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Measurement and Beam Instrumentation Course at Jefferson Laboratory, January 15-26th 2018

Lecture: ’s

F. Marhauser

Monday, January 15, 2018 This Lecture

－ This lecture provides theoretical basics useful for follow-up lectures on and waveguides

－ Introduction to Maxwell’s Equations • Sources of electromagnetic • of Maxwell’s • Stokes’ and ’ law to derive form of Maxwell’s equation • Some clarifications on all four equations • Time-varying fields  equation • Example: Plane wave － and Group －

2 Maxwell’s Equations

A dynamical theory of the electromagnetic , 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 ’. Phil. Trans. R. Soc. A 373: 20140473. Sources of Electromagnetic Fields

－ Electromagnetic fields arise from 2 sources: • Electrical (Q) Stationary charge creates 푑푄 • Electrical current (퐼 = ) 푑푡 Moving charge creates － Typically charge and current are utilized in Maxwell’s equations to quantify the effects of fields: 푑푄 • ρ = – total electric charge per unit V 푑푉

(or 푄 = 푉 휌 푑푉) 퐼(푆) • 퐽 = lim electric – total per unit 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

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 these equations 퐹 = 푞(퐸 + v × 퐵) form the basic of the classic ρ = electric (As/m3) 퐷 = 휖 퐸 0 J = electric current density (A/m2) 휖0= of free space D = electric 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 = density (Unit: =Vs/m2) 6 (Gauss’) Theorem

{ div

(훻 ∙ 퐹 ) 푑푉 = (퐹 ∙ 푛 )푑푆 = 퐹 ∙ 푑푆 푉 푆 푆

Integral of divergence of (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 (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 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 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 훻 × 퐸 = − 퐸 ∙ 푑푙 = − ∙ 푑푆 Faraday’s law of induction 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 = 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 푖 푞푖 ; 퐷 = 휖0퐸 퐸= 휖0 Example: Metallic plate, assume only surface charges on one side

2. Non-Uniform field 2 푑퐸 = 퐸 ∙ 푑푆 = 퐸 ∙ 푛 푑푆 휖0퐸 ∙ 푑푆 = 휖0 퐸 휋푅 = 푞푖 = 푄푐푖푟푐푙푒 푆 푖 ; 푐𝑖푟푐푙푒 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:

2. Non-Uniform field 퐸 = 0 퐸 = 0

푑퐸 = 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

푑퐵 = 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 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 푑퐵 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 - (EMF) Ɛ: 퐵(푡 = 푡1)

Ɛ = 퐸 ∙ 푑푙 [푉] 휕푆 - Ɛ equivalent to 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 푑퐵 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 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 , 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 ; 푐𝑖푟푐푢푚푓푒푟푒푛푐푒 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 푤ℎ𝑖푙푒 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 I 휕푆 푑 퐸 tangential to a circle at any 퐻 ∙ 푑푙 = 퐼 + 휖0 ; 퐵 = 휇0퐻 radius r of integration 푑푡 휕푆 ; 푐𝑖푟푐푢푚푓푒푟푒푛푐푒 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 and

20 퐵 Time-Varying E-Field in Free Space

- Assume charge-free, homogeneous, linear, and isotropic medium - We can derive a : 휕퐵 훻 × 퐸 = − ;Faraday’s law of induction 휕푡

휕퐵 휕 훻 × 훻 × 퐸 = −훻 × = − 훻 × 퐵 ; || curl 휕푡 휕푡 훻2= Δ = Laplace 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 medium - We can derive a wave equation:

휕퐸 훻 × 퐵 = 휇퐽 + 휇휖 ; Ampère’s law 휕푡

휕퐸 훻 × 훻 × 퐵 = 휇 훻 × 퐽 + 휇휖 훻 × ; || curl 휕푡

휕2퐵 훻 훻 ∙ 퐵 − 훻2퐵 = 휇 훻 × 퐽 − 휇휖 ; curl of curl 훻 × 훻 × 퐴 = 훻 훻 ∙ 퐴 − 훻2퐴 휕푡2 ; Faraday’s law

휕2퐵 훻2퐵 − 휇휖 = 0 Similar homogeneous wave equation as for E-Field 휕푡2

; Gauss’s law for magnetism 훻 ∙ 퐵 = 0 ; no moving charge (퐽 =0)

22 Time-Harmonic Fields

- In many cases one has to deal with purely harmonic fields (~푒푖휔푡)

휕2퐵 훻2퐵 − 휇휖 = 0 훻2퐵 = −휇휖휔2퐵 휕푡2 휕2퐸 훻2퐸 − 휇휖 = 0 훻2퐸 = −휇휖휔2퐸 휕푡2

23 Example: Plane Wave in Free Space

−푖 푘∙푟 푖휔푡 푖(휔푡−푘∙푟 ) 퐴 (푟 , 푡) = 푅푒 퐴 퐴 (푟 , 푡) = 퐴 0 ∙ 푒 ∙ 푒 = 퐴 0 ∙ 푒 - k is a wave vector pointing in direction of wave propagation - Wave is unconstrained in plane orthogonal to wave direction, i.e. has surfaces of constant phase (wavefronts), wave vector k is perpendicular to the wavefront

- Magnitude of field (whether it is E or B) is constant everywhere on plane, but varies with time and in direction of propagation - One may align propagation of wave (k) with z-direction, which simplifies the equation - In Cartesian coordinates: −푖푘푧 푖휔푡 퐴 (푥, 푦, 푧, 푡) = 퐴 0 ∙ 푒 ∙ 푒 휕퐴 휕퐴 휕퐴 - Applying homogeneous wave equation 훻2퐴 = −휇휖휔2퐴 (with 훻2퐴 = 푥 + 푦 + 푧 휕푥2 휕푦2 휕푧2

2 2 2 2 훻 퐴 = −푘 퐴 = −휇휖휔 퐴 푘2 = 휇휖휔2 2 2 휔 푘 = 휇휖휔 = 2 1 혷 - We know speed of in linear medium: 혷 = 휇휖 Example: Plane Wave in Free Space

Wikipedia CC BY-SA 2.0 Example: Plane Wave in Free Space

- Acknowledging that k is generally a vector: 푘 = 푘 ∙ 푘푧 휔2 - Inserting the just derived equation 푘2 = , i.e. a dependency with the angular frequency, 혷2 we can denote the relation of k with the 2휋푓 2휋 푘 = 푘 = 푘 혷 푧 λ 푧 2휋 2휋푓 푘 = = λ 혷 k is the [1/m] 휔 1 1 혷푝ℎ ≡ = = 푐0 Phase velocity 푘 휇휖 휇푟휖푟 푑휔 1 혷 ≡ = = 혷 𝑔푟 푑푘 휇휖 푝ℎ Group velocity = Phase velocity =

26 Wave Impedance

- Furthermore for plane wave, due to 3rd Maxwell equation we know that magnetic field is orthogonal to electrical field and can derive for time-harmonic field: 휕퐻 훻 × 퐸 = −휇 = −휇휔퐻 휕푡 - Considering the absence of charges in free space and 4th Maxwell equation, we find: 휕퐸 훻 × 퐻 = 휀 = 휀휔퐸 휕푡 - We then can find for the electrical field components considering 휕퐹 휕퐹 휕퐹푧 푦 휕퐹푥 휕퐹푧 푦 휕퐹푥 휕퐸푦 ;훻 × 퐹 = − 푖 + − 푗 + − 푘 = 𝑖휇휔퐻 휕푦 휕푧 휕푧 휕푥 휕푧 휕푦 휕푧 푥 휕퐸 휕퐸 − 푦 푖 + 푥 푗 = −𝑖휇휔퐻 = −𝑖휇휔퐻 푖 − 𝑖휇휔퐻 푗 휕퐸 휕푧 휕푧 푥 푦 푥 = −𝑖휇휔퐻 휕푧 푦 - Similarly for the magnetic field considering 휕퐻푦 휕퐻푦 휕퐻 − 푖 + 푥 푗 = 𝑖휀휔퐸 = 𝑖휀휔퐸 푖 + 𝑖휀휔퐸 푗 = −𝑖휀휔퐸푥 휕푧 휕푧 푥 푦 휕푧 휕퐻 푥 = 𝑖휀휔퐸 휕푧 푦 - All field components are orthogonal to propagation direction  this means that the plane wave is a Transverse-Electric-Magnetic (TEM) wave

27 Wave Impedance

- We obtained two sets of independent equations, that lead to two linearly independent solutions 1a) 1b) 2a) 2b) 휕퐸 휕퐻푦 휕퐸푦 휕퐻 푥 = −𝑖휇휔퐻 = −𝑖휀휔퐸 = 𝑖휇휔퐻 푥 = 𝑖휀휔퐸 휕푧 푦 휕푧 푥 휕푧 푥 휕푧 푦 - The wave equation for the electric field components yields: 2 2 휕 퐸 휕 퐸푦 2 휕퐴푥 휕퐴푦 휕퐴푧 푥 2 2 ;훻 퐴 = 2 + 2 + 2 = −푘 퐸 = −푘 퐸 휕푥 휕푦 휕푧 휕푥2 푥 휕푥2 푦

−푖푘푧 +푖푘푧 −푖푘푧 +푖푘푧 - Utilizing the Ansatz: 퐸푥 = 퐸푥,푝푒 + 퐸푥,푟푒 퐸푦 = 퐸푦,푝푒 + 퐸푦,푟푒 we can derive the corresponding magnetic field components: 푘 푘 퐻 = 퐸 푒−푖푘푧 − 퐸 푒+푖푘푧 ;1a) 퐻 = − 퐸 푒−푖푘푧 − 퐸 푒+푖푘푧 ;2a) 푦 휇휔 푥,푝 푥,푟 푥 휇휔 푦,푝 푦,푟 푘 - Using the substitution 푍 = : 휇휔 1 −푖푘푧 +푖푘푧 1 −푖푘푧 +푖푘푧 퐻푦 = 퐸푥,푝푒 − 퐸푥,푟푒 퐻 = − 퐸 푒 − 퐸 푒 푍 푥 푍 푦,푝 푦,푟

휇휔 휇휔 휇 휇0 휇푟 휇0 푍 = = = ≈ 2 2 푍0 = ≈ 120휋 Ω ≈ 376.73 Ω ; 푘 = 휇휖휔 휀 푘 휇휖휔 휖 휀0 휀푟 28 0 Z is the wave impedance in impedance Appendix Presence of Material

- For linear materials

휖 = 휖푟휖0 r is

휇 = 휖푟휖0 r is relative permeability

- Particularly, the displacement current was conceived by Maxwell as the separation (movement) of the (bound) charges due to the of the medium (bound charges slightly separate inducing electric moment) P is (‘polarization’) is the density of 퐷 = 휖퐸 = 휖0퐸 + 푃 permanent and induced electric dipole moments - For homogeneous, linear isotropic dielectric material

푃 = 휖0(휖푟−1)퐸 (휖푟−1) = c푒 c푒=

- For anisotropic dielectric material 푃 = 휖0 c푖,푗퐸푗 푗 - Material may be non-linear, i.e. P is not proportional to E( in ferroelectric materials) - Generally P(ω) is a function of frequency, since the bound charges cannot act immediately to the applied field (c푒(ω)  this gives rise to losses

30 Similar Expressions for

- For magnetic fields the presence of magnetic material can give rise to a magnetization by microscopic electric currents or the of - Example: If a ferromagnet (e.g. ) is exposed to a magnetic field, the microscopic align with the field and remain aligned to some extent when the magnetic field vanishes (magnetization vector M)  a non-linear dependency between H and M occurs - Magnetization may occur in directions other than that of the applied magnetic field - The magnetization vector describes the density of the permanent or induced moments in a magnetic material

퐵 = 휇0 ∙ 퐻 = 휇0 퐻 + 푀 = 휇0 1 + 풳푣 퐻

- Herein 풳푣 is the , which described whether is material if appealed or retracted by the presence of a magnetic field - The relative permeability of the material can then be denoted as:

휇푟 = 1 + 풳푣