝑬 𝟐 Power source if 𝐽 βˆ™ 𝑬 < 0 𝑃out = S 𝑬 Γ— 𝑯 βˆ™ d𝒂 = S 𝑺 βˆ™ d𝒂 𝑃out + dπ‘Š d𝑑 = βˆ’π‘ƒd 𝑀e = 1 2 πœ€ 𝑬 2 Electric energy density in Joules m3 𝑀m = 1 2 πœ‡ 𝑯 2 Magnetic energy density in Joules m3
4 Figure 1: The circuit power into an N terminal network π‘˜=1 𝑁 π‘‰π‘˜πΌπ‘˜ equals the electromagnetic power flow into the surface surrounding the network, βˆ’ s 𝑬 Γ— 𝑯 Β· d𝑺 B. Power In Electric Circuits E H 𝑆 = 𝐸 Γ— 𝐻 𝑉1 𝑉2 𝑉3 π‘‰π‘βˆ’1 𝑉𝑁 𝐼1 𝐼2 𝐼3 πΌπ‘βˆ’1 𝐼𝑁 𝑆
5 Outside circuit elements C 𝑬 βˆ™ d𝒍 β‰ˆ 0, 𝛻 Γ— 𝑬 = 0 β‡’ 𝑬 = βˆ’π›»Π€ (Kirchoff’s Voltage Law π‘˜ π‘£π‘˜ = 0) 𝛻 Γ— 𝑯 = 𝑱 β‡’ 𝛻 βˆ™ 𝑱 = 0, S 𝑱 βˆ™ d𝑺 = 0 (Kirchoff’s Current Law π‘˜ π‘–π‘˜ = 0) 𝑃in = βˆ’ S 𝑬 Γ— 𝑯 βˆ™ d𝑺 = βˆ’ V 𝛻 βˆ™ 𝑬 Γ— 𝑯 d𝑉 𝛻 βˆ™ 𝑬 Γ— 𝑯 = 𝑯 βˆ™ 𝛻 Γ— 𝑬 βˆ’ 𝑬 βˆ™ 𝛻 Γ— 𝑯 = βˆ’π‘¬ βˆ™ 𝑱 = 𝛻Ѐ βˆ™ 𝑱 𝛻 βˆ™ 𝑱Ѐ = Ѐ𝛻 βˆ™ 𝑱 + 𝑱 βˆ™ 𝛻Ѐ 𝛻 βˆ™ 𝑬 Γ— 𝑯 = 𝑱 βˆ™ 𝛻Ѐ = 𝛻 βˆ™ Ѐ𝑱 𝑃in = βˆ’ V 𝛻 βˆ™ 𝑬 Γ— 𝑯 d𝑉 = βˆ’ V 𝛻 βˆ™ 𝑱Ѐ d𝑉 = βˆ’ S 𝑱Ѐ βˆ™ d𝑺 On 𝑆, Π€ = voltages on each wire, 𝑱 is non-zero only on wires. 𝑃in = βˆ’ S 𝑱Ѐ βˆ™ d𝑺 = βˆ’ π‘˜=1 𝑁 π‘£π‘˜ S 𝑱 βˆ™ d𝑺 = π‘˜=1 𝑁 π‘£π‘˜π‘–π‘˜ 0 0 βˆ’π‘–π‘˜
C. Complex Poynting’s Theorem (Sinusoidal Steady State, ejπœ”π‘‘ ) 6 𝑬 𝒓, 𝑑 = Re 𝑬 𝒓 ejπœ”π‘‘ = 1 2 π‘¬βˆ— 𝒓 ejπœ”π‘‘ + π‘¬βˆ— 𝒓 eβˆ’jπœ”π‘‘ 𝑯 𝒓, 𝑑 = Re 𝑯 𝒓 ejπœ”π‘‘ = 1 2 π‘―βˆ— 𝒓 ejπœ”π‘‘ + π‘―βˆ— 𝒓 eβˆ’jπœ”π‘‘ The real part of a complex number is one-half of the sum of the number and its complex conjugate
7 Maxwell’s Equations In Sinusoidal Steady State 𝛻 Γ— 𝑬 𝒓 = βˆ’jπœ”πœ‡π‘― 𝒓 𝛻 Γ— 𝑯 𝒓 = 𝑱 𝒓 + jπœ”πœ€ 𝒓 𝛻 βˆ™ 𝑬 𝒓 = 𝝆f 𝒓 πœ€ 𝛻 βˆ™ 𝑯 𝒓 = 0
8 𝑺 𝒓, 𝑑 = 𝑬 𝒓, 𝑑 Γ— 𝑯 𝒓, 𝑑 = 1 4 𝑬 𝒓 ejπœ”π‘‘ + 𝑬 𝒓 eβˆ’jπœ”π‘‘ Γ— 𝑯 𝒓 ejπœ”π‘‘ + π‘―βˆ— 𝒓 eβˆ’jπœ”π‘‘ = 1 4 𝑬 𝒓 Γ— 𝑯 𝒓 e2jπœ”π‘‘ +π‘¬βˆ— 𝒓 Γ— 𝑯 𝒓 + 𝑬 𝒓 Γ— π‘―βˆ— 𝒓 + π‘¬βˆ— 𝒓 Γ— π‘―βˆ— 𝒓 eβˆ’2jπœ”π‘‘ 𝑺 = 1 4 π‘¬βˆ— 𝒓 Γ— 𝑯 𝒓 + 𝑬 𝒓 Γ— π‘―βˆ— 𝒓 = 1 2 Re 𝑬 𝒓 Γ— π‘―βˆ— 𝒓 = 1 2 Re π‘¬βˆ— 𝒓 Γ— 𝑯 𝒓 (A complex number plus its complex conjugate is twice the real part of that number)
9 𝑺 = 1 2 Re 𝑬 𝒓 Γ— π‘―βˆ— 𝒓 𝛻 βˆ™ 𝑺 = 𝛻 βˆ™ 1 2 𝑬 𝒓 Γ— π‘―βˆ— 𝒓 = 1 2 π‘―βˆ— 𝒓 βˆ™ 𝛻 Γ— 𝑬 𝒓 βˆ’ 𝑬 𝒓 βˆ™ 𝛻 Γ— π‘―βˆ— 𝒓 = 1 2 π‘―βˆ— 𝒓 βˆ’jπœ”πœ‡π‘― 𝒓 βˆ’ 𝑬 𝒓 βˆ™ π‘±βˆ— 𝒓 βˆ’ jπœ”πœ€π‘¬βˆ— 𝒓 = 1 2 βˆ’jπœ”πœ‡ 𝑯 𝒓 2 + jπœ”πœ€ 𝑬 𝒓 2 βˆ’ 1 2 𝑬 𝒓 βˆ™ π‘±βˆ— 𝒓 𝑀m = 1 4 πœ‡ 𝑯 𝒓 2 , 𝑀e = 1 4 πœ€ 𝑬 𝒓 2 𝑷d = 1 2 𝑬 𝒓 βˆ™ π‘±βˆ— 𝒓 𝛻 βˆ™ 𝑺 + 2jπœ” 𝑀m βˆ’ 𝑀e = βˆ’π‘·d
III. Transverse Electromagnetic Waves (𝑱 = 0, 𝜌f = 0) 10 The great success of Maxwell’s equations lies partly in the simple prediction of electromagnetic waves and their simple characterization of materials in terms of conductivity 𝜎, permittivity πœ€, and permeability πœ‡. In vacuum we find 𝜎 = 0, πœ€ = πœ€o and πœ‡ = πœ‡o. Therefore, 𝑱 = πœŽπ‘¬ = 0 and 𝜌f = 0. A. Wave Equation 𝛻 Γ— 𝑬 = βˆ’πœ‡ πœ•π‘― πœ•π‘‘ 𝛻 Γ— 𝑯 = πœ€ πœ•π‘¬ πœ•π‘‘ 𝛻 βˆ™ 𝑬 = 0 𝛻 βˆ™ 𝑯 = 0 In contrast the nano-structure of media can be quite complex and requires quantum mechanics and for its full explanation. Fortunately, simple classical approximations to atoms and molecules suffice to understand the origin of 𝜎, πœ€ and πœ‡.
11 𝛻 Γ— 𝛻 Γ— 𝑬 = βˆ’πœ‡ πœ• πœ•π‘‘ 𝛻 Γ— 𝐻 = βˆ’πœ‡ πœ• πœ•π‘‘ πœ€ πœ•π‘¬ πœ•π‘‘ 𝛻 Γ— 𝛻 Γ— 𝑬 = 𝛻 𝛻 βˆ™ 𝑬 βˆ’ 𝛻2 𝑬 = βˆ’πœ€πœ‡ πœ•2 𝑬 πœ•π‘‘2 Wave equation, 𝛻2 𝑬 = 1 𝑐2 πœ•2 𝑬 πœ•π‘‘2 where, 𝑐 = 1 πœ€πœ‡ is the velocity of the electromagnetic wave. In free space (i.e. vacuum), πœ‡ = πœ‡o = 4Ο€ Γ— 10βˆ’7 henries m and πœ€ = πœ€o β‰ˆ 10βˆ’9 36Ο€ farads m , which leads to 𝑐o = 1 πœ€oπœ‡o β‰ˆ 3 Γ— 108 m s . Similarly, 𝛻 Γ— 𝛻 Γ— 𝑯 = 𝛻 𝛻 βˆ™ 𝑯 βˆ’ 𝛻2 𝑯 = πœ€ πœ• πœ•π‘‘ 𝛻 Γ— 𝑬 = βˆ’πœ€πœ‡ πœ•2 𝑯 πœ•π‘‘2 𝛻2 𝑯 = 1 𝑐2 πœ•2 𝑯 πœ•π‘‘2 , 𝑐 = 1 πœ€πœ‡ 0 0
12 x z y πœ€1, πœ‡1 𝐸π‘₯βˆ’ = Re 𝑬π‘₯βˆ’ 𝑧 ejπœ”π‘‘ π»π‘¦βˆ’ = Re 𝑯yβˆ’ 𝑧 ejπœ”π‘‘ 𝐻𝑦+ = Re 𝑯y+ 𝑧 ejπœ”π‘‘ 𝐸π‘₯+ = Re 𝑬π‘₯+ 𝑧 ejπœ”π‘‘ 𝐾π‘₯+ = Re 𝑲0 𝑧 ejπœ”π‘‘ πœ€2, πœ‡2 B. Plane Waves 𝐸π‘₯ 𝑧, 𝑑 = Re 𝑬π‘₯ 𝑧 ejπœ”π‘‘ d2 𝑬π‘₯ d𝑧2 = βˆ’ πœ”2 𝑐2 𝑬π‘₯ d2𝑬π‘₯ d𝑧2 + π‘˜2𝑬π‘₯ = 0
13 where we have, π‘˜2 = πœ”2 𝑐2 = πœ”2 πœ€πœ‡ π‘˜ = 2Ο€ πœ† is the wavenumber, πœ† is the wavelength π‘˜ = Β± πœ” 𝑐 β‡’ πœ” = π‘˜π‘ πœ” = 2π𝑓 = 2Ο€ πœ† 𝑐 β‡’ π‘“πœ† = 𝑐 𝑬π‘₯ = 𝐴1ejπ‘˜π‘§ + 𝐴2eβˆ’jπ‘˜π‘§ 𝐸π‘₯ = Re 𝐴1ej πœ”π‘‘+π‘˜π‘§ + 𝐴2ej πœ”π‘‘βˆ’π‘˜π‘§ For the wave in the βˆ’π‘§ direction we have: πœ”π‘‘ + π‘˜π‘§ = constant, πœ”d𝑑 + π‘˜d𝑧 = 0 d𝑧 d𝑑 = βˆ’ πœ” π‘˜ = βˆ’π‘ travelling wave in the βˆ’ 𝑧 direction travelling wave in the +𝑧 direction
14 For the wave in the +𝑧 direction we have: πœ”π‘‘ βˆ’ 𝑧 = constant, πœ”d𝑑 βˆ’ π‘˜d𝑧 = 0, d𝑧 d𝑑 = πœ” π‘˜ = +𝑐 𝑬π‘₯ 𝑧 = 𝑬π‘₯+eβˆ’jπ‘˜π‘§ 𝑧 > 0 𝑬π‘₯βˆ’ejπ‘˜π‘§ 𝑧 < 0 𝛻 Γ— 𝑬 = βˆ’πœ‡ πœ•π‘― πœ•π‘‘ β‡’ d𝑬π‘₯ d𝑧 = βˆ’jπœ”πœ‡π‘―π‘¦ β‡’ 𝑯𝑦 = βˆ’ 1 jπœ”πœ‡ d𝑬π‘₯ d𝑧 𝛻 Γ— 𝑯 = πœ€ πœ•π‘¬ πœ•π‘‘ β‡’ d𝑯𝑦 d𝑧 = βˆ’jπœ”πœ€π‘¬π‘₯ 𝑯𝑦 = π‘˜ πœ”πœ‡ 𝑬π‘₯+eβˆ’jπ‘˜π‘§ 𝑧 > 0 βˆ’ π‘˜ πœ”πœ‡ 𝑬π‘₯βˆ’ejπ‘˜π‘§ 𝑧 < 0 π‘˜ πœ”πœ‡ = πœ” πœ€πœ‡ πœ”πœ‡ = πœ€ πœ‡ , πœ‚ = πœ‡ πœ€ is the wave impedance 𝑯𝑦 = 𝑬π‘₯+ πœ‚ eβˆ’jπ‘˜π‘§ 𝑧 > 0 βˆ’ 𝑬π‘₯βˆ’ πœ‚ ejπ‘˜π‘§ 𝑧 < 0
15 Now we look at the boundary conditions: 𝑬π‘₯ 𝑧 = 0+, 𝑑 = 𝑬π‘₯ 𝑧 = 0βˆ’, 𝑑 β‡’ 𝑬π‘₯+ = 𝑬π‘₯βˆ’ 𝑯𝑦 𝑧 = 0βˆ’, 𝑑 βˆ’ 𝑯𝑦 𝑧 = 0+, 𝑑 = 𝑲π‘₯ 𝑧 = 0, 𝑑 β‡’ βˆ’π‘¬π‘₯βˆ’ βˆ’ 𝑬π‘₯+ πœ‚ = 𝑲0 𝑬π‘₯+ = 𝑬π‘₯βˆ’ = βˆ’ πœ‚π‘²0 2 𝑬π‘₯ 𝑧 = βˆ’ πœ‚π‘²0 2 eβˆ’jπ‘˜π‘§ 𝑧 > 0 βˆ’ πœ‚π‘²0 2 ejπ‘˜π‘§ 𝑧 < 0 𝑯𝑦 𝑧 = βˆ’ 𝑲0 2 eβˆ’jπ‘˜π‘§ 𝑧 > 0 𝑲0 2 ejπ‘˜π‘§ 𝑧 < 0 𝑺 = 1 2 𝑬 Γ— π‘―βˆ— = πœ‚πΎ0 2 8 π’Šπ‘§ 𝑧 > 0 βˆ’ πœ‚πΎ0 2 8 π’Šπ‘§ 𝑧 < 0 (𝑲0 real)
16 𝐸π‘₯ 𝑧, 𝑑 = Re 𝑬π‘₯ 𝑧 ejπœ”π‘‘ = βˆ’ πœ‚π‘²0 2 cos πœ”π‘‘ βˆ’ π‘˜π‘§ 𝑧 > 0 βˆ’ πœ‚π‘²0 2 cos πœ”π‘‘ + π‘˜π‘§ 𝑧 < 0 𝐻𝑦 𝑧, 𝑑 = Re 𝑯𝑦 𝑧 ejπœ”π‘‘ = βˆ’ 𝑲0 2 cos πœ”π‘‘ βˆ’ π‘˜π‘§ 𝑧 > 0 𝑲0 2 cos πœ”π‘‘ + π‘˜π‘§ 𝑧 < 0 𝑆𝑧 = 𝐸π‘₯𝐻𝑦 = πœ‚πΎ0 2 4 cos2 πœ”π‘‘ βˆ’ π‘˜π‘§ 𝑧 > 0 βˆ’ πœ‚πΎ0 2 4 cos2 πœ”π‘‘ + π‘˜π‘§ 𝑧 < 0 𝑆𝑧 = πœ‚πΎ0 2 8 𝑧 > 0 βˆ’ πœ‚πΎ0 2 8 𝑧 < 0
17 C. Normal Incidence Onto A Perfect Conductor Incident Fields: 𝑬i 𝑧, 𝑑 = Re 𝑬iej πœ”π‘‘βˆ’π‘˜π‘§ π’Šπ‘₯ 𝑯i 𝑧, 𝑑 = Re 𝑬i πœ‚ ej πœ”π‘‘βˆ’π‘˜π‘§ π’Šπ‘¦ Reflected Fields: 𝑬r 𝑧, 𝑑 = Re 𝑬rej πœ”π‘‘+π‘˜π‘§ π’Šπ‘₯ 𝑯r 𝑧, 𝑑 = Re βˆ’ 𝑬r πœ‚ ej πœ”π‘‘+π‘˜π‘§ π’Šπ‘¦ π‘˜ = πœ” πœ€πœ‡, πœ‚ = πœ‡ πœ€ The boundary conditions require that, 𝐸π‘₯ 𝑧 = 0, 𝑑 = 𝐸π‘₯, i 𝑧 = 0, 𝑑 + 𝐸π‘₯, r 𝑧 = 0, 𝑑 = 0 𝑬i + 𝑬r = 0 β‡’ 𝑬r = βˆ’π‘¬i
18 Figure 2: Time varying electromagnetic phenomena differ only in the scaling of time (frequency) and size (wavelength). In linear dielectric media the frequency and wavelength are related as 𝑓λ = 𝑐 (πœ” = π‘˜π‘), where 𝑐 = 1 πœ€πœ‡ is the velocity of light in the medium. sin π‘˜z πœ† = 2Ο€ π‘˜ 𝑇 = 2Ο€ πœ” 2Ο€ π‘˜ 2Ο€ πœ” 𝑧 𝑑 Ο€ πœ” Ο€ π‘˜ βˆ’1 βˆ’1 1 1 sin πœ”π‘‘ 0 102 104 106 108 1010 1012 1014 1016 1018 1020 Circuit Theory Microwaves f (Hz) Visible Light Ultraviolet X-Rays Gamma Rays Power Infrared (Heat) Radio and Television AM FM 3 Γ— 106 Ξ» meters 3 Γ— 104 3 Γ— 102 3 3 Γ— 10βˆ’2 3 Γ— 10βˆ’4 3 Γ— 10βˆ’6 3 Γ— 10βˆ’8 3 Γ— 10βˆ’10 3 Γ— 10βˆ’12
19 For 𝑬i = 𝐸i real we have: 𝐸π‘₯ 𝑧, 𝑑 = 𝐸π‘₯, i 𝑧, 𝑑 + 𝐸π‘₯, r 𝑧, 𝑑 = Re 𝑬i eβˆ’jπ‘˜π‘§ βˆ’ ejπ‘˜π‘§ ejπœ”π‘‘ = 2𝐸i sin π‘˜π‘§ sin πœ”π‘‘ 𝐻𝑦 𝑧, 𝑑 = 𝐻𝑦, i 𝑧, 𝑑 + 𝐻𝑦, r 𝑧, 𝑑 = Re 𝑬i πœ‚ eβˆ’jπ‘˜π‘§ βˆ’ ejπ‘˜π‘§ ejπœ”π‘‘ = 2𝐸i πœ‚ cos π‘˜π‘§ cos πœ”π‘‘ 𝐾𝑧 𝑧 = 0, 𝑑 = 𝐻𝑦 𝑧 = 0, 𝑑 = 2𝐸i πœ‚ cos πœ”π‘‘ Radiation pressure in free space πœ‡ = πœ‡o, πœ€ = πœ€o Force𝑧 Area 𝑧=0 = 1 2 𝑲 Γ— πœ‡o𝑯 = 1 2 πœ‡o𝐾π‘₯𝐻𝑦 𝑧=0 π’Šπ‘§ = 1 2 πœ‡o𝐻𝑦 2 𝑧 = 0 π’Šπ‘§ = 2πœ‡o𝐸i 2 πœ‚o 2 cos2 πœ”π‘‘ π’Šπ‘§ = 2πœ‡o πœ‚o πœ€o 𝐸i 2 cos2 πœ”π‘‘ π’Šπ‘§ = 2πœ€o 𝐸i 2 cos2 πœ”π‘‘ π’Šπ‘§
20 Figure 3: A uniform plane wave normally incident upon a perfect conductor has zero electric field at the conducting surface thus requiring a reflected wave. The source of this reflected wave is the surface current at 𝑧 = 0, which equals the magnetic field there. The total electric and magnetic fields are 90Β° out of phase in time and space. x z y 𝐸i = Re 𝑬iej πœ”π‘‘βˆ’π‘˜π‘§ π’Šπ‘₯ 𝐻i = Re 𝑬i 𝑍o ej πœ”π‘‘βˆ’π‘˜π‘§ π’Šπ‘¦ π‘˜ = πœ” 𝑐 π’Šπ‘§ 𝐻r = Re βˆ’ 𝑬r 𝑍o ej πœ”π‘‘+π‘˜π‘§ π’Šπ‘¦ 𝐸r = Re 𝑬rej πœ”π‘‘+π‘˜π‘§ π’Šπ‘₯ π‘˜r = βˆ’ πœ” c1 π’Šπ‘§ πœ€o, πœ‡o πœ‚o = 𝑍o= πœ‡o πœ€o 𝐻𝑦 𝑠, 𝑑 = 2𝑬i 𝑍o cos π‘˜π‘  cos πœ”π‘‘ 𝐸π‘₯ 𝑠, 𝑑 = 2𝑬isin π‘˜π‘  sin πœ”π‘‘ 𝜎 β†’ ∞
21 Figure 4: A uniform plane wave normally incident upon a dielectric interface separating two different materials has part of its power reflected and part transmitted. IV. Normal Incidence Onto A Dielectric x z y πœ€1, πœ‡1 πœ‚1 = 𝑍1 = πœ‡1 πœ€1 , 𝑐1 = 1 πœ€1πœ‡1 πœ€2, πœ‡2 𝑍2 = πœ‡2 πœ€2 , 𝑐2 = 1 πœ€2πœ‡2 𝐸i = Re 𝑬iej πœ”π‘‘βˆ’π‘˜1𝑧 π’Šπ‘₯ 𝐸t = Re 𝑬tej πœ”π‘‘βˆ’π‘˜2𝑧 π’Šπ‘₯ 𝐻t = Re 𝑬t 𝑍2 ej πœ”π‘‘βˆ’π‘˜2𝑧 π’Šπ‘¦ 𝐻r = Re βˆ’ 𝑬r 𝑍1 ej πœ”π‘‘+π‘˜1𝑧 π’Šπ‘¦ 𝐻i = Re 𝑬i 𝑍1 ej πœ”π‘‘βˆ’π‘˜1𝑧 π’Šπ‘¦ 𝐸r = Re 𝑬rej πœ”π‘‘+π‘˜1𝑧 π’Šπ‘₯ π‘˜i = π‘˜1π’Šπ‘§ = πœ” 𝑐1 π’Šπ‘§ π‘˜r = βˆ’π‘˜1π’Šπ‘§ = βˆ’πœ” 𝑐1 π’Šπ‘§ π‘˜t = π‘˜2π’Šπ‘§ = πœ” 𝑐2 π’Šπ‘§
22 𝑬i 𝑧, 𝑑 = Re 𝑬iej πœ”π‘‘βˆ’π‘˜1𝑧 𝑖π‘₯ , π‘˜1 = πœ” πœ€1πœ‡1 𝑯i 𝑧, 𝑑 = Re 𝑯i πœ‚1 ej πœ”π‘‘βˆ’π‘˜1𝑧 𝑖𝑦 , πœ‚1 = πœ” πœ‡1 πœ€1 𝑬r 𝑧, 𝑑 = Re 𝑬rej πœ”π‘‘+π‘˜1𝑧 𝑖π‘₯ 𝑯r 𝑧, 𝑑 = Re βˆ’ 𝑬r πœ‚1 ej πœ”π‘‘+π‘˜1𝑧 𝑖𝑦 𝑬t 𝑧, 𝑑 = Re 𝑬tej πœ”π‘‘βˆ’π‘˜2𝑧 𝑖π‘₯ , π‘˜2 = πœ” πœ€2πœ‡2 𝑯t 𝑧, 𝑑 = Re 𝑬t πœ‚2 ej πœ”π‘‘βˆ’π‘˜2𝑧 𝑖𝑦 , πœ‚2 = πœ” πœ‡2 πœ€2
23 𝐸π‘₯ 𝑧 = 0βˆ’ = 𝐸π‘₯ 𝑧 = 0+ β‡’ 𝑬i + 𝑬r = 𝑬t 𝐻𝑦 𝑧 = 0βˆ’ = 𝐻𝑦 𝑧 = 0+ β‡’ 𝑬i βˆ’ 𝑬r πœ‚1 = 𝑬t πœ‚2 𝑅 = 𝐸r 𝐸i = πœ‚2 βˆ’ πœ‚1 πœ‚1 + πœ‚2 is the Reflection coefficient 𝑇 = 𝐸t 𝐸i = 2πœ‚2 πœ‚1 + πœ‚2 is the Transmission coefficient 1 + 𝑅 = 𝑇
24 𝑆𝑧, i = 1 2 Re 𝑬π‘₯ 𝑧 𝑯𝑦 βˆ— 𝑧 = 1 2πœ‚1 Re 𝑬ieβˆ’jπ‘˜1𝑧 + 𝑬rejπ‘˜1𝑧 𝑬i βˆ— ejπ‘˜1𝑧 βˆ’ 𝑬r βˆ— eβˆ’jπ‘˜1𝑧 = 1 2πœ‚1 𝑬i 2 βˆ’ 𝑬r 2 + 1 2πœ‚1 Re 𝑬r𝑬i βˆ— e2jπ‘˜1𝑧 βˆ’ 𝑬r βˆ— 𝑬ieβˆ’2jπ‘˜1𝑧 = 1 2πœ‚1 𝑬i 2 βˆ’ 𝑬r 2 = 𝑬i 2 2πœ‚1 1 βˆ’ 𝑅2 𝑆𝑧, t = 1 2πœ‚2 𝑬t 2 = 𝑬i 2 𝑇2 2πœ‚2 = 𝑬i 2 1 βˆ’ 𝑅2 2πœ‚2 = 𝑆𝑧, i pure imaginary
25 V. Lossy Dielectrics, 𝑱 = πœŽπ‘¬ Ampereβ€² s Law: 𝛻 Γ— 𝑯 = 𝑱 + πœ€ πœ•π‘¬ πœ•π‘‘ = πœŽπ‘¬ + πœ€ πœ•π‘¬ πœ•π‘‘ For ejπœ”π‘‘ fields, 𝛻 Γ— 𝑯 = jπœ”πœ€ + 𝜎 𝑬 = jπœ”πœ€ 1 + 𝜎 jπœ”πœ€ 𝑬 where, 𝜎 = conductivity of a medium ( Siemens m )
26 Define complex permittivity by 𝜺 = πœ€ 1 + 𝜎 jπœ”πœ€ . Then complex amplitude solutions are the same as real amplitude solutions if we replace πœ€ by 𝜺: π‘˜ = πœ” πœΊπœ‡ πœ‚ = πœ‡ 𝜺 = πœ‡ πœ€ 1 + 𝜎 jπœ”πœ€ π‘˜ = πœ” πœ€πœ‡ 1 + 𝜎 jπœ”πœ€
27 A. Low Loss Limit: 𝜎 πœ”πœ€ β‰ͺ 1 π‘˜ = πœ” πœ€πœ‡ 1 + 𝜎 jπœ”πœ€ β‰ˆ πœ” πœ€πœ‡ 1 + 1 2 𝜎 jπœ”πœ€ β‰ˆ πœ” πœ€πœ‡ βˆ’ j𝜎 2 πœ‡ πœ€ β‰ˆ πœ” πœ€πœ‡ βˆ’ j πœŽπœ‚ 2 eβˆ’jπ‘˜π‘§ = eβˆ’jπœ” πœ€πœ‡π‘§ e βˆ’j βˆ’j πœŽπœ‚ 2 𝑧 = eβˆ’jπœ” πœ€πœ‡π‘§ eβˆ’ πœŽπœ‚ 2 𝑧 slow exponential decay
28 B. Large Loss Limit: 𝜎 πœ”πœ€ ≫ 1 π‘˜ = πœ” πœ€πœ‡ 1 + 𝜎 jπœ”πœ€ β‰ˆ πœ” πœ€πœ‡ 𝜎 πœ”πœ€ βˆ’j we know that, πœ” πœ” = πœ” β‰ˆ πœ”πœ‡πœŽ 2 1 βˆ’ j β‰ˆ 1 βˆ’ j 𝛿 𝛿 = 2 πœ”πœ‡πœŽ is the skin depth eβˆ’jπ‘˜π‘§ = e βˆ’j 1βˆ’j 𝑧 𝛿 = e βˆ’j 𝑧 𝛿 e βˆ’ 𝑧 𝛿 fast exponential decay 1 βˆ’ j 2
(1) Markus Zahn, Electromagnetics and Applications, Massachusetts Institute of Technology: MIT Open Course Ware, 2005. References 29

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  • 2. Maxwell’s Equation I. Maxwell’s Equation For Linear Media 𝛻 Γ— 𝑬 = βˆ’πœ‡ πœ•π‘― πœ•π‘‘ Faradayβ€² s Law 𝛻 Γ— 𝑯 = 𝑱 + πœ€ πœ•π‘¬ πœ•π‘‘ Ampereβ€² s Law 𝛻 βˆ™ 𝑬 = 𝜌f πœ€ Gaussβ€² s Law 𝛻 βˆ™ 𝑯 = 0 Gaussβ€² s Law where, 𝛻 = vector differential (del) operator 𝑬 = electric field intensity 𝑯 = magnetic field intensity 𝑱 = electric current density 𝜌f = free electric charge density 1
  • 3. 2 II. Pointing’s Theorem A. Power Flow And Electromagnetic Energy 𝛻 βˆ™ 𝑬 Γ— 𝑯 = 𝑯 βˆ™ 𝛻 Γ— 𝑬 βˆ’ 𝑬 βˆ™ 𝛻 Γ— 𝑯 = βˆ’πœ‡π‘― βˆ™ πœ•π‘― πœ•π‘‘ βˆ’ 𝑬 βˆ™ 𝑱 + πœ€ πœ•π‘¬ πœ•π‘‘ = βˆ’ πœ‡ 2 πœ• πœ•π‘‘ 𝑯 2 βˆ’ πœ€ 2 πœ• πœ•π‘‘ 𝑬 2 βˆ’ 𝑬 βˆ™ 𝑱 𝛻 βˆ™ 𝑬 Γ— 𝑯 + πœ• πœ•π‘‘ 1 2 πœ€ 𝑬 2 + 1 2 πœ‡ 𝑯 2 = βˆ’π‘¬ βˆ™ 𝑱
  • 4. 3 V 𝛻 βˆ™ 𝑬 Γ— 𝑯 d𝑉 = S 𝑬 Γ— 𝑯 βˆ™ d𝑺 S 𝑬 Γ— 𝑯 βˆ™ d𝒂 + d d𝑑 V 1 2 πœ€ 𝑬 2 + 1 2 πœ‡ 𝑯 2 d𝑉 = βˆ’ 𝑉 𝑬 βˆ™ 𝑱 d𝑉 𝑺 = 𝑬 Γ— 𝑯 Poynting Vector Watt m2 π‘Š = V 1 2 πœ€ 𝑬 2 + 1 2 πœ‡ 𝑯 2 d𝑉 Electromagnetic Stored Energy 𝑃d = V 𝑬 βˆ™ 𝑱 d𝑉 Power dissipated if 𝐽 βˆ™ 𝑬 > 0 e.g., 𝑱 = πœŽπ‘¬ β‡’ 𝐽 βˆ™ 𝑬 = 𝜎 𝑬 𝟐 Power source if 𝐽 βˆ™ 𝑬 < 0 𝑃out = S 𝑬 Γ— 𝑯 βˆ™ d𝒂 = S 𝑺 βˆ™ d𝒂 𝑃out + dπ‘Š d𝑑 = βˆ’π‘ƒd 𝑀e = 1 2 πœ€ 𝑬 2 Electric energy density in Joules m3 𝑀m = 1 2 πœ‡ 𝑯 2 Magnetic energy density in Joules m3
  • 5. 4 Figure 1: The circuit power into an N terminal network π‘˜=1 𝑁 π‘‰π‘˜πΌπ‘˜ equals the electromagnetic power flow into the surface surrounding the network, βˆ’ s 𝑬 Γ— 𝑯 Β· d𝑺 B. Power In Electric Circuits E H 𝑆 = 𝐸 Γ— 𝐻 𝑉1 𝑉2 𝑉3 π‘‰π‘βˆ’1 𝑉𝑁 𝐼1 𝐼2 𝐼3 πΌπ‘βˆ’1 𝐼𝑁 𝑆
  • 6. 5 Outside circuit elements C 𝑬 βˆ™ d𝒍 β‰ˆ 0, 𝛻 Γ— 𝑬 = 0 β‡’ 𝑬 = βˆ’π›»Π€ (Kirchoff’s Voltage Law π‘˜ π‘£π‘˜ = 0) 𝛻 Γ— 𝑯 = 𝑱 β‡’ 𝛻 βˆ™ 𝑱 = 0, S 𝑱 βˆ™ d𝑺 = 0 (Kirchoff’s Current Law π‘˜ π‘–π‘˜ = 0) 𝑃in = βˆ’ S 𝑬 Γ— 𝑯 βˆ™ d𝑺 = βˆ’ V 𝛻 βˆ™ 𝑬 Γ— 𝑯 d𝑉 𝛻 βˆ™ 𝑬 Γ— 𝑯 = 𝑯 βˆ™ 𝛻 Γ— 𝑬 βˆ’ 𝑬 βˆ™ 𝛻 Γ— 𝑯 = βˆ’π‘¬ βˆ™ 𝑱 = 𝛻Ѐ βˆ™ 𝑱 𝛻 βˆ™ 𝑱Ѐ = Ѐ𝛻 βˆ™ 𝑱 + 𝑱 βˆ™ 𝛻Ѐ 𝛻 βˆ™ 𝑬 Γ— 𝑯 = 𝑱 βˆ™ 𝛻Ѐ = 𝛻 βˆ™ Ѐ𝑱 𝑃in = βˆ’ V 𝛻 βˆ™ 𝑬 Γ— 𝑯 d𝑉 = βˆ’ V 𝛻 βˆ™ 𝑱Ѐ d𝑉 = βˆ’ S 𝑱Ѐ βˆ™ d𝑺 On 𝑆, Π€ = voltages on each wire, 𝑱 is non-zero only on wires. 𝑃in = βˆ’ S 𝑱Ѐ βˆ™ d𝑺 = βˆ’ π‘˜=1 𝑁 π‘£π‘˜ S 𝑱 βˆ™ d𝑺 = π‘˜=1 𝑁 π‘£π‘˜π‘–π‘˜ 0 0 βˆ’π‘–π‘˜
  • 7. C. Complex Poynting’s Theorem (Sinusoidal Steady State, ejπœ”π‘‘ ) 6 𝑬 𝒓, 𝑑 = Re 𝑬 𝒓 ejπœ”π‘‘ = 1 2 π‘¬βˆ— 𝒓 ejπœ”π‘‘ + π‘¬βˆ— 𝒓 eβˆ’jπœ”π‘‘ 𝑯 𝒓, 𝑑 = Re 𝑯 𝒓 ejπœ”π‘‘ = 1 2 π‘―βˆ— 𝒓 ejπœ”π‘‘ + π‘―βˆ— 𝒓 eβˆ’jπœ”π‘‘ The real part of a complex number is one-half of the sum of the number and its complex conjugate
  • 8. 7 Maxwell’s Equations In Sinusoidal Steady State 𝛻 Γ— 𝑬 𝒓 = βˆ’jπœ”πœ‡π‘― 𝒓 𝛻 Γ— 𝑯 𝒓 = 𝑱 𝒓 + jπœ”πœ€ 𝒓 𝛻 βˆ™ 𝑬 𝒓 = 𝝆f 𝒓 πœ€ 𝛻 βˆ™ 𝑯 𝒓 = 0
  • 9. 8 𝑺 𝒓, 𝑑 = 𝑬 𝒓, 𝑑 Γ— 𝑯 𝒓, 𝑑 = 1 4 𝑬 𝒓 ejπœ”π‘‘ + 𝑬 𝒓 eβˆ’jπœ”π‘‘ Γ— 𝑯 𝒓 ejπœ”π‘‘ + π‘―βˆ— 𝒓 eβˆ’jπœ”π‘‘ = 1 4 𝑬 𝒓 Γ— 𝑯 𝒓 e2jπœ”π‘‘ +π‘¬βˆ— 𝒓 Γ— 𝑯 𝒓 + 𝑬 𝒓 Γ— π‘―βˆ— 𝒓 + π‘¬βˆ— 𝒓 Γ— π‘―βˆ— 𝒓 eβˆ’2jπœ”π‘‘ 𝑺 = 1 4 π‘¬βˆ— 𝒓 Γ— 𝑯 𝒓 + 𝑬 𝒓 Γ— π‘―βˆ— 𝒓 = 1 2 Re 𝑬 𝒓 Γ— π‘―βˆ— 𝒓 = 1 2 Re π‘¬βˆ— 𝒓 Γ— 𝑯 𝒓 (A complex number plus its complex conjugate is twice the real part of that number)
  • 10. 9 𝑺 = 1 2 Re 𝑬 𝒓 Γ— π‘―βˆ— 𝒓 𝛻 βˆ™ 𝑺 = 𝛻 βˆ™ 1 2 𝑬 𝒓 Γ— π‘―βˆ— 𝒓 = 1 2 π‘―βˆ— 𝒓 βˆ™ 𝛻 Γ— 𝑬 𝒓 βˆ’ 𝑬 𝒓 βˆ™ 𝛻 Γ— π‘―βˆ— 𝒓 = 1 2 π‘―βˆ— 𝒓 βˆ’jπœ”πœ‡π‘― 𝒓 βˆ’ 𝑬 𝒓 βˆ™ π‘±βˆ— 𝒓 βˆ’ jπœ”πœ€π‘¬βˆ— 𝒓 = 1 2 βˆ’jπœ”πœ‡ 𝑯 𝒓 2 + jπœ”πœ€ 𝑬 𝒓 2 βˆ’ 1 2 𝑬 𝒓 βˆ™ π‘±βˆ— 𝒓 𝑀m = 1 4 πœ‡ 𝑯 𝒓 2 , 𝑀e = 1 4 πœ€ 𝑬 𝒓 2 𝑷d = 1 2 𝑬 𝒓 βˆ™ π‘±βˆ— 𝒓 𝛻 βˆ™ 𝑺 + 2jπœ” 𝑀m βˆ’ 𝑀e = βˆ’π‘·d
  • 11. III. Transverse Electromagnetic Waves (𝑱 = 0, 𝜌f = 0) 10 The great success of Maxwell’s equations lies partly in the simple prediction of electromagnetic waves and their simple characterization of materials in terms of conductivity 𝜎, permittivity πœ€, and permeability πœ‡. In vacuum we find 𝜎 = 0, πœ€ = πœ€o and πœ‡ = πœ‡o. Therefore, 𝑱 = πœŽπ‘¬ = 0 and 𝜌f = 0. A. Wave Equation 𝛻 Γ— 𝑬 = βˆ’πœ‡ πœ•π‘― πœ•π‘‘ 𝛻 Γ— 𝑯 = πœ€ πœ•π‘¬ πœ•π‘‘ 𝛻 βˆ™ 𝑬 = 0 𝛻 βˆ™ 𝑯 = 0 In contrast the nano-structure of media can be quite complex and requires quantum mechanics and for its full explanation. Fortunately, simple classical approximations to atoms and molecules suffice to understand the origin of 𝜎, πœ€ and πœ‡.
  • 12. 11 𝛻 Γ— 𝛻 Γ— 𝑬 = βˆ’πœ‡ πœ• πœ•π‘‘ 𝛻 Γ— 𝐻 = βˆ’πœ‡ πœ• πœ•π‘‘ πœ€ πœ•π‘¬ πœ•π‘‘ 𝛻 Γ— 𝛻 Γ— 𝑬 = 𝛻 𝛻 βˆ™ 𝑬 βˆ’ 𝛻2 𝑬 = βˆ’πœ€πœ‡ πœ•2 𝑬 πœ•π‘‘2 Wave equation, 𝛻2 𝑬 = 1 𝑐2 πœ•2 𝑬 πœ•π‘‘2 where, 𝑐 = 1 πœ€πœ‡ is the velocity of the electromagnetic wave. In free space (i.e. vacuum), πœ‡ = πœ‡o = 4Ο€ Γ— 10βˆ’7 henries m and πœ€ = πœ€o β‰ˆ 10βˆ’9 36Ο€ farads m , which leads to 𝑐o = 1 πœ€oπœ‡o β‰ˆ 3 Γ— 108 m s . Similarly, 𝛻 Γ— 𝛻 Γ— 𝑯 = 𝛻 𝛻 βˆ™ 𝑯 βˆ’ 𝛻2 𝑯 = πœ€ πœ• πœ•π‘‘ 𝛻 Γ— 𝑬 = βˆ’πœ€πœ‡ πœ•2 𝑯 πœ•π‘‘2 𝛻2 𝑯 = 1 𝑐2 πœ•2 𝑯 πœ•π‘‘2 , 𝑐 = 1 πœ€πœ‡ 0 0
  • 13. 12 x z y πœ€1, πœ‡1 𝐸π‘₯βˆ’ = Re 𝑬π‘₯βˆ’ 𝑧 ejπœ”π‘‘ π»π‘¦βˆ’ = Re 𝑯yβˆ’ 𝑧 ejπœ”π‘‘ 𝐻𝑦+ = Re 𝑯y+ 𝑧 ejπœ”π‘‘ 𝐸π‘₯+ = Re 𝑬π‘₯+ 𝑧 ejπœ”π‘‘ 𝐾π‘₯+ = Re 𝑲0 𝑧 ejπœ”π‘‘ πœ€2, πœ‡2 B. Plane Waves 𝐸π‘₯ 𝑧, 𝑑 = Re 𝑬π‘₯ 𝑧 ejπœ”π‘‘ d2 𝑬π‘₯ d𝑧2 = βˆ’ πœ”2 𝑐2 𝑬π‘₯ d2𝑬π‘₯ d𝑧2 + π‘˜2𝑬π‘₯ = 0
  • 14. 13 where we have, π‘˜2 = πœ”2 𝑐2 = πœ”2 πœ€πœ‡ π‘˜ = 2Ο€ πœ† is the wavenumber, πœ† is the wavelength π‘˜ = Β± πœ” 𝑐 β‡’ πœ” = π‘˜π‘ πœ” = 2π𝑓 = 2Ο€ πœ† 𝑐 β‡’ π‘“πœ† = 𝑐 𝑬π‘₯ = 𝐴1ejπ‘˜π‘§ + 𝐴2eβˆ’jπ‘˜π‘§ 𝐸π‘₯ = Re 𝐴1ej πœ”π‘‘+π‘˜π‘§ + 𝐴2ej πœ”π‘‘βˆ’π‘˜π‘§ For the wave in the βˆ’π‘§ direction we have: πœ”π‘‘ + π‘˜π‘§ = constant, πœ”d𝑑 + π‘˜d𝑧 = 0 d𝑧 d𝑑 = βˆ’ πœ” π‘˜ = βˆ’π‘ travelling wave in the βˆ’ 𝑧 direction travelling wave in the +𝑧 direction
  • 15. 14 For the wave in the +𝑧 direction we have: πœ”π‘‘ βˆ’ 𝑧 = constant, πœ”d𝑑 βˆ’ π‘˜d𝑧 = 0, d𝑧 d𝑑 = πœ” π‘˜ = +𝑐 𝑬π‘₯ 𝑧 = 𝑬π‘₯+eβˆ’jπ‘˜π‘§ 𝑧 > 0 𝑬π‘₯βˆ’ejπ‘˜π‘§ 𝑧 < 0 𝛻 Γ— 𝑬 = βˆ’πœ‡ πœ•π‘― πœ•π‘‘ β‡’ d𝑬π‘₯ d𝑧 = βˆ’jπœ”πœ‡π‘―π‘¦ β‡’ 𝑯𝑦 = βˆ’ 1 jπœ”πœ‡ d𝑬π‘₯ d𝑧 𝛻 Γ— 𝑯 = πœ€ πœ•π‘¬ πœ•π‘‘ β‡’ d𝑯𝑦 d𝑧 = βˆ’jπœ”πœ€π‘¬π‘₯ 𝑯𝑦 = π‘˜ πœ”πœ‡ 𝑬π‘₯+eβˆ’jπ‘˜π‘§ 𝑧 > 0 βˆ’ π‘˜ πœ”πœ‡ 𝑬π‘₯βˆ’ejπ‘˜π‘§ 𝑧 < 0 π‘˜ πœ”πœ‡ = πœ” πœ€πœ‡ πœ”πœ‡ = πœ€ πœ‡ , πœ‚ = πœ‡ πœ€ is the wave impedance 𝑯𝑦 = 𝑬π‘₯+ πœ‚ eβˆ’jπ‘˜π‘§ 𝑧 > 0 βˆ’ 𝑬π‘₯βˆ’ πœ‚ ejπ‘˜π‘§ 𝑧 < 0
  • 16. 15 Now we look at the boundary conditions: 𝑬π‘₯ 𝑧 = 0+, 𝑑 = 𝑬π‘₯ 𝑧 = 0βˆ’, 𝑑 β‡’ 𝑬π‘₯+ = 𝑬π‘₯βˆ’ 𝑯𝑦 𝑧 = 0βˆ’, 𝑑 βˆ’ 𝑯𝑦 𝑧 = 0+, 𝑑 = 𝑲π‘₯ 𝑧 = 0, 𝑑 β‡’ βˆ’π‘¬π‘₯βˆ’ βˆ’ 𝑬π‘₯+ πœ‚ = 𝑲0 𝑬π‘₯+ = 𝑬π‘₯βˆ’ = βˆ’ πœ‚π‘²0 2 𝑬π‘₯ 𝑧 = βˆ’ πœ‚π‘²0 2 eβˆ’jπ‘˜π‘§ 𝑧 > 0 βˆ’ πœ‚π‘²0 2 ejπ‘˜π‘§ 𝑧 < 0 𝑯𝑦 𝑧 = βˆ’ 𝑲0 2 eβˆ’jπ‘˜π‘§ 𝑧 > 0 𝑲0 2 ejπ‘˜π‘§ 𝑧 < 0 𝑺 = 1 2 𝑬 Γ— π‘―βˆ— = πœ‚πΎ0 2 8 π’Šπ‘§ 𝑧 > 0 βˆ’ πœ‚πΎ0 2 8 π’Šπ‘§ 𝑧 < 0 (𝑲0 real)
  • 17. 16 𝐸π‘₯ 𝑧, 𝑑 = Re 𝑬π‘₯ 𝑧 ejπœ”π‘‘ = βˆ’ πœ‚π‘²0 2 cos πœ”π‘‘ βˆ’ π‘˜π‘§ 𝑧 > 0 βˆ’ πœ‚π‘²0 2 cos πœ”π‘‘ + π‘˜π‘§ 𝑧 < 0 𝐻𝑦 𝑧, 𝑑 = Re 𝑯𝑦 𝑧 ejπœ”π‘‘ = βˆ’ 𝑲0 2 cos πœ”π‘‘ βˆ’ π‘˜π‘§ 𝑧 > 0 𝑲0 2 cos πœ”π‘‘ + π‘˜π‘§ 𝑧 < 0 𝑆𝑧 = 𝐸π‘₯𝐻𝑦 = πœ‚πΎ0 2 4 cos2 πœ”π‘‘ βˆ’ π‘˜π‘§ 𝑧 > 0 βˆ’ πœ‚πΎ0 2 4 cos2 πœ”π‘‘ + π‘˜π‘§ 𝑧 < 0 𝑆𝑧 = πœ‚πΎ0 2 8 𝑧 > 0 βˆ’ πœ‚πΎ0 2 8 𝑧 < 0
  • 18. 17 C. Normal Incidence Onto A Perfect Conductor Incident Fields: 𝑬i 𝑧, 𝑑 = Re 𝑬iej πœ”π‘‘βˆ’π‘˜π‘§ π’Šπ‘₯ 𝑯i 𝑧, 𝑑 = Re 𝑬i πœ‚ ej πœ”π‘‘βˆ’π‘˜π‘§ π’Šπ‘¦ Reflected Fields: 𝑬r 𝑧, 𝑑 = Re 𝑬rej πœ”π‘‘+π‘˜π‘§ π’Šπ‘₯ 𝑯r 𝑧, 𝑑 = Re βˆ’ 𝑬r πœ‚ ej πœ”π‘‘+π‘˜π‘§ π’Šπ‘¦ π‘˜ = πœ” πœ€πœ‡, πœ‚ = πœ‡ πœ€ The boundary conditions require that, 𝐸π‘₯ 𝑧 = 0, 𝑑 = 𝐸π‘₯, i 𝑧 = 0, 𝑑 + 𝐸π‘₯, r 𝑧 = 0, 𝑑 = 0 𝑬i + 𝑬r = 0 β‡’ 𝑬r = βˆ’π‘¬i
  • 19. 18 Figure 2: Time varying electromagnetic phenomena differ only in the scaling of time (frequency) and size (wavelength). In linear dielectric media the frequency and wavelength are related as 𝑓λ = 𝑐 (πœ” = π‘˜π‘), where 𝑐 = 1 πœ€πœ‡ is the velocity of light in the medium. sin π‘˜z πœ† = 2Ο€ π‘˜ 𝑇 = 2Ο€ πœ” 2Ο€ π‘˜ 2Ο€ πœ” 𝑧 𝑑 Ο€ πœ” Ο€ π‘˜ βˆ’1 βˆ’1 1 1 sin πœ”π‘‘ 0 102 104 106 108 1010 1012 1014 1016 1018 1020 Circuit Theory Microwaves f (Hz) Visible Light Ultraviolet X-Rays Gamma Rays Power Infrared (Heat) Radio and Television AM FM 3 Γ— 106 Ξ» meters 3 Γ— 104 3 Γ— 102 3 3 Γ— 10βˆ’2 3 Γ— 10βˆ’4 3 Γ— 10βˆ’6 3 Γ— 10βˆ’8 3 Γ— 10βˆ’10 3 Γ— 10βˆ’12
  • 20. 19 For 𝑬i = 𝐸i real we have: 𝐸π‘₯ 𝑧, 𝑑 = 𝐸π‘₯, i 𝑧, 𝑑 + 𝐸π‘₯, r 𝑧, 𝑑 = Re 𝑬i eβˆ’jπ‘˜π‘§ βˆ’ ejπ‘˜π‘§ ejπœ”π‘‘ = 2𝐸i sin π‘˜π‘§ sin πœ”π‘‘ 𝐻𝑦 𝑧, 𝑑 = 𝐻𝑦, i 𝑧, 𝑑 + 𝐻𝑦, r 𝑧, 𝑑 = Re 𝑬i πœ‚ eβˆ’jπ‘˜π‘§ βˆ’ ejπ‘˜π‘§ ejπœ”π‘‘ = 2𝐸i πœ‚ cos π‘˜π‘§ cos πœ”π‘‘ 𝐾𝑧 𝑧 = 0, 𝑑 = 𝐻𝑦 𝑧 = 0, 𝑑 = 2𝐸i πœ‚ cos πœ”π‘‘ Radiation pressure in free space πœ‡ = πœ‡o, πœ€ = πœ€o Force𝑧 Area 𝑧=0 = 1 2 𝑲 Γ— πœ‡o𝑯 = 1 2 πœ‡o𝐾π‘₯𝐻𝑦 𝑧=0 π’Šπ‘§ = 1 2 πœ‡o𝐻𝑦 2 𝑧 = 0 π’Šπ‘§ = 2πœ‡o𝐸i 2 πœ‚o 2 cos2 πœ”π‘‘ π’Šπ‘§ = 2πœ‡o πœ‚o πœ€o 𝐸i 2 cos2 πœ”π‘‘ π’Šπ‘§ = 2πœ€o 𝐸i 2 cos2 πœ”π‘‘ π’Šπ‘§
  • 21. 20 Figure 3: A uniform plane wave normally incident upon a perfect conductor has zero electric field at the conducting surface thus requiring a reflected wave. The source of this reflected wave is the surface current at 𝑧 = 0, which equals the magnetic field there. The total electric and magnetic fields are 90Β° out of phase in time and space. x z y 𝐸i = Re 𝑬iej πœ”π‘‘βˆ’π‘˜π‘§ π’Šπ‘₯ 𝐻i = Re 𝑬i 𝑍o ej πœ”π‘‘βˆ’π‘˜π‘§ π’Šπ‘¦ π‘˜ = πœ” 𝑐 π’Šπ‘§ 𝐻r = Re βˆ’ 𝑬r 𝑍o ej πœ”π‘‘+π‘˜π‘§ π’Šπ‘¦ 𝐸r = Re 𝑬rej πœ”π‘‘+π‘˜π‘§ π’Šπ‘₯ π‘˜r = βˆ’ πœ” c1 π’Šπ‘§ πœ€o, πœ‡o πœ‚o = 𝑍o= πœ‡o πœ€o 𝐻𝑦 𝑠, 𝑑 = 2𝑬i 𝑍o cos π‘˜π‘  cos πœ”π‘‘ 𝐸π‘₯ 𝑠, 𝑑 = 2𝑬isin π‘˜π‘  sin πœ”π‘‘ 𝜎 β†’ ∞
  • 22. 21 Figure 4: A uniform plane wave normally incident upon a dielectric interface separating two different materials has part of its power reflected and part transmitted. IV. Normal Incidence Onto A Dielectric x z y πœ€1, πœ‡1 πœ‚1 = 𝑍1 = πœ‡1 πœ€1 , 𝑐1 = 1 πœ€1πœ‡1 πœ€2, πœ‡2 𝑍2 = πœ‡2 πœ€2 , 𝑐2 = 1 πœ€2πœ‡2 𝐸i = Re 𝑬iej πœ”π‘‘βˆ’π‘˜1𝑧 π’Šπ‘₯ 𝐸t = Re 𝑬tej πœ”π‘‘βˆ’π‘˜2𝑧 π’Šπ‘₯ 𝐻t = Re 𝑬t 𝑍2 ej πœ”π‘‘βˆ’π‘˜2𝑧 π’Šπ‘¦ 𝐻r = Re βˆ’ 𝑬r 𝑍1 ej πœ”π‘‘+π‘˜1𝑧 π’Šπ‘¦ 𝐻i = Re 𝑬i 𝑍1 ej πœ”π‘‘βˆ’π‘˜1𝑧 π’Šπ‘¦ 𝐸r = Re 𝑬rej πœ”π‘‘+π‘˜1𝑧 π’Šπ‘₯ π‘˜i = π‘˜1π’Šπ‘§ = πœ” 𝑐1 π’Šπ‘§ π‘˜r = βˆ’π‘˜1π’Šπ‘§ = βˆ’πœ” 𝑐1 π’Šπ‘§ π‘˜t = π‘˜2π’Šπ‘§ = πœ” 𝑐2 π’Šπ‘§
  • 23. 22 𝑬i 𝑧, 𝑑 = Re 𝑬iej πœ”π‘‘βˆ’π‘˜1𝑧 𝑖π‘₯ , π‘˜1 = πœ” πœ€1πœ‡1 𝑯i 𝑧, 𝑑 = Re 𝑯i πœ‚1 ej πœ”π‘‘βˆ’π‘˜1𝑧 𝑖𝑦 , πœ‚1 = πœ” πœ‡1 πœ€1 𝑬r 𝑧, 𝑑 = Re 𝑬rej πœ”π‘‘+π‘˜1𝑧 𝑖π‘₯ 𝑯r 𝑧, 𝑑 = Re βˆ’ 𝑬r πœ‚1 ej πœ”π‘‘+π‘˜1𝑧 𝑖𝑦 𝑬t 𝑧, 𝑑 = Re 𝑬tej πœ”π‘‘βˆ’π‘˜2𝑧 𝑖π‘₯ , π‘˜2 = πœ” πœ€2πœ‡2 𝑯t 𝑧, 𝑑 = Re 𝑬t πœ‚2 ej πœ”π‘‘βˆ’π‘˜2𝑧 𝑖𝑦 , πœ‚2 = πœ” πœ‡2 πœ€2
  • 24. 23 𝐸π‘₯ 𝑧 = 0βˆ’ = 𝐸π‘₯ 𝑧 = 0+ β‡’ 𝑬i + 𝑬r = 𝑬t 𝐻𝑦 𝑧 = 0βˆ’ = 𝐻𝑦 𝑧 = 0+ β‡’ 𝑬i βˆ’ 𝑬r πœ‚1 = 𝑬t πœ‚2 𝑅 = 𝐸r 𝐸i = πœ‚2 βˆ’ πœ‚1 πœ‚1 + πœ‚2 is the Reflection coefficient 𝑇 = 𝐸t 𝐸i = 2πœ‚2 πœ‚1 + πœ‚2 is the Transmission coefficient 1 + 𝑅 = 𝑇
  • 25. 24 𝑆𝑧, i = 1 2 Re 𝑬π‘₯ 𝑧 𝑯𝑦 βˆ— 𝑧 = 1 2πœ‚1 Re 𝑬ieβˆ’jπ‘˜1𝑧 + 𝑬rejπ‘˜1𝑧 𝑬i βˆ— ejπ‘˜1𝑧 βˆ’ 𝑬r βˆ— eβˆ’jπ‘˜1𝑧 = 1 2πœ‚1 𝑬i 2 βˆ’ 𝑬r 2 + 1 2πœ‚1 Re 𝑬r𝑬i βˆ— e2jπ‘˜1𝑧 βˆ’ 𝑬r βˆ— 𝑬ieβˆ’2jπ‘˜1𝑧 = 1 2πœ‚1 𝑬i 2 βˆ’ 𝑬r 2 = 𝑬i 2 2πœ‚1 1 βˆ’ 𝑅2 𝑆𝑧, t = 1 2πœ‚2 𝑬t 2 = 𝑬i 2 𝑇2 2πœ‚2 = 𝑬i 2 1 βˆ’ 𝑅2 2πœ‚2 = 𝑆𝑧, i pure imaginary
  • 26. 25 V. Lossy Dielectrics, 𝑱 = πœŽπ‘¬ Ampereβ€² s Law: 𝛻 Γ— 𝑯 = 𝑱 + πœ€ πœ•π‘¬ πœ•π‘‘ = πœŽπ‘¬ + πœ€ πœ•π‘¬ πœ•π‘‘ For ejπœ”π‘‘ fields, 𝛻 Γ— 𝑯 = jπœ”πœ€ + 𝜎 𝑬 = jπœ”πœ€ 1 + 𝜎 jπœ”πœ€ 𝑬 where, 𝜎 = conductivity of a medium ( Siemens m )
  • 27. 26 Define complex permittivity by 𝜺 = πœ€ 1 + 𝜎 jπœ”πœ€ . Then complex amplitude solutions are the same as real amplitude solutions if we replace πœ€ by 𝜺: π‘˜ = πœ” πœΊπœ‡ πœ‚ = πœ‡ 𝜺 = πœ‡ πœ€ 1 + 𝜎 jπœ”πœ€ π‘˜ = πœ” πœ€πœ‡ 1 + 𝜎 jπœ”πœ€
  • 28. 27 A. Low Loss Limit: 𝜎 πœ”πœ€ β‰ͺ 1 π‘˜ = πœ” πœ€πœ‡ 1 + 𝜎 jπœ”πœ€ β‰ˆ πœ” πœ€πœ‡ 1 + 1 2 𝜎 jπœ”πœ€ β‰ˆ πœ” πœ€πœ‡ βˆ’ j𝜎 2 πœ‡ πœ€ β‰ˆ πœ” πœ€πœ‡ βˆ’ j πœŽπœ‚ 2 eβˆ’jπ‘˜π‘§ = eβˆ’jπœ” πœ€πœ‡π‘§ e βˆ’j βˆ’j πœŽπœ‚ 2 𝑧 = eβˆ’jπœ” πœ€πœ‡π‘§ eβˆ’ πœŽπœ‚ 2 𝑧 slow exponential decay
  • 29. 28 B. Large Loss Limit: 𝜎 πœ”πœ€ ≫ 1 π‘˜ = πœ” πœ€πœ‡ 1 + 𝜎 jπœ”πœ€ β‰ˆ πœ” πœ€πœ‡ 𝜎 πœ”πœ€ βˆ’j we know that, πœ” πœ” = πœ” β‰ˆ πœ”πœ‡πœŽ 2 1 βˆ’ j β‰ˆ 1 βˆ’ j 𝛿 𝛿 = 2 πœ”πœ‡πœŽ is the skin depth eβˆ’jπ‘˜π‘§ = e βˆ’j 1βˆ’j 𝑧 𝛿 = e βˆ’j 𝑧 𝛿 e βˆ’ 𝑧 𝛿 fast exponential decay 1 βˆ’ j 2
  • 30. (1) Markus Zahn, Electromagnetics and Applications, Massachusetts Institute of Technology: MIT Open Course Ware, 2005. References 29