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Maxwell's Equations

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Maxwell's Equations Microwave Measurement and Beam Instrumentation Course at Jefferson Laboratory, January 15-26th 2018 Lecture: Maxwell’s Equations F. Marhauser Monday, January 15, 2018 This Lecture － This lecture provides theoretical basics useful for follow-up lectures on resonators and waveguides － Introduction to Maxwell’s Equations • Sources of electromagnetic fields • Differential form of Maxwell’s equation • Stokes’ and Gauss’ law to derive integral form of Maxwell’s equation • Some clarifications on all four equations • Time-varying fields wave equation • Example: Plane wave － Phase and Group Velocity － Wave impedance 2 Maxwell’s Equations A dynamical theory of the electromagnetic field James Clerk Maxwell, F. R. S. Philosophical Transactions of the Royal Society of London, 1865 155, 459-512, published 1 January 1865 Maxwell’s Equations - Originally there were 20 equations Taken from Longair, M. 2015 ‘...a paper ...I hold to be great guns’: a commentary on Maxwell (1865) ‘A dynamical theory of the electromagnetic field’. Phil. Trans. R. Soc. A 373: 20140473. Sources of Electromagnetic Fields － Electromagnetic fields arise from 2 sources: • Electrical charge (Q) Stationary charge creates electric field 푑푄 • Electrical current (퐼 = ) 푑푡 Moving charge creates magnetic field － Typically charge and current densities are utilized in Maxwell’s equations to quantify the effects of fields: 푑푄 • ρ = electric charge density – total electric charge per unit volume V 푑푉 (or 푄 = 푉 휌 푑푉) 퐼(푆) • 퐽 = lim electric current density – total electric current per unit area S 푆→0 푆 (or 퐼 = 푆 퐽 ∙ 푑푆) － If either the magnetic or electrical fields vary in time, both fields are coupled and the resulting fields follow Maxwell’s equations 5 Maxwell’s Equations Differential Form 휌 (1) 훻 ∙ 퐷 = 휌 or 훻 ∙ 퐸 = Gauss’s law 휖0 (2) 훻 ∙ 퐵 = 0 Gauss’s law for magnetism 휕퐵 (3) 훻 × 퐸 = − Faraday’s law of induction 휕푡 휕퐷 휕퐸 (4) 훻 × 퐻 = 퐽 + or 훻 × 퐵 = 휇 퐽 + 휇 휖 Ampère’s law 휕푡 0 0 0 휕푡 - Together with the Lorentz force these equations 퐹 = 푞(퐸 + v × 퐵) Lorentz Force form the basic of the classic electromagnetism ρ = electric charge density (As/m3) 퐷 = 휖 퐸 0 J = electric current density (A/m2) 휖0=permittivity of free space D = electric flux density/displacement field (Unit: As/m2) E = electric field intensity (Unit: V/m) 퐵 = 휇0퐻 H = magnetic field intensity (Unit: A/m) µ0=permeability of free space B = magnetic flux density (Unit: Tesla=Vs/m2) 6 Divergence (Gauss’) Theorem { div (훻 ∙ 퐹 ) 푑푉 = (퐹 ∙ 푛 )푑푆 = 퐹 ∙ 푑푆 푉 푆 푆 Integral of divergence of vector field (퐹 ) over volume V inside closed boundary S equals outward flux of vector field (퐹 ) through closed surface S 휕 휕 휕 휕퐹 휕퐹푦 휕퐹 훻 ∙ 퐹 = , , ∙ 퐹 , 퐹 , 퐹 = 푥 + + 푧 휕푥 휕푦 휕푧 푥 푦 푧 휕푥 휕푦 휕푧 7 Curl (Stokes’) Theorem { curl (훻 × 퐹 ) · 푑푆 = ((훻 × 퐹 ) ∙ 푛 )푑푆 = 퐹 ∙ 푑푙 푆 푆 휕푆 Integral of curl of vector field (퐹 ) over surface S equals line integral of vector field (퐹 ) over closed boundary dS defined by surface S 푖 푗 푘 휕퐹 휕퐹 휕퐹 휕퐹 휕퐹 휕퐹 훻 × 퐹 = 휕 휕 휕 = 푧 − 푦 푖 + 푥 − 푧 푗+ 푦 − 푥 푘 휕푥 휕푦 휕푧 휕푦 휕푧 휕푧 휕푥 휕푧 휕푦 퐹푥 퐹푦 퐹푧 휕퐹푦(푥, 푦) 휕퐹 (푥, 푦) 풆. 품. : 푭 = ퟎ → 훻 × 퐹 = − 푥 푘 풛 휕푥 휕푦 Curl vector is perpendicular to surface S ; 푘 = 푛 휕퐹푦(푥, 푦) 휕퐹 (푥, 푦) ( − 푥 ) 푑푆 = 퐹 푑푥 + 퐹 푑푦 Green’s Theorem 휕푥 휕푦 푥 푦 푆 휕푆 8 Example: Curl (Stokes’) Theorem curl { (훻 × 퐹 ) · 푑푆 = ((훻 × 퐹 ) ∙ 푛 )푑푆 = 퐹 ∙ 푑푙 푆 푆 휕푆 Integral of curl of vector field (퐹 ) over surface S equals line integral of vector field (퐹 ) over closed boundary dS defined by surface S 9 Example: Curl (Stokes) Theorem curl { (훻 × 퐹 ) · 푑푆 = ((훻 × 퐹 ) ∙ 푛 )푑푆 = 퐹 ∙ 푑푙 푆 푆 휕푆 Integral of curl of vector field (퐹 ) over surface S equals line integral of vector field (퐹 ) over closed boundary dS defined by surface S Example: Closed line integrals of various vector fields No curl Some curl Stronger curl No net curl 10 Maxwell’s Equations Differential Form Integral Form 훻 ∙ 퐷 = 휌 퐷 ∙ 푑푆 = 휌 푑푉 (훻 ∙ 퐹 ) 푑푉 = 퐹 ∙ 푑푆 Gauss’s law 푉 푆 푆 푉 } Gauss’ theorem 훻 ∙ 퐵 = 0 퐵 ∙ 푑푆 = 0 Gauss’s law for magnetism 푆 휕퐵 휕퐵 Faraday’s law of induction 훻 × 퐸 = − 퐸 ∙ 푑푙 = − ∙ 푑푆 휕푡 (훻 × 퐹 ) · 푑푆 = 퐹 ∙ 푑푙 휕푡 푆 휕푆 휕푆 푆 휕퐷 } Stokes’ theorem 휕퐷 Ampère’s law 훻 × 퐻 = 퐽 + 퐻 ∙ 푑푙 = 퐽 ∙ 푑푆 + ∙ 푑푆 휕푡 휕푡 휕푆 푆 푆 ρ = electric charge density (C/m3=As/m3) 퐷 = 휖0퐸 J = electric current density (A/m2) 휖0=permittivity of free space D = electric flux density/displacement field (Unit: As/m2) E = electric field intensity (Unit: V/m) 퐵 = 휇0퐻 H = magnetic field intensity (Unit: A/m) 2 µ0=permeability of free space B = magnetic flux density (Unit: Tesla=Vs/m ) 11 Electric Flux & 1st Maxwell Equation Definition of Electric Flux Gauss: Integration over closed surface 1. Uniform field 퐸 = 퐸 ∙ 푆 = 퐸 ∙ 푛 푆 = 퐸 ∙ 푆 ∙ 푐표푠 휃 [푉푚] 휖0퐸 ∙ 푑푆 = 휖0 푬 = 휌 푑푉 = 푞푖 - angle between field and normal vector 푆 푉 푖 to surface matters 푖 푞푖 ; 퐷 = 휖0퐸 퐸= 휖0 Example: Metallic plate, assume only surface charges on one side 2. Non-Uniform field 2 푑퐸 = 퐸 ∙ 푑푆 = 퐸 ∙ 푛 푑푆 휖0퐸 ∙ 푑푆 = 휖0 퐸 휋푅 = 푞푖 = 푄푐푖푟푐푙푒 푆 푖 ; 푐푟푐푙푒 푆 = 휋푅2 퐸 = 퐸 ∙ 푑푆 푄푐푖푟푐푙푒 퐸 = 2 푆 휖0휋푅 12 Electric Flux & 1st Maxwell Equation Definition of Electric Flux Gauss: Integration over closed surface 1. Uniform field 퐸 = 퐸 ∙ 푆 = 퐸 ∙ 푛 푆 = 퐸 ∙ 푆 ∙ 푐표푠 휃 [푉푚] 휖0퐸 ∙ 푑푆 = 휖0 푬 = 휌 푑푉 = 푞푖 - angle between field and normal vector 푆 푉 푖 to surface matters 푖 푞푖 ; 퐷 = 휖0퐸 퐸= 휖0 Example: Capacitor 2. Non-Uniform field 퐸 = 0 퐸 = 0 푑퐸 = 퐸 ∙ 푑푆 = 퐸 ∙ 푛 푑푆 푄푐푖푟푐푙푒 −푄푐푖푟푐푙푒 퐸= + = 0 퐸 = 퐸 ∙ 푑푆 휖0 휖0 푆 13 Electric Flux & 1st Maxwell Equation Examples of non-uniform fields Point charge Q Add charges 휖0퐸 ∙ 푑푆 = 휌 푑푉 = 푞푖 = 푄푠푝ℎ푒푟푒 푆 푉 푖 퐸 퐸 Integration of over closed spherical surface S 2 휖0퐸 ∙ 푑푆 = 휖0푬(푟)4휋푟 = 푄 푆 푞 푞 −푞 3푞 푄 ; 푠푝ℎ푒푟푒 푆 = 4휋푟2 푖 푖 푠푝ℎ푒푟푒 pointing out radially 퐸= = + = 0 퐸= = 휖0 휖0 휖0 휖0 휖0 Principle of Superposition holds: 푄 푬(푟) = ∙ 풓 2 1 푞1 푞2 푞3 휖04휋푟 퐸(푟) = 2 푟푐 1+ 2 푟푐2 + 2 푟푐3 + ⋯ 휖04휋 푟푐1 − 푟 푟푐2 − 푟 푟푐3 − 푟 14 Magnetic Flux & 2nd Maxwell Equation Definition of Magnetic Flux Gauss: Integration over closed surface Uniform field = 퐵 ∙ 푆 = 퐵 ∙ 푛 푆 = 퐵 ∙ 푆 푐표푠 휃 [푊푏 = 푉푠] 퐵 퐵 ∙ 푑푆 = 0 푀= 0 푆 - There are no magnetic monopoles - All magnetic field lines form loops 퐵 퐵 Non-Uniform field 푑퐵 = 퐵 ∙ 푑푆 = 퐵 ∙ 푛 푑푆 Closed surface: What about this case? Flux lines out = flux lines in Flux lines out > flux lines in ? = 퐵 ∙ 푑푆 퐵 - No. In violation of 2nd Maxwell’s law, i.e. 푆 integration over closed surface, no holes allowed - Also: One cannot split magnets into separate poles, i.e. there always will be a 15 North and South pole Magnetic Flux & 3rd Maxwell Equation If integration path is not changing in time 푑 푑퐵 퐸 ∙ 푑푙 = − 퐵 ∙ 푑푆 = − ; = 퐵 ∙ 푑푆 푑푡 푑푡 퐵 푆 휕푆 푆 - Change of magnetic flux induces an electric field along a closed loop - Note: Integral of electrical field over closed loop may be non-zero, when induced by a time-varying magnetic field - Electromotive force (EMF) Ɛ: 퐵(푡 = 푡1) Ɛ = 퐸 ∙ 푑푙 [푉] 휕푆 - Ɛ equivalent to energy per unit charge traveling once around loop Faraday’s law of induction 16 Magnetic Flux & 3rd Maxwell Equation If integration path is not changing in time 푑 푑퐵 퐸 ∙ 푑푙 = − 퐵 ∙ 푑푆 = − ; = 퐵 ∙ 푑푆 푑푡 푑푡 퐵 푆 휕푆 푆 - Change of magnetic flux induces an electric field along a closed loop - Note: Integral of electrical field over closed loop may be non-zero, when induced by a time-varying magnetic field - Electromotive force (EMF) Ɛ: 퐵(푡) Ɛ = 퐸 ∙ 푑푙 [푉] 휕푆 - Ɛ equivalent to energy per unit charge traveling once around loop - or voltage measured at end of open loop Faraday’s law of induction 17 Ampère's (circuital) Law or 4th Maxwell Equation Right hand side of equation: - Note that 퐽 ∙ 푑푆 is a surface integral, but S may 휕퐷 푆 퐻 ∙ 푑푙 = 퐽 ∙ 푑푆 + ∙ 푑푆 have arbitrary shape as long as ∂S is its closed 휕푡 휕푆 푆 푆 boundary { conduction current I 퐽 ∙ 푑푆 = 퐼 Example: 퐵 푆 - What if there is a capacitor? Left hand side of equation: - While current is still be flowing (charging capacitor): 퐵 ∙ 푑푙 = 푩 푟 2휋푟 = 휇0퐼 푑푄 휕푆 퐽 ∙ 푑푆 = 퐼 = tangential to a circle at any 푑푡 푆 ; 퐵 = 휇0퐻 radius r of integration ; 푐푟푐푢푚푓푒푟푒푛푐푒 퐶 = 2휋푟 휇 퐼 |푩 푟 | = 0 2휋푟 18 Ampère's (circuital) Law or 4th Maxwell Equation - But one may also place integration surface S between plates current does not flow through surface here 휕퐷 퐻 ∙ 푑푙 = 퐽 ∙ 푑푆 + ∙ 푑푆 휕푡 휕푆 푆 푆 퐽 ∙ 푑푆 = 0 { 푆 conduction current I 푤ℎ푙푒 퐵 ≠ 0 ? Example: 퐵 - This is when the displacement field is required as a corrective 2nd source term for the magnetic fields 휕퐷 푑 푑푄 푑 ∙ 푑푆 = 퐷 ∙ 푑푆 = = 퐼 = 휖 퐸 Left hand side of equation: 휕푡 푑푡 푑푡 퐷 0 푑푡 푆 푆 ; Gauss’s law { 퐵 ∙ 푑푙 = 푩 푟 2휋푟 = 휇0퐼 displacement current I 휕푆 푑퐸 tangential to a circle at any 퐻 ∙ 푑푙 = 퐼 + 휖0 ; 퐵 = 휇0퐻 radius r of integration 푑푡 휕푆 ; 푐푟푐푢푚푓푒푟푒푛푐푒 퐶 = 2휋푟 퐸 휇0퐼 |푩 푟 | = 휇 퐼 2휋푟 |푩 푟 | = 0 퐷 19 퐵 2휋푟 Presence of Resistive Material 휕퐷 퐻 ∙ 푑푙 = 퐽 ∙ 푑푆 + ∙ 푑푆 휕푡 휕푆 푆 푆 { { conduction current displacement current - In resistive materials the current density J is proportional to the electric field 1 퐽 = 퐸 = 퐸 with the electric conductivity (1/(Ω·m) or S/m), respectively 휌 =1/ the electric resistivity (Ω·m) - Generally (ω, T) is a function of frequency and temperature 20 퐵 Time-Varying E-Field in Free Space - Assume charge-free, homogeneous, linear, and isotropic medium - We can derive a wave equation: 휕퐵 훻 × 퐸 = − ;Faraday’s law of induction 휕푡 휕퐵 휕 훻 × 훻 × 퐸 = −훻 × = − 훻 × 퐵 ; || curl 휕푡 휕푡 훻2= Δ = Laplace operator 휕 휕퐷 훻 훻 ∙ 퐸 − 훻2퐸 = −휇 퐽 + ; curl of curl 훻 × 훻 × 퐴 = 훻 훻 ∙ 퐴 − 훻2퐴 휕푡 휕푡 휕퐷 ; 훻 × 퐻 = 퐽 + ; Ampère’s law 휕푡 휌 휕 휕퐸 훻 − 훻2퐸 = −휇 퐸 + 휖 휖0 휕푡 휕푡 ; Gauss’s law ; 퐽 = 퐸 휕2퐸 훻2퐸 − 휇휖 = 0 Homogeneous wave equation 휕푡2 ; we presumed no charge 21 Time-Varying B-Field in Free Space - Assume charge-free, homogeneous, linear, and isotropic
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