Lecture: Maxwell’s Equations - Fermilab

1 Lecture: maxwell s EquationsMicrowave Measurement and Beam Instrumentation Courseat Jefferson Laboratory, January 15-26th2018F. MarhauserMonday, January 15, 2018 This Lecture This lecture provides theoretical basics useful for follow-up lectureson 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 impedance2 maxwell s EquationsA dynamical theory of the electromagnetic fieldJames Clerk maxwell , F. R. Transactions of the Royal Society of London, 1865 155, 459-512, published 1 January 1865 maxwell s EquationsTaken from Longair, M.

2 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: there were 20 equationsSources of Electromagnetic Fields5 Electromagnetic fields arise from 2 sources: Electrical charge (Q) Electrical current ( = ) Typically charge and current densities are utilized in maxwell s Equations to quantify the effects of fields: = electric charge density total electric charge per unit volume V(or = ) =lim 0 ( ) electric current density total electric current per unit area S(or = )Stationary charge creates electric fieldMoving charge creates magnetic field If either the magnetic or electrical fields vary in time, both fields are coupled and the resulting fields follow maxwell s equationsMaxwell s Equations6 = = =0 = + = 0 = 0 DifferentialFormD= electric flux density/displacement field (Unit: As/m2)E= electric field intensity (Unit: V/m) = electric charge density (As/m3)H= magnetic field intensity (Unit: A/m)B= magnetic flux density (Unit.)

3 Tesla=Vs/m2)J= electric current density (A/m2) = 0 0=permittivity of free space = 0 + 0 0 0=permeability of free spaceororGauss s lawGauss s law for magnetismAmp re s lawFaraday s law of induction(1)(2)(3)(4) = ( +v )-Together with the Lorentz force these equationsform the basic of the classic electromagnetismLorentz ForceDivergence (Gauss ) Theorem7 Integralofdivergenceofvectorfield( )overvolumeVinsideclosedboundarySequalso utwardfluxofvectorfield( )throughclosedsurfaceS ( ) = ( ) = {div = , , , , = + + ( ( , ) ( , ) ) = + Curl (Stokes ) Theorem8 ( ) = (( ) ) = Green s Theorem{curl ..: = = ( , ) ( , ) Integralofcurlofvectorfield( )oversurfaceSequalslineintegralofvectorf ield( )overclosedboundarydSdefinedbysurfaceS.} }

4 = CurlvectorisperpendiculartosurfaceS = = + + Example: Curl (Stokes ) Theorem9 Integralofcurlofvectorfield( )oversurfaceSequalslineintegralofvectorf ield( )overclosedboundarydSdefinedbysurfaceS ( ) = (( ) ) = {curlExample: Curl (Stokes) Theorem10 ( ) = (( ) ) = Example: Closed line integrals of various vector fields{curlIntegralofcurlofvectorfield( )oversurfaceSequalslineintegralofvectorf ield( )overclosedboundarydSdefinedbysurfaceSNo curlSome curlStronger curlNo net curlMaxwell s Equations11 = = =0 = + DifferentialFormIntegralFormD= electric flux density/displacement field (Unit: As/m2)E= electric field intensity (Unit: V/m)}H= magnetic field intensity (Unit: A/m)B= magnetic flux density (Unit: Tesla=Vs/m2)J= electric current density (A/m2)Gauss theoremStokes theorem = 0 = 0 0=permittivity of free space 0=permeability of free space = =0 = = + Gauss s lawGauss s law for magnetismAmp re s lawFaraday s law of induction ( ) = ( ) = = electric charge density (C/m3=As/m3)}12; = 0 1.

5 Uniform fieldElectric Flux & 1stMaxwell Equation = = = [ ]; = 2 0 = 0 2= = = 0 2-angle between field and normal vectorto surface matters 0 = 0 = = Gauss: Integration over closedsurface = 02. Non-Uniform field = = = Example: Metallic plate,assume only surface charges on one sideDefinition of Electric Flux13 Gauss: Integration over closedsurface =0 = 0+ 0=0 =0 Example: CapacitorElectric Flux & 1stMaxwell Equation1. Uniform field = = = [ ]-angle between field and normal vectorto surface matters2. Non-Uniform field = = = Definition of Electric Flux; = 0 0 = 0 = = = 014 Integration of over closed spherical surface S ( )= 04 2.

6 =4 2 Examples of non-uniform fieldsPoint charge Q 0 = 0 ( )4 2= 0 = = = = 0= 0+ 0=0 =3 0= 0 Principle of Superposition holds: ( )=1 04 1 1 2 1+ 2 2 2 2+ 3 3 2 3+ Electric Flux & 1stMaxwell Equation pointing out radiallyAdd charges = = = [ = ]15 Uniform fieldMagnetic Flux & 2ndMaxwell EquationGauss: Integration over closedsurface =0 Non-Uniform field = = = Definition of Magnetic Flux =0-There are no magnetic monopoles-All magnetic field lines form loops Closed surface:Flux lines out = flux lines inWhat about this case?Flux lines out > flux lines in ?-No. In violation of 2ndMaxwell s law, integration over closed surface, no holes allowed -Also: One cannot split magnets into separate poles, there always will be a North and South pole16 = = Magnetic Flux & 3rdMaxwell Equation ( = 1)Faraday s law of inductionIf integration path is not changing in time; = -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) :- equivalent to energy per unitcharge traveling once around loop 17.

7 = -Change of magnetic flux induces an electric field along a closed loop = = Magnetic Flux & 3rdMaxwell Equation = [ ]-Electromotive force (EMF) :-Note: Integral of electrical field over closed loop may be non-zero, when induced by a time-varying magnetic field ( )If integration path is not changing in time- equivalent to energy per unitcharge traveling once around loop -or voltage measured at end of open loopFaraday s law of induction18 Amp re's (circuital) Law or 4thMaxwell Equation-Note that is a surface integral, but Smay have arbitrary shape as long as S is its closed boundary-What if there is a capacitor? = -While current is still be flowing (charging capacitor): = = = + ; = 0 ; =2 = 2 = 0 Example:tangential to a circle at any radius r of integration{conduction current I| |= 0 2 Right hand side of equation:Left hand side of equation:19 Amp re's (circuital) Law or 4thMaxwell Equation = + ; = 0 ; =2 = 2 = 0 | |= 0 2 {displacement current I = = = = 0 -But one may also place integration surface Sbetween plates current does not flow through surface here =0 0?}}

8 -This is when the displacement field is required as a corrective 2ndsource term for the magnetic fieldstangential to a circle at any radius r of integration; Gauss s law | |= 0 2 {conduction current I Example:Left hand side of equation: = + 0 20 = + {conduction current {displacement current-In resistive materials the current density Jis proportional to the electric field = =1 with the electric conductivity (1/( m) or S/m), respectively =1/ the electric resistivity ( m)-Generally ( , T) is a function of frequency and temperaturePresence of Resistive Material21-We can derive a wave equation:Time-Varying E-Field in Free Space = ;Faraday s law of induction = = ; curl of curl = 2 ; || curl 2 = + ; Amp re s law 0 2 = + ; Gauss s law; = 2= = Laplace operator 2 2 2=0; we presumed no charge-Assume charge-free, homogeneous, linear, and isotropic medium; = + Homogeneous wave equation22 Time-Varying B-Field in Free Space; Amp re s law = + 2 = 2 2; Faraday s law = + ; Gauss s law for magnetism = 0 ;no moving charge ( =0) 2 2 2=0 Similar homogeneous wave equation as for E-Field-We can derive a wave equation:-Assume charge-free, homogeneous, linear, and isotropic medium; || curl.}}}

9 Curl of curl = 2 23-In many cases one has to deal with purely harmonic fields (~ )Time-Harmonic Fields 2 2 2=0 2 = 2 2 = 2 2 2 2=0 Example: Plane Wave in Free Space ( , )= 0 = 0 ( )-kis a wave vector pointing in direction of wave propagation ( , )= -Wave is unconstrained in plane orthogonal to wave direction, has surfaces of constant phase (wavefronts), wave vector kis perpendicular to the wavefront -In Cartesian coordinates: ( , , , )= 0 -One may align propagation of wave (k) with z-direction, which simplifies the equation 2 = 2 = 2 -Magnitude of field (whether it is Eor B) is constant everywhere on plane, but varies with time and in direction of propagation 2= 2 =1 -Applying homogeneous wave equation 2 = 2 (with 2 = 2+ 2+ 2-We know speed of light in linear medium: 2= 2= 2 2 Example: Plane Wave in Free SpaceWikipedia CC BY-SA : Plane Wave in Free Spacekis the wavenumber [1/m] =1 = =1 = 01 = Phase velocityGroup velocity =2 =2 -Acknowledging that kis generally a vector.)

10 =2 =2 -Inserting the just derived equation 2= 2 2, a dependency with the angular frequency, we can denote the relation of kwith the wavelength= Phase velocity = speed of light27 Wave Impedance; = + + = = + = = + -Similarly for the magnetic field considering-All field components are orthogonal to propagation direction this means that the plane wave is a Transverse-Electric-Magnetic (TEM) wave = = = = -We then can find for the electrical field components considering-Considering the absence of charges in free space and 4thMaxwell equation, we find:-Furthermore for plane wave, due to 3rdMaxwell equation we know that magnetic field is orthogonal to electrical field and can derive for time-harmonic field.