Physics 220: Classical Electrodynamics

0. General

Instructor:

Roger Blandford: rdb3@stanford.edu, PAB 207. Office Hours: Tuesday 12-1p or by appointment

Teaching Assistants

Weejee Cho: weejee.cho@stanford.edu, McCullough 311, Office Hours: Tuesday 4-5p at Varian 4th floor (beware that it is not at my office!) or by appointment
Dominik Rastawicki: dominikr@stanford.edu, Varian 014, Office Hours: Wednesday 4-5p or by appointment

Lectures:

Tue, Thur  9-30a-10-45a Hewlett 102

Context:

This will be a major revision of a two quarter course traditionally taught from Jackson's famous text "Classical Electrodynamics". It will differ from prior offerings in several ways:
--It will only be taught for one quarter
--It will use a textbook that has only been out for four weeks
--It will emphasize principles and applications of local interest
--It will de-emphasize traditional applied mathematics through the use of Mathematica.

Textbook:

Zangwill: Modern Electrodynamics. 
Its strengths appear to be clarity, scholarship and interesting, contemporary applications.  However, it seems strange to defer the introduction of special relativity till the very end of the book 108 years after the publication of the "On the Electrodynamics of Moving Bodies" and when we teach four vectors to freshmen. (The use of ict also seems anachronistic.) 

Supplementary texts in rough order of relevance are:
Jackson: Classical Electrodynamics
Griffiths: Introduction to Electrodynamics
Landau and Lifshitz: Classical Theory of Fields
Landau, Lifshitz and Pitaevski: Electrodynamics of Continuous Media
Fetter: Class Notes
Panofsky & Phillips: Classical Electricity and Magnetism
Purcell: Electricity and Magnetism
Brau: Modern Problems in Classical Electrodynamics
There are many more sources on this topic which you may prefer.

Zangwill contains "more material than can be comfortably covered in a two-semester course" and we only have one quarter. However, as this is likely to be your last formal instruction in this area, I judge it more important for you to see all of the subject superficially than some of it in depth.  (I am presuming that you have all had an undergraduate E&M course covering much of the first 12 chapters.) Accordingly, my plan is to try to discuss most of the material covered  by Zangwill at the foundational level, though quite unevenly, roughly in his order (except for the relativity), in the Tuesday lectures and to devote the Thursday lectures to a few pertinent applications allowing some development of the formalism and introducing a few useful calculational techniques.

Synopsis

2/4 1. Maxwell's equations, relativity, stress-energy tensor, particle dynamics. Z 1, 2, 3, 12, 22
9/4 2. Electrostatics, magnetostatics Z  4, 5,  7,  8, 11
16/4 3. Dielectrics and magnets Z 6, 13
23/4 4. Currents, vector potential, multipoles, Z 9, 10, 11  
30/4 5. Time-dependent fields Z 14, 15
7/5 6. Electromagnetic waves Z 16, 17
14/5 7. Dispersion and waveguides Z 18, 19
21/5 8. Multipole radiation Z 20, 21
28/5 9. Relativistic radiation Z 23
4/6 10. Summary. Electromagnetism as classical field theory and the path to QED Z 24

I can guarantee that this will change.

Applications of local interest  that I am considering include particle traps, magnets, radio telescopes, ionosphere, solar corona, cosmic rays, pulsars, quasars, tokamaks, SLAC LINAC, LCLS, electromagnetic induced transparency, superconductivity, meta materials, fiber optics, plasmonics, photonics. Other suggestions welcome.

Requirements:

I will set reading ahead of the Tuesday lecture and begin with a random student summarizing the material and identifying issues for discussion.
I shall assign one homework set per week on Thursday, due in class the following Thursday.
There will be a final exam carrying equal weight with the homeworks, format TBD.
I shall make available a Mathematica file containing the main points and sketch solutions to problems. I hope this will be useful for review. However, it will be terse and NOT a substitute for a textbook.

Disclaimer and Plea

You should know that I have never taught electromagnetism before at any level. Class participation and feedback will be vital as the TAs and I learn what you have and have not already seen and how best to present the material.

1. Basics

Charges and Currents (Z30-32)

Charge

Changeable quality of a body - positive and negative - measureable by Coulomb force ∝+(((Subscript[Q, 1])(Subscript[Q, 2]))/(r^2)) 
Lavoisier, Dalton et al found that mass conserved and matter made of atoms whose masses were in fixed, integer ratios.
Faraday found that charge transferred per atom during electrolysis was 1,2,3… times a fixed charge.
We now know Avogadro’s number, or equivalently the amu, implying that free charge (no quarks!) is quantized in units of e = 1.6×(10^(-19))C. 
(We use SI units throughout.)

Charge density

Volume and surface: dQ=ρdV=σdS

Current

Charge is conserved but can flow eg along conducting wire. dQ=I dt

Units

Theory and experiment tell us that the force per length between parallel currents separated by d ∝(((Subscript[I, 1])(Subscript[I, 2]))/d). 
We make an arbitrary choice for the constant of proportionality as ((Subscript[μ, 0])/(2π)) where (Subscript[μ, 0])=4π ×(10^(-7))N (A^(-2)) the vacuum permeability. 
Use this to define the unit of current, the Ampere (A) and hence the unit of charge, the  A s or Coulomb (C).
We can also use this to define the unit of magnetic flux density B=(((Subscript[μ, 0])I)/(2πd)) in T.
The force between two charges is written as (((Subscript[Q, 1])(Subscript[Q, 2]))/(4(Subscript[πϵ, 0])(r^2))).
Historically, we used this to measure vacuum permittivity: (Subscript[ϵ, 0])=8.85 (10^(-12))F (m^(-1))and verify a prediction of the theory that ((((Subscript[μ, 0])(Subscript[ϵ, 0])))^(-1/2)) =c = 300 Mm (s^(-1)), the measured speed of light.
Today, we trust the theory and use (Subscript[ϵ, 0])=(Subscript[μ, 0]^(-1))(c^(-2)).
The electric field E=(Q/(4(Subscript[πϵ, 0])(r^2))) is measured in (Vm^(-1)).

Current density

dI = j dS where area dS is perpendicular to flow of charge

Fields and their Description (Z34, 1-10, 15-19)

Lines of Force

Faraday’s creation of fields is conceptually superior to Newton’s “action at a distance” and essential to explain variation and waves. Even more important in QM!
Maxwell took the idea even further, de-emphasizing  the sources.

Scalar

e.g. electric potential, φ, number defined initially at a point and then generalized to everywhere in some volume of space at a time t (in Newtonian language) as a field.

Vector

Position Vector

Elementary vector connectan origin O to a point P in Euclidean space - r(P). r has a length or norm |r|.
Can add, subtract, multiply by number.
Two approaches, both useful

Analysis

Three numbers e.g. x≡(Subscript[r, i]), i=1,2,3 = x,y,z- use  Roman for 3-vectors in space. 
Presupposes existence of a basis (Subscript[e, (i)]) - Cartesian for the moment. In this approach a vector can be defined as a list of 3 numbers that transforms in a particular way under rotations, say.
Distinguish active rotation of vector in fixed basis from passive rotation of basis with fixed vector.
Introduce orthogonal transformation matrix (Subscript[A, ij]) with (A^T)=(A^(-1)). For passive case (Subscript[e', (i)])=(Subscript[A, ij])(Subscript[e, (j)]) , (Subscript[r', i])=(Subscript[A, ij])(Subscript[r, j])
nb|r|≡r=((((Subscript[r, i])(Subscript[r, i])))^(1/2))adopting Einstein summation convention.
Useful for calculation. 
Generalizes to abstract spaces of QM.

Geometry

Physical quantity that exists independent of the basis. 
Prototype is an infinitesimal displacement from a point in Euclidean space to neighboring point. Lives in tangent space. 
Measure length as multiple of unit length. 
Useful conceptually.

Relationship

Interpret component (Subscript[r, i]) geometrically as projection of vector r along basis vector (Subscript[e, (i)]).

Physical Vectors

Electromagnetic vectors

Take an infinitesimal displacement dr and multiply by (dt^(-1))to get v after taking the limit. Repeat to get a. Multiply by m to get F, divide by q to get E take ratio to Lorentz force to get B and so on, either analyically or geometrically

Connection

In order to calculate a we need to transport v(t+dt) from one point to a neighboring point and subtract it from v(t) there. 
The transport is geometrically unambiguous in flat space (Euclid's 5th axiom!) and operationally trivial if we use Cartesian coordinates. 
If we are working in polar coordinates, we can always get an answer by transforming into Cartesian coordinates and then back again.
However if the space is not flat - eg the surface of a sphere or general relativity, we have to define how to perform this transport using a connection. 
We will discuss some general ways of doing this.

Vector Field

As with scalars we can imagine a vector field e.g. E(r;t) defined everywhere in some volume at a given time t.

Tensors

Analytical

Juxtaposition of  r (for rank) vectors - (3^r) numbers, define by transformation properties under rotation eg define a form of inertia tensor (Subscript[I, ij])=(Subscript[mr, i])(Subscript[r, j]) for one particle.
Second rank tensors are sometimes called dyadics.

Geometrical

Real function of vectors. T[a,b]≡a·T·b independent of basis.

Relationship

(Subscript[I, ij]) = I[(Subscript[e, (i)]),(Subscript[e, (j)])]. Order of slots matters unless symmetric. Slots can be left open.

Special tensors

Unit tensor I[(Subscript[e, (i)]),(Subscript[e, (j)]) ]= (Subscript[I, ij])=(Subscript[δ, ij]); metric tensor for Cartesian coordinates, which we will focus on now
Levi-Civita tensor ϵ[(Subscript[e, (1)]),(Subscript[e, (2)]),(Subscript[e, (3)])]=+1, etc and (Subscript[ϵ, ijm])(Subscript[ϵ, klm])=(Subscript[δ, ik])(Subscript[δ, jl])-(Subscript[δ, il])(Subscript[δ, jk]) is very useful.

Products

Scalar (dot) product.

Analytical: define cE·B=(Subscript[cE, i])(Subscript[B, i ])
. This is invariant under rotation. 
Geometrical: cE·B = (1/4)(|E+cB((|)^2)-|E-cB((|)^2)) or E·(E+cB)-(E^2), which has a simple interpretation in terms of area.
An equivalent way to think about this is to treat either vector as a function that accepts the other vector to create a real number e.g. E(B)=B(A)≡E·B

Vector (cross) product.

Analytical: (Subscript[(E×B), i])=(Subscript[ϵ, ijk])(Subscript[E, j])(Subscript[B, k]); express as determinant.
Geometrical: Area of parallelogram formed by E, B 
Either way a vector along direction perpendicular to plane containing E, B with right-handed convention.
nb dS=ϵ[ , (Subscript[dx, (1)]),(Subscript[dx, (2)])]; dV=ϵ[(Subscript[dx, (1)]),(Subscript[dx, (2)]),(Subscript[dx, (3)])].

Outer (Direct, Tensor) Product

I = mr⊗r =(Subscript[I, ij]) (Subscript[e, (i)])⊗(Subscript[e, (j)]) etc

Contraction

Reduce rank of tensor by 2.
Analytical: I=(Subscript[I, ii])
Geometrical: replace ⊗ in vector expansion with · . E·B is contraction of E⊗B.

Differentiation

Gradient

Vectors can be created by differentiating scalars.
Analytical: (Subscript[E, i])=-((∂φ)/(∂(Subscript[r, i])))≡-∇φ≡-(Subscript[φ, ,i]) in electrostatics
Geometrical: Introduce directional derivative along vector v :W= -q(∂_v)φ =qv·E≡qv(E) - argument of scalar product that produces the scalar result.  
Either way, E=-∇φ

Divergence

Create scalar by differentiating vector field.
Analytical: Treat (∂/(∂(Subscript[r, i])))≡(∂_i) as vector - it transforms under rotation like a vector - and form scalar product with another vector, e.g. E. to form a scalar that is invariant under rotation of coordinates.
Geometrical: Form tensor ∇E≡∇⊗E and then contract to form ∇·E=(ρ/(Subscript[ϵ, 0])).

Curl

Create vector by differentiating vector field j=(1/(Subscript[μ, 0]))∇×B in magnetostatics. Treat just like cross product.

Integration

Charge and Current

Q =∫dVρ; I=∫dS·j

Gauss’ Theorem

∫dV∇·E=∫dS·E

Stokes’ Theorem

∫dS·∇×B=∮ dr·B if region compact.

Dynamics (Z29)

Charged particles

Ironically, Maxwell saw charge as derivative of the fields, not as particles. It was 30 yr before Heaviside and Lorentz understood the dynamics of charged particles! 
((dp)/dt)=q(E+v×B); ((de)/dt)=q(E·v)
If we smooth the charges, ρ=(Subscript[Σ, i])(Subscript[q, i])δ(r-(Subscript[r, i]))  j=(Subscript[Σ, i])(Subscript[q, i])(Subscript[v, i])δ(r-(Subscript[r, i]))

Force and Power density

The force density - exerted on surroundings - is f=ρE+j×B and the power density - rate of doing work on surroundings per unit volume - is W=E·j

Maxwell' s equations (Z33-37)

Start with vacuum version using E, B.

M1

∇·E =(( ρ)/(Subscript[ϵ, 0]))

Inertial Frames

nb Maxwell's equations, like Newton's laws of motion, are valid in inertial frames. Extra terms arise in rotating/accelerating frames eg (Subscript[ρ, rot])=-2(Subscript[ϵ, 0])Ω·B

M2

∇·B=0;

Monopoles

Could restate Maxwell with rotated  fields: (2^(-1/2))(E±cB) to make symmetric but this is not helpful!  Cabrera event; Dirac quantization; Inflation argument

Magnetic Flux

Φ=∫dS·B, measured in Wb vanishes for closed surface.

M3

((∂B)/(∂t))=-∇×E; Faraday’s Law
=>((∂∇·B)/(∂t))=0 so M2 is just an initial condition!

Electrostatics

If stationary, ∃φ: E=-∇φ and M1=>Poisson:(∇^2)φ=-(ρ/(Subscript[ϵ, 0])) (Laplace when ρ=0).

M4

∇×B =(Subscript[μ, 0]) j+(Subscript[μ, 0])(Subscript[ϵ, 0])((∂E)/(∂t))

Charge conservation

Combining M1+M4: ((∂ρ)/(∂t))+∇·j=0. Integrate over volume and use Gauss' Th:  (dQ/(dT  ))+I=0

Magnetic Field

If stationary, Stokes Th. =>Ampere: ∮ dr·B=(Subscript[μ, 0])I and Biot-Savart: B=((Subscript[μ, 0])/(4π))∫dV'((j(r')(r-r'))/(|r-r'((|)^3)))

Vector Potential

Introduce A: B=∇×A. Gauge freedom must be curtailed, exploited. In electro-, magnetostatics, usually choose Coulomb gauge: ∇·A=0.

Maxwell in Media

Electric Displacement

D=ϵE = (Subscript[ϵ, 0])E+P=(1+χ)(Subscript[ϵ, 0])E, where ϵ is the permittivity and χ is the electric susceptibility. These are usually scalars but may be complex, tensors or nonlinear.
D is measured in C (m^(-2)).

Magnetic Field

B = μH=(Subscript[μ, 0])(H+M)=(1+(Subscript[χ, m]))(Subscript[μ, 0])H, where μ is the permeability and (Subscript[χ, m])the magnetic susceptibility. 
H is measured in (Am^(-1))

Modified Equations

In terms of "free" charges and currents, 
∇·D=(Subscript[ρ, f]);  ∇×H=(Subscript[j, f])+((∂D)/(∂t)); M2, M3 unchanged.

Energy and Momentum (Z508-513)

Poynting Theorem

-W= -E·j = E·((Subscript[ϵ, 0]) ((∂E)/(∂t))-((∇×B)/(Subscript[μ, 0]))) =(Subscript[ϵ, 0])E·((∂E)/(∂t))  +∇·(((E×B)/(Subscript[μ, 0]))) -(B/(Subscript[μ, 0]))·∇×E= ((∂U)/(∂t))+∇·S, where U=(((Subscript[ϵ, 0]) (E^2))/2)+((B^2)/(2(Subscript[μ, 0])))is the energy density and S=((E×B)/(Subscript[μ, 0])) is the energy flux.

Maxwell Stress Tensor

-f = -(Subscript[ϵ, 0])E∇·E-(((∇×B)/((Subscript[μ, 0]) ))-(Subscript[ϵ, 0])((∂E)/(∂t)))×B=(Subscript[ϵ, 0])((∂E×B)/(∂t))-(Subscript[ϵ, 0])(E∇·E-E×∇×E)-(1/(Subscript[μ, 0]))(B∇·B-B×∇×B)=((∂g)/(∂t))+∇·(Subscript[T, M]), 
where (Subscript[T, M])=(Subscript[ϵ, 0])(((E^2)/2)I-E⊗E)+(1/(Subscript[μ, 0]))(((B^2)/2)I-Β⊗B); is the stress tensor and g=(Subscript[ϵ, 0])E×B =(Subscript[μ, 0])(Subscript[ϵ, 0])S is the momentum density so the stress-energy tensor is anisotropicas it should be.
The stress tensor exhibits isotropic pressure and anisotropic tension.

Special Relativistic Electrodynamics (Z959-963)

Foundations

Principle of Relativity

Laws of physics same form in all reference frames
Maxwell predicts speed of light in vacuo from electrostatic and magnetostatic experiments - same in all inertial frames: (∇^2)E=(Subscript[μ, 0])(Subscript[ϵ, 0])(((∂^2)E)/(∂(t^2)))- so not really a separate principle! 
Henceforth set c=1 when working relativistically.

Kinematics

Coordinates

Label a point in spacetime by 4-vector (x^α)=[t,r]=[t,x,y,z].  
Label coordinates by α, β = 0,1,2,3. Greek for SR; Roman for Newtonian
Distinguish super/subscripts, covariant/contravariant forms/vectors...

Lorentz transformation

Lorentz coordinate transformation x'=Λ·x. where matrix Λ=(({{γ, -γβ, 0, 0}, {-γβ, γ, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}})) for motion along x, where γ=(((1-(β^2)))^(-1/2)). 
All 4-vectors (V^α) transform in this manner. Active and passive transformations.

Metric

Introduce Minkowski tensor (Subscript[η, αβ])=(({{-1, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}})) for Cartesian coordinates so that (ds^2)=(Subscript[η, αβ])(dx^α)(dx^β) is invariant.
Use for raising and lowering indices e.g.(Subscript[j, α])=(Subscript[η, αβ])(j^β)etc. nb ((Subscript[η, α])^β)≡ ((Subscript[δ, α])^β).

Dynamics

(4-)momentum vector: p =mu=m(dx/(dτ));  Components (p^α)≡[ϵ,p], (Subscript[p, α])≡[-ϵ,p]

Current

(j^α)=[ρ,j],  ((∂ρ)/(∂t))+(∂/(∂r))·j ≡∇· j≡(∂_α)(j^α)≡(Subscript[j, ,α]^α)=0

Vector Potential

E, B live in space of specific observers and are not 4-vectors.
(A^α)=[φ,A] is 4-vector with B=∇×A. E=-((∂φ)/(∂r))-((∂A)/(∂t))
Gauge freedom: Often use Lorenz gauge: ∇·A=(Subscript[(A^α), ,α])=0

Field Tensor

(Subscript[F, αβ])=(Subscript[A, β,α])-(Subscript[A, α,β]); (Subscript[(F^(αβ)), ,β])=-(Subscript[(A^(α,β)), ,β])=(Subscript[μ, 0])(j^α); (Subscript[F, [αβ,γ]])=0, where [ ] denotes a symmetric combination.

Equation of Motion and Force Density

(dp/(dτ))=qF·u which is equivalent to the expressions given above 
4-force density: f=F·j

Stress-Energy Tensor

(Subscript[T, em])=(({{U, S}, {g, (Subscript[T, M])}})); (T^(αβ))=(1/(Subscript[μ, 0]))((1/4)(Subscript[F, γδ])(F^(δγ))(η^(αβ)) -(Subscript[(F^α), γ])(F^(γβ))). 
Recall that energy flux S is the same as the momentum density g so (Subscript[T, EM ])is symmetric.
In addition f=-∇·(Subscript[T, em]).
Conservation Law: ∇·T=∇·((Subscript[T, em])+(Subscript[T, everything else]))=0

Homework I

1. a) Use the Levi-Civita tensor to show that ∇x(vxr)=2v+r(∂_r)v -r∇·v 
b) Show that ((∫_0)^∞)dxδ(cos x) (e^(-x))=(1/(2 ))cosech(π/2)
c) Show that ∮ dsxv=∫(dSx∇)xv and hence deduce that ∮ dsxr=-2∫dS.
2. If the photon were massive, the Poisson equation would be modified to: (∇^2)φ = -(ρ/(Subscript[ϵ, 0]))+(λ^2)φ, where λ=mc/ℏ.
Limits on m can be set using a Cavendish-type experiment comprising a solid, conducting sphere of radius a surrounded by a concentric, conducting thin shell of radius b maintained at a common potential V by a connecting wire.
By expanding to quadratic order in λ, show that the charge residing on the inner sphere is given approximately by Q = 4 π (Subscript[ϵ, 0])V (λ^2)(a^3) f(b/a), where f is a function that you should determine.
3. Use the transformation laws for E, B to demonstrate the invariance of E·B, (c^2) (B^2) -(E^2).
4. Explain why the (relativistic) stress-energy tensor for a collection of charged particles can be written in the manifestly symmetric form (∑_n)(((Subscript[p, n])⊗(Subscript[p, n]))/(Subscript[p, n]^0)) δ(x-(Subscript[x, n])) where p, x denote 4-vectors.
Hence verify that the combined (electromagnetic plus particle) stress-energy tensor has vanishing divergence. (This exercise is a bit of a challenge involving maniplating delta functions. Don’t spend too long on it.)

Sketch Solutions

1. a) (Subscript[ϵ, ijk]) (∂_j)(Subscript[ϵ, klm]) (Subscript[v, l])(Subscript[r, m])= ((Subscript[δ, il])(Subscript[δ, jm])-(Subscript[δ, im])(Subscript[δ, jl]))(∂_j)(Subscript[v, l])(Subscript[r, m])=(∂_j)(Subscript[v, i])(Subscript[r, j])-(∂_j)(Subscript[v, j])(Subscript[r, i])=3(Subscript[v, i])+r(∂_r)(Subscript[v, i])-(Subscript[r, i])∇·v-(Subscript[v, i])
b) =((e^(-π/2))/(|sinπ/2|))+((e^(-3π/2))/(|sin3π/2|))+((e^(-5π/2))/(|sin5π/2|))+…=((e^(-π/2))/(1-(e^(-π))))
c) ∮ds·(k×v)=-k·∮ ds×v=∫dS·∇×(k×v)=k·∫dS∇·v -k·∫(Subscript[dS, i])∇(Subscript[v, i])=k·∫(dS×∇)×v, ∀k.
=(Subscript[ϵ, ijk])(Subscript[ϵ, jlm])∫(Subscript[dS, l])(Subscript[∇, m])(Subscript[r, k])
2. φ=((Ae^(λ(r-a)))/r)+((Be^(-λ(r-a)))/r)=(((A+B))/a) at r=a 
φ’=(((Aλe)^(λ(r-a)))/r)-(((Bλe)^(-λ(r-a)))/r)-((Ae^(λ(r-a)))/(r^2))-((Be^(-λ(r-a)))/(r^2))=(((A-B)λ)/a)-(((A+B))/(a^2)) at r=a
A+B=aV; (A+B)(1+((λ^2)/2)(((b-a))^2))+(A-B)λ(b-a)=bV=>A-B=(V/λ)-((λaV(b-a))/2)
Gauss=>4((πa)^2)φ'(a)=-(Q/(Subscript[ϵ, 0]))+((4((πa)^3))/3)φ(a)=> Q=((2(Subscript[πϵ, 0])(λ^2) (Va^2)(3 b - a))/3)?
3. (Subscript[E, ⊥]^('))=γ((Subscript[E, ⊥])+v×B), (Subscript[E, ||]^('))=(Subscript[E, ||]),(Subscript[B, ⊥]^('))=γ((Subscript[B, ⊥])-v×E),(Subscript[B, ||]^('))=(Subscript[B, ||])
4. (Subscript[T, p])=(∑_n)(Subscript[p, n])⊗[1,(Subscript[v, n])] δ(x-(Subscript[x, n]))
(1/(Subscript[μ, 0])) (Subscript[((1/4)(Subscript[F, γδ])(F^(δγ)) (η^(αβ))-(Subscript[(F^α), γ])(F^(γβ))), ,β])=-(Subscript[(F^α), β])(J^β)
(Subscript[((Subscript[T, p])^(αi)), ,i])= (∑_n)(Subscript[p, n]^α)((Subscript[dx, n]^i)/dt)(∂/(∂(x^i)))δ(x -(Subscript[x, n]))=-(∑_n)(Subscript[p, n]^α)((Subscript[dx, n]^i)/dt)(∂/(∂(Subscript[x, n]^i)))δ(x -(Subscript[x, n]))=-(∑_n)(Subscript[p, n]^α)(∂/(∂t))δ(x -(Subscript[x, n]))=-(∂_t)(∑_n)(Subscript[p, n]^α)δ(x -(Subscript[x, n]))+(∑_n)((dp^α)/dt)δ(x -(Subscript[x, n]))=>(Subscript[((Subscript[T, p])^(αβ)), ,β])=(∑_n)((Subscript[dp, n]^α)/dt)δ(x -(Subscript[x, n]))=(∑_n)(Subscript[q, n])(Subscript[(F^α), β])((Subscript[dr, n]^β)/dt)δ(x -(Subscript[x, n]))=(Subscript[(F^α), β])(J^β)

2. Statics

Reading

Z  90-92, 96-97, 108-111, 130-131, 133-134, 199, 203-205, 212-213, 215-217, 237-238, 252-254.
Just look these over and note places where material is new to you.

Electric Field

Field Strength

Breakdown of Air 1-3 MV (m^(-1))
Lightning: 100C, 100MV, 10 GJ for example
Person: C~200pF, φ ~ 30kV, U~100mJ for example

Force and Energy

(Subscript[F, 12]) = ∫(Subscript[dq, 1])(Subscript[E, 2])= (1/(4(Subscript[πϵ, 0])))∫(Subscript[dq, 1])(Subscript[dq, 2])(((Subscript[r, 1])-(Subscript[r, 2]))/(|(Subscript[r, 1])-(Subscript[r, 2])((|)^3))); U=(1/2)∫dqφ=((Subscript[ϵ, 0])/2)∫(dVE^2)using Gauss if integrate out to infinity.

Electric Stress Tensor

Consider two opposite charges q separated by d. F=-((q^2)/(4(Subscript[πϵ, 0])(d^2))). At midplane, (Subscript[E, ⊥])=2 ×(( q)/(4(Subscript[πϵ, 0])((((d/2))^2)+(ρ^2))))× ((d/2 )/((((((d/2))^2)+(ρ^2)))^(1/2))), (Subscript[E, ||])=0.
Electric pressure=(((Subscript[ϵ, 0])(Subscript[E, ⊥]^2))/2), electric tension=(Subscript[ϵ, 0])(Subscript[E, ⊥]^2), so (Subscript[T, M])=-(((Subscript[ϵ, 0])(Subscript[E, ⊥]^2))/2)=> F = (∫_0^∞)(Subscript[dρ2πρT, M])

Laplace’s Equation

Boundary Conditions and Green Theorem

Solve (∇^2)φ=0 plus boundary conditions, such as conductors, isolated charges, surface charge distribution.
Solution unique with Dirichlet, Neumann and mixed boundary conditions. 
Analytic approach is to find eigenfunctions appropropriate to the geometry and to exploit linearity to sum them to match the boundary conditions.
Simple solution: φ varies linearly, E const.
G(r,r')=(1/(4(Subscript[πϵ, 0])|r-r'|)); φ=∫dq’G+surface terms that reflect the boundary conditions.

Cartesian Separation of Variables

Eigenfunctions

φ=X(x)Y(y)Z(z); X'' =-(Subscript[k, x]^2) X etc, where  (k^2)=(Subscript[k, x]^2)+(Subscript[k, y]^2)+(Subscript[k, z]^2)=0.
Choose exponential or hyperbolic/trigonometric for convenience.

Square Duct

φ(x,±π)=±1, φ(±π,y)=0.
Natural eigenfunctions are cosh nx sin ny.

φ0=(( y)/π) ;
φ1= -.05Cosh[x]Sin[y];
φ2=.001Cosh[2x]Sin[2y];
φp=φ0+φ1+φ2;
φsp=φ0+φ1+φ2/.x->π;
(∂_(x,x))φp+(∂_(y,y))φp;
ContourPlot[φp,{x,-π,π},{y,-π,π}];
Plot[{φ0,φ1/.x->π,φ2/.x->π,φsp},{y,-π,π},Frame->True,PlotRange->{-1,1},PlotStyle->{Dashed,Dashed,Dashed,Thick},FrameLabel->{"y","φ(π,y)"},RotateLabel->False]

More formally, (y/π) + (∑_(n=1)^∞)(Subscript[a, n])cosh nπ sin ny=0 which is easy to invert for (Subscript[a, n]).

nmax=50;
a=N[Table[-(((∫_(-π)^π)y Sin[n y]\[DifferentialD]y)/((π^2)Cosh[n π])),{n,1,nmax}]];
φ=(y/π)+(∑_(n=1)^nmax)a[[n]]Cosh[n x]Sin[n y];φs=(y/π)+(∑_(n=1)^nmax)a[[n]]Cosh[n π]Sin[n y];vE=-{(∂_x)φ,(∂_y)φ};
ContourPlot[φ,{x,-π,π},{y,-π,π}]
Plot[{φ,φs},{y,-π,π},Frame->True,PlotRange->{-1,1},FrameLabel->{"y","φ(π,y)"},RotateLabel->False,FrameLabel->{"y","φ(π,y)"},RotateLabel->False]
StreamDensityPlot[{vE,φ},{x,-2.8,2.8},{y,-2.8,2.8},ColorFunction->Hue];
VectorDensityPlot[{vE,φ},{x,-2.8,2.8},{y,-2.8,2.8},ColorFunction->Hue]

Cylindrical Separation of Variables

2D: φ=Ρ(ρ)Φ(ϕ); P= const.+  ∑(Subscript[a, m]) (ρ^m),((ρ^0)->lnρ); Φ=(e^(imϕ))
3D: φ=Ρ(ρ)Φ(ϕ)Z(z);(((ρΡ')')/((k^2)ρ))+(1-((m^2)/((k^2)(ρ^2))))Ρ=0; Ρ = (Subscript[a, m])(Subscript[J, m]) (kρ)+(Subscript[b, m])(Subscript[Y, m])(kρ),  (Subscript[a, m])(Subscript[I, m]) (ikρ)+(Subscript[b, m])(Subscript[K, m])(ikρ) for k real imaginary; Φ=(e^(imϕ));Z=(e^ikz).

n=2;Plot[{BesselJ[n,x],BesselY[n,x],BesselI[n,x],BesselK[n,x]},{x,0,10},Frame->True,PlotRange->{-1,1},FrameLabel->{"x","J,Y,I,K"},RotateLabel->False]

Spherical  Separation of Variables

Spherical Harmonics

φ=R(r)(Subscript[Y, lm])(θ,ϕ);((r^2)R')'+l(l+1)R=0; sinθ (∂_θ)sinθ (∂_θ)(Subscript[Y, lm])-(m^2)(Subscript[Y, lm])+l(l+1)(sin^2)(Subscript[θY, lm])=0; (Subscript[Y, lm])∝(e^(imϕ)); ∫(Subscript[dΩY, l'm']^( * ))(Subscript[Y, lm]) = (Subscript[δ, ll'])(Subscript[δ, mm']);(Subscript[Y, l-m])=(((-))^m)(Subscript[Y, lm]^( * )).
(Subscript[Y, 00])=(1/(2(π^(1/2))));
(Subscript[Y, 1m])=((((3/(8π))))^(1/2))[sinθ (e^(-iϕ)),(2^(1/2))cosθ,-((sinθe)^(iϕ))];
(Subscript[Y, 2m])=-((((15/(32π))))^(1/2))[(sin^2)θ (e^(-2iϕ)),2cosθ ((sinθe)^(-iϕ)),((((2/3)))^(1/2))(3(cos^2)θ-1),-2cosθ ((sinθe)^(iϕ)),(sin^2)θ (e^(2iϕ))]

l=3;m=2;SphericalHarmonicY[l,m,θ,ϕ];
(∫_0^π)(∫_0^(2π))Sin[θ]SphericalHarmonicY[l,m,θ,ϕ]SphericalHarmonicY[l,m,θ,ϕ]\[Conjugate]\[DifferentialD]ϕ\[DifferentialD]θ;
ContourPlot[Re[SphericalHarmonicY[l,m,θ,ϕ]],{ϕ,0,2π},{θ,0,π},FrameLabel->{"ϕ","θ"},RotateLabel->False]

Expansion of Potential

φ=(1/(4(Subscript[πϵ, 0])))∫((dq')/(|r-r'|))=(1/(4(Subscript[πϵ, 0])))∫((dq')/((((r^2)+((r')^2)-2rr'cosγ))^(1/2)))φ=(1/(4(Subscript[πϵ, 0])(r^(l+1))))∫dq'(∑_l)((r')^l)(Subscript[P, l])(cosγ)= (((Subscript[Y, lm])(θ,ϕ))/((2l+1)(Subscript[ϵ, 0])(r^(l+1))))∫dq'((r')^l)(Subscript[Y, lm]^( * ))(θ',ϕ'), r>r' (=(((r^l)(Subscript[Y, lm]^( * ))(θ,ϕ))/((2l+1)(Subscript[ϵ, 0])))∫dq'(((Subscript[Y, lm])(θ',ϕ'))/((r')^(l+1))),r<r').
φ= ((q')/(4(Subscript[πϵ, 0])r))+(1/(4(Subscript[πϵ, 0])(r^2)))∫dq'r'(cosθcosθ'+sinθsinθ'cos(ϕ-ϕ'))+(1/(4(Subscript[πϵ, 0])(r^3)))∫dq'((r')^2)…
=((q')/(4(Subscript[πϵ, 0])r))+((p·r)/(4(Subscript[πϵ, 0])(r^3)))+((3r·(OverTilde[Q])·r-Tr(OverTilde[Q])(r^2))/(4(Subscript[πϵ, 0])(r^5)))+… where p=∫dq'r', (OverTilde[Q])=(1/2)∫dq'r'⊗r'
=((q')/(4(Subscript[πϵ, 0])r))+((p·r)/(4(Subscript[πϵ, 0])(r^3)))+((r·Q·r)/(4(Subscript[πϵ, 0])(r^5)))+…where Q=3(OverTilde[Q])-Tr[(OverTilde[Q]])I  using (Subscript[r, i]^('))(Subscript[r, j]^('))(r^2)(Subscript[δ, ij])=(Subscript[r, i])(Subscript[r, j])((r')^2)(Subscript[δ, ij]). Q is traceless and has 5, not 6 elements.

l=2;Assuming[θ'∈Reals&&ϕ'∈Reals,FullSimplify[(∑_(m=-l)^l) SphericalHarmonicY[l,m,θ,ϕ]SphericalHarmonicY[l,m,θ',ϕ']\[Conjugate]]]

Multipoles

Dipole

Field Lines

φ=((p·r)/(4(Subscript[πϵ, 0])(r^3))); E=((3(p·r)r-((pr)^2))/(4(Subscript[πϵ, 0])(r^5))); 
Field lines: (1/r)(dr/(dθ))=2cotθ=>r∝(sin^2)θ; flux conservation

φ=If[(x^2)+(y^2)>1,(y/((((x^2)+(y^2)))^(3/2))),y];vE=-{(∂_x)φ,(∂_y)φ};
Show[StreamDensityPlot[{vE,φ},{x,-5,5},{y,-5,5},ColorFunction->Hue],Graphics[Circle[] ]]
Show[VectorDensityPlot[{vE,φ},{x,-5,5},{y,-5,5},ColorFunction->Hue],Graphics[Circle[] ]]

Force, Energy and Torque

F=p·∇E; U=-p·(Subscript[E, ext])= -(1/2)(Subscript[p, i])·E((Subscript[r, i])); N=p×E

Quadrupole

φ=((r·Q·r)/(4(Subscript[πϵ, 0])(r^5))); Q traceless.

φ=((-(x^2)-(y^2)+2(z^2))/((((x^2)+(y^2)+(z^2)))^(5/2)));vE=Simplify[-{(∂_x)φ,(∂_y)φ,(∂_z)φ}];
Show[StreamPlot[{vE[[1]],vE[[3]]}/.y->0,{x,-5,5},{z,-5,5}],Graphics[Disk[]]]

Magnetic Field

Field Strength

Examples

Magnetar - 30GT; regular neutron star - 100 MT; largest permanent magnet - 45 T (nb ((B^2)/(2(Subscript[μ, 0])))~1 GJ ~(10^4)atm ~(10^(-2 ))Elastic Moduli ~ yield stress - be careful!), sunspot ~ 100mT, brain activity ~ 1pT, smallest detectable field - GP-B squid ~ 1 aT

Magnetic Potential

If source currents are localized, can use magnetic potential to solve for field - B=-∇ψ. 
Also possible for current-free region around extended current, but if not ``simply-connected’’, ψ is multiple-valued - better to use A. 
Often when unlocalized current is small, as in magnetospheres, coronae, potential field is a fair approximation.

Multipoles

ψ=((Subscript[μ, 0])/(4π))(((m·r)/(r^3))+((r·M·r)/(r^5))+…)

LHC

Magnets

Designed with ~4.3 km radius for 7 TeV protons with 8T magnets; ~1400 dipoles, ~400 quadrupoles... ((p^0)/B)~ 3km

Quadrupolar Lens

Consider a short quadrupolar potential magnet ψ=-(((Subscript[B, 0])x y)/λ). 
A particle traveling almost parallel to the z-direction receives a (linear) impulse (Subscript[Δp, ⊥])=(((Subscript[B, 0])t)/(λ ))[-x,y], where t is the thickness. 
Place a second magnet a distance d along the beam, rotate through (90^o) and you have a circularly symmetric converging lens. 
Demonstrate using matrices to describe the influence of the magnets and the propagation.
Used in electron microscopes too.

Dark Matter?

Fermi

Gamma ray satellite in 550 km altitude orbit, (29^o)inclination.

AMS

150 mT magnet on ISS  to distinguish matter from antimatter, positrons from electrons

Positron Spectrum

Recent results. Possible signature of dark matter or pulsars or propagation.

Fermi measurement uses geomagnetic field to separate charges.

Geomagnetic Field

Earth

Mean radius (Subscript[R, E])=6371km, - unit of length (a=6384km, c=6353 km) field in μT. On surface,  23<|B|<67μT. Magnetopause at 10 (Subscript[R, E]).

Multipole Expansion

2010 expansion in associated Legendre polynomials.
Convert to Cartesian coordinates.

Assuming[0≤θ<π&&0≤ϕ<2π,FullSimplify[Table[{1,m,LegendreP[1,m,Cos[θ]],SphericalHarmonicY[1,m,θ,ϕ]},{m,0,1}]]];
Assuming[0≤θ<π&&0≤ϕ<2π,FullSimplify[Table[{2,m,LegendreP[2,m,Cos[θ]],SphericalHarmonicY[2,m,θ,ϕ]},{m,0,2}]]];
ψ=(r^(-2))(-29.496Cos[θ]+Sin[θ](-1.586Cos[ϕ]+4.945Sin[ϕ]))+(r^(-3))(-2.397(1.5((Cos[θ])^2)-.5)+(3^(1/2))Cos[θ]Sin[θ](3.026Cos[ϕ]-2.708Sin[ϕ])+.5 (3^(1/2))((Sin[θ])^2)(1.669Cos[2ϕ]-.575Sin[2ϕ]));

Bs=Assuming[r>0,FullSimplify[Chop[-{(∂_r)ψ,(1/r)(∂_θ)ψ,(1/(r Sin[θ]))(∂_ϕ)ψ}]]]
Show[ContourPlot[Norm[Bs]/.{θ->θ,r->1.},{ϕ,-π,π},{θ,0,π},FrameLabel->{ϕ,θ},RotateLabel->False]]
Rcs={{(x/((((x^2)+(y^2)+(z^2)))^(1/2))),((x z)/(((((x^2)+(y^2)))^(1/2))((((x^2)+(y^2)+(z^2)))^(1/2)))),((-y)/((((x^2)+(y^2)))^(1/2)))},{(y/((((x^2)+(y^2)+(z^2)))^(1/2))),((y z)/(((((x^2)+(y^2)))^(1/2))((((x^2)+(y^2)+(z^2)))^(1/2)))),(x/((((x^2)+(y^2)))^(1/2)))},{(z/((((x^2)+(y^2)+(z^2)))^(1/2))),-(((((x^2)+(y^2)))^(1/2))/((((x^2)+(y^2)+(z^2)))^(1/2))),0}};
Bc=Chop[FullSimplify[Rcs.Bs/.{Cos[θ]->(z/((((x^2)+(y^2)+(z^2)))^(1/2))),Sin[θ]->(((((x^2)+(y^2)))^(1/2))/((((x^2)+(y^2)+(z^2)))^(1/2))),Cos[ϕ]->(x/((((x^2)+(y^2)))^(1/2))),Sin[ϕ]->(y/((((x^2)+(y^2)))^(1/2))),Cos[2ϕ]->(((x^2)-(y^2))/((x^2)+(y^2))),Sin[2ϕ]->((2x y)/((x^2)+(y^2))),r->((((x^2)+(y^2)+(z^2)))^(1/2))}]];
Show[StreamPlot[{Bc[[1]],Bc[[3]]}/.y->0,{x,-5,5},{z,-5,5}],Graphics[Disk[] ]]

Equation of motion for relativistic cosmic rays

((dx)/(dτ))=u; ((du)/(dτ))=((quxB)/m)

Historical Record

In 1900, the magnetic potential was

ψ=(r^(-2))(-31.543Cos[θ]-Sin[θ](-2.298Cos[ϕ]+5.922Sin[ϕ]))+(r^(-3))(-.677(1.5((Cos[θ])^2)-.5)-3Cos[θ]Sin[θ](2.905Cos[ϕ]-1.061Sin[ϕ])+3((Sin[θ])^2)(.924Cos[2ϕ]-1.121Sin[2ϕ]));

Penning Trap

Uniform field,(Subscript[B, 0]), two hyperboloidal end caps and hyperboloid of revolution ring oppositely charged.
φ = K((z^2)-(1/2)((x^2)+(y^2)))

Homework

1. Evaluate the magnetic stress tensor on the midplane between two unequal but parallel dipoles. Integrate the stress over the plane and interpret carefully your answer.
2. Consider the description of the Earth’s magnetic potential above and just use the first term. Give the magnetic dipole moment in SI units and evaluate the magnetic field in μT adopting Cartesian coordinates.
Write a routine that will integrate backward in time the trajectories of cosmic ray electrons and positrons with energies similar to those measured by Fermi as the satellite crosses the Earth’s equatorial plane.
Show a few trajectories to demonstrate how charge discrimination is possible. (If this is too challenging, just explain how to do this in principle.) Comment on the difference between the 1900 and 2010 data.
3. Consider a charged particle in a Penning trap as described above. Show that the axial motion is harmonic and find its frequency. Next, let K=0 and solve for the motion.
Finally write down the radial equation of motion with K≠0 and show that there are two eigenfrequencies. Describe the trajectories and show that there is a maximum value of K if the particle is to be bound.
4. Zangwill 4.19

3. Media

Reading

Z  158-9, 160-167, 175-177, 184-186, 407-411, 421-423, 430-432, 443-446.
Just look these over and note places where material is new to you.

Polarization and Magnetization

Influence of the Medium

Insulators: polarization charge contributes an additional field to (Subscript[E, ext ])(but does not cancel it completely as in a conductor). 
Magnetization current contributes additional B
We describe the average effect using Polarization P and Magnetization M which can be interpreted as the dipole moments per unit volume.
We combine with the actual E, B to define the Displacement D=(Subscript[ϵ, 0])E+P and the Magnetic Field H=(B/(Subscript[μ, 0]))-M.
The polarization charge density and magnetization current density are (Subscript[ρ, p])=-∇·P, (Subscript[j, m])= ∇xM   while the surface charges and currents are (Subscript[σ, p])=P·n, (Subscript[J, m])=Mxn  
D and H sourced by free charge, current: ∇·D = (Subscript[ρ, f]), ∇xH=(Subscript[j, f]); H=-∇ψ when (Subscript[j, f])=0
Unnecessary but possible  to introduce surface magnetic charge density.

Constitutive relations: P=(Subscript[χϵ, 0])E; D=(1+χ)(Subscript[ϵ, 0])E =ϵE, where ϵ is the permittivity, χ= κ-1 is the susceptibility which is often assumed to be linear but may be a tensor and κ is the dielectric constant.
Likewise, M=(Subscript[χ, m])H;  B = (Subscript[κ, m])(Subscript[μ, o])H=(1+(Subscript[χ, m]))(Subscript[μ, 0])H=μH, where (Subscript[χ, m])= (Subscript[κ, m])-1 is the magnetic susceptibility  and μ is the permeability.

Examples

All materials will create P, M when exposed to E, H. χ>0 but (Subscript[χ, m])<,>0 for diamagnetic (orbital-dominant) , paramagnetic (spin-dominant) materials. 
Ferromagnets with μ>>1(can be 10,000) dominated by local field below Curie temperature - organized in domains with moving boundaries. Subject to hysteresis, B~(Subscript[μ, 0])M , H~0.
Heat dissipated when cycle (Subscript[H, ext]); M not simple function of (Subscript[H, ext]).
Also have permanent electrets. Most materials have κ~ 2-30; can be 10,000.
Type 1 superconductors expel flux density - Meissner effect.

Sphere

Polarized

Unit radius, uniformly polarized, E->0 as r->∞.
 P=P(OverHat[z]), (Subscript[ρ, p])=0; (Subscript[σ, p])=P·(OverHat[r])=(Subscript[ϵ, 0])(Subscript[ΔE, ⊥]); (Subscript[E, int])=-(P/(3(Subscript[ϵ, 0])));(Subscript[D, int])=((2P)/(3(Subscript[ϵ, 0]))); (Subscript[E, ext])=((3(r·P)r-(r^2)P)/(3(Subscript[ϵ, 0]) (r^5)));(Subscript[D, ext])=((3(r·P)r-(r^2)P)/(3 (r^5))); p= 4 ((πa)^2)σ<(cos^2)θ>a=PV; (Subscript[E, ||])=-((Psinθ)/(3(Subscript[ϵ, 0]))), (Subscript[D, ⊥])=((2Pcosθ)/3) continuous,
Can superpose external field; for r<1, E =(Subscript[E, 0])+(Subscript[E, int]) =>P = (Subscript[χϵ, 0])E =((Subscript[χϵ, 0])/(1 + χ/3))(Subscript[E, 0])

E0=0;φ=If[(x^2)+(z^2)<1,(z/3)-E0 z,(z/(3((((x^2)+(z^2)))^(3/2))))-E0 z];vE=If[(x^2)+(z^2)<1,{0,E0-(1/3)},{((x z)/((((x^2)+(z^2)))^(3/2))),E0-((x^2)/(3((((x^2)+(z^2)))^(3/2))))+((2(z^2))/(3((((x^2)+(z^2)))^(3/2))))}];
plt1=Show[StreamDensityPlot[{vE,φ},{x,-5,5},{z,-5,5},ColorFunction->Hue],Graphics[Circle[]]];
plt2=Show[VectorDensityPlot[{vE,φ},{x,-5,5},{z,-5,5},ColorFunction->Hue],Graphics[Circle[]]];
GraphicsRow[{plt1,plt2},ImageSize->500]

Magnetized

Uniform magnetization. M=M(OverHat[z]); (Subscript[J, m])=Msinθ; m =(1/2)∫dSrxJ= (1/2)((∫_0)^π)2 ((πsinθdθr)^3)(Subscript[J, m])sinθ=(Subscript[J, m])V; (Subscript[H, int])=-(1/3)M; (Subscript[B, int])=(2/3)(Subscript[μ, 0])M; (Subscript[H, ext])=((3(r·M)r-(r^2)M)/(3 (r^5))), (Subscript[B, ext])=(Subscript[μ, 0])((3(r·M)r-(r^2)M)/(3 (r^5)))
Magnetic shielding; high permeability spherical shell; (Subscript[κ, m])>>1;  ((Subscript[B, in])/(Subscript[B, ext]))=((4(b^2))/((Subscript[κ, m])((b^2)-(a^2))))

Atoms

Back of envelope estimates

(Subscript[E, atom])~(R/ea)~100GV >> (Subscript[E, ext]). 
I~(eR/ℏ)~100μA; j~(I/(a^2))~10PA (m^(-2)); B~(Subscript[μ, 0])ja~((Subscript[μ, B])/(a^3))~1T
Polarization, magnetization involve averages over large fluctuations; intrinsically quantum mechanical; liquids and solids are many-body
Handle using current which creates a “gauge” problem when inferring M.
Big difference if dipoles are localized spins or there are large scale magnetization conduction currents
Ultimately, the average is well-defined.

Individual atoms

Conducting sphere radius a, φ=((p·r)/(4(Subscript[πϵ, 0])(r^3)))-E·r=>p=(Subscript[αϵ, 0])E, where α is the polarizibility = 3V in this case.
Approximate effect of other atoms by making a small sphere at the center of which the field is (Subscript[E, local])= (Subscript[E, ext])+(P/(3(Subscript[ϵ, 0])))=(P/(Subscript[nαϵ, 0])).
=>((κ - 1)/(κ+2)) = ((nα)/3)=(Subscript[nV, atom]) Clausius-Mossotti.

Polar Molecules

Large dipole moments leading to larger κ.

Plasmas

Free electrons and ions
Larmor frequency (Subscript[Ω, e]) =(qB/m)=1.8(x10^11)((B/(1T))) rad (s^(-1)); (Subscript[Ω, i])= (10^8)((Z/A))((B/(1T))) rad (s^(-1)); 
Larmor radius (Subscript[r, Le])=((Subscript[v, e])/(Subscript[Ω, e]))=1.7(((Subscript[v, e])/c))((((B/(1T))))^(-1))~1.7(((((Subscript[T, e])/(6(x10^9)K))))^(1/2))((((B/(1T))))^(-1))mm;  (Subscript[r, Li])=((Subscript[v, i])/(Subscript[Ω, i]))=3.1((A/Z))(((Subscript[v, i])/c))((((B/(1T))))^(-1))~3.1(((((Subscript[T, i])/(1(x10^13)K))))^(1/2))((((B/(1T))))^(-1))m 
Except for collisions, electrons are confined perpendicular to field but can be large magnetization currents due to drift and collisions
ITER -24 12T magnets ;
Space plasmas; yT? -30 GT fields; neutron stars are not ferromagnets because nuclei are magic

Force and Energy

Poynting theorem with media ∇.(ExH)+E·((∂D)/(∂t))+H·((∂B)/(∂t))=-E.(Subscript[j, f])
Total: (Subscript[u, E])=∫dD·E when build up along with currents
Potential: (Subscript[V, E])=-∫dE·D when fixed external currents and likewise for magnetic energy

Energy Density

Fourier analyze in time; also possible in space
E=∫((dω)/(2π))(e^(-iωt))(OverTilde[E]); (OverTilde[E])=∫dt (e^(iωt))E;  for real variables, (OverTilde[E])(ω)=(OverTilde[(SuperStar[E])])(-ω)
Convolution: C (t) = ∫dt' E (t' + t) D (t') = (Overscript[(∫_(-∞)), ∞])((dω)/(2π))(e^(-iωt ))(OverTilde[E])(ω)(OverTilde[D])(-ω)
(Subscript[P, E])(ω)=((| (OverTilde[E])((|)^2))/(πT)) is the “spectral power” for a duration T and positive frequencies 
E·(∂_t)D =(∂_t)((ϵ/2)(E^2)) +((E^2)/2)(∂_t)ϵ=(∂_t)(Overscript[(∫_0), ∞]) dω(1/2)(∂_ω)((Subscript[ωϵ, r]))(Subscript[P, E])+(Overscript[(∫_0), ∞]) dω (Subscript[ωϵ, i])(Subscript[P, E(ω)])

Causality

Linear dielectric: P(t)=(Subscript[ϵ, 0])(∫_(-∞)^∞)dt'χ(t-t')E(t').
Using (OverTilde[E])=∫(dte^(iωt))E etc (OverTilde[P])(ω)=(Subscript[ϵ, 0])(OverTilde[χ])(ω)(OverTilde[E(])ω) and (OverTilde[χ])(ω)=(OverTilde[χ])*(-ω) for real response. Re(χ)->dispersion, Im(χ)->absorption.
Causality =>χ(t)=0 for t<0=>(OverTilde[χ])(ω)=(∫_0^∞)dtχ(t)(Subscript[(e^(iωt)), ]), converges for Im(ω)>0, analytic function. 
Cauchy residue theorem =>∮ dω’(((OverTilde[χ])(ω'))/(ω'-ω))=P(∫_(-∞)^∞)dω'(((OverTilde[χ])(ω'))/(ω'-ω))-iπ(OverTilde[χ])(ω)=0 where contour passes over pole at ω on the real axis.
Kramers-Krönig: Re[(OverTilde[χ])(ω)]=(1/π)P(∫_(-∞)^∞)dω'((Im[(OverTilde[χ])(ω')])/(ω'-ω))=(2/π)P(∫_0^∞)dω'((ω'Im[(OverTilde[χ])(ω')])/(((ω')^2)-(ω^2))), Im[(OverTilde[χ])(ω)]=-(1/π)P(∫_(-∞)^∞)dω'((Re[(OverTilde[χ])(ω')])/(ω'-ω))=((2ω)/π)P(∫_0^∞)dω'((Re[(OverTilde[χ])(ω')])/(((ω')^2)-(ω^2)))

Plasma Physics

Solar Dynamics Observatory

Launched in 2010 to study solar corona and photosphere.

Corona

T~(10^6)K, n~(10^16)(m^(-3)), B (dipole)~100μT, B(sunspot) ~ 1-100mT...

Magneto-fluid-dynamics

Equation of motion for solar plasma, treated as a magnetized fluid: ρa=ρg-∇·P+(Subscript[ρ, e])E+jxB.  
Solve for (Subscript[j, ⊥]) =((Bx (B·∇) B)/(B^4))(Subscript[P, ||])-(Subscript[[∇x((((Subscript[P, ⊥]) B)/(B^2)))], ⊥])+(Subscript[P, ⊥])((Bx∇B)/(B^3))+(Subscript[ρ, e])((ExB)/(B^2))+ρ((gxB)/(B^2))+ρ((axB)/(B^2)), interpretable as curvature, magnetization, gradient, ExB, gravity and acceleration drifts.
Current along magnetic field which depends upon collision frequency (Subscript[j, ||]) = (Overscript[((ne^2)/(mν)), ])(Subscript[E, ||]).

Gravity-Probe B

Launched in 2004. Four spinning (80 Hz) gyros in a polar orbit that undergo a small relativistic precession

Gravitomagnetism

Express GR metric in terms of  ϕ=(1/2)(1-(Subscript[g, 00])), (Subscript[a, i])=(Subscript[g, 0i]) and define gravitoelectric field (or gravity!), e=-∇ϕ, and gravitomagnetic field b=∇xa.
e, b satisfy ∇·e=-4πGρ, ∇·b=0, ∇xe=0,∇xb=-4(4πGρv-(∂_t)e)  ((dv)/dt)=e+vxb  to post-Newtonian (v<<1) order; note signs; extra 4 derives from tensor not vector character of fields.

Geodetic precession

The gyro orbiting through a gravitoelectric field e=-(ω^2)r sees a gravitomagnetic field b=-vxe. It undergoes spin-orbit precession with frequency (Subscript[Ω, so])=-(1/2)b =(1/(2 ))(ω^2)l.
It turns out that there is an additional geometric term due to the fact that the space is curved arising from parallel transporting the gyro around a complete orbit, equal to twice this with a resultant geodetic precession of  (Subscript[Ω, geo])=(3/(2 ))(ω^2)l =6.6" (yr^(-1)), measure to ±0.018 " (yr^(-1)) (Everitt et al).

Lense-Thirring Precession

A second component of precession is induced by the spin of the Earth. The Earth has a dipole moment m=-2GS.
The precession is given by (Subscript[Ω, LT]) = <(m/(2(r^3)))-((3(m·r)r)/(2(r^5)))>=(GS/(2(r^3)))=0.039"\" \!\(\n\*SuperscriptBox[\(yr\), \(-1\)]\)".
Measured to ~±0.007 “ (yr^(-1)).

London moment

Four superconducting (T~2K) Nb-coated 40mm quartz spheres (to 10nm!)  generate aligned field ((-2(Subscript[x2m, e]) Ω)/(2e))=6nT.
Magnetically screen using Pb foil and read out with Squids that can detect a single flux quantum (h/(2e))=2fWb. 

Patch effect (non-uniform coating leading to variable electric dipole moment) was big problem.

Homework

1. With the considerable benefit of hindsight...
In class, we discussed briefly how Faraday discovered the laws of electrolysis in the 1830s, knew of Dalton’s discoveries and atomic ideas and Ampere’s suspicion that atoms were electrical in nature. 
All of this was before it was appreciated that accelerating charges radiate energy. 
What second piece of classical physics could Faraday have interpreted electrically to derive an estimate of the size of atoms? 
(I am not thinking of Brownian motion although Faraday had met Brown and knew of his discoveries.)
If you don’t see this quickly, move on.

2. Gravity Probe-B
Starting from the pseudo-Maxwell formulation of post-Newtonian relativity described above derive the formula for the rate of Lense-Thirring precession.

3. Free electrons in a plasma
A (very) simple model of a metal or a gaseous plasma posits that the electrons are free and the ions provide a uniform, neutralizing, positive background. 
An electromagnetic wave propagates through this plasma. 
The equation of motion for an individual electron can be written as
(Overscript[x, ..])+ν (OverDot[x])=-(eE/m), 
where E(t) is the electric field. (Subscript[ω, P])=(((((ne^2)/(Subscript[mϵ, 0]))))^(1/2))is the plasma frequency, ν<<(Subscript[ω, p]) is the electron collision frequency and n is the electron density. Interpret each of the terms.
Now suppose that a single monochromatic wave propagates through the plasma with angular frequency ω and solve for the current density, the permittivity and the susceptibility.
Next use Maxwell’s equations to show that the phase speed of the wave is (((1-((Subscript[ω, P]^2)/(ω^2))))^(-1/2))c.
Evaluate and interpret the wave energy density and the wave damping rate.
Calculate the group velocity and relate it to the energy density?
Are your answers consistent with the Kramers-Kronig relation?

4. Magnetization current.
Consider a two dimensional magnetic field (Subscript[B, z])(x,y) permeating a plasma with perpendicular pressure tensor components (Subscript[P, xx])(x,y)=(Subscript[P, yy])(x,y) =(Subscript[P, ⊥]).
Derive the second and third terms in the expression for (Subscript[j, ⊥])above assuming that a=g=E=(Subscript[P, ||])=0.
Explain what is going on physically using simple diagrams.

Sketch Solutions

1. Electrolysis => q/m = k for atoms; crystal structure => atoms close-packed (Kepler knew this!!) => m/(a^3) ∼ ρ; elastic (eg Young' s) modulus E~(q^2)/4(Subscript[πϵ, 0])(a^4) =>a~(((4(Subscript[πϵ, 0])E/(ρ^2) k))^(1/2))
2. Magnetic dipole moment for orbiting ring charge q  is q(r^2)ω/2 => (Subscript[m, ⊕])=-4 x (Subscript[S, ⊕])/2=-2(Subscript[S, ⊕]).  N=∫dmrx(vxb)=(1/2)sxb=(Subscript[Ω, LT])xs; b = ((3 ((Subscript[m, ⊕])·r) r)/(r^5))-((Subscript[m, ⊕])/(r^3))=><b>=((Subscript[m, ⊕])/(2(r^3))) =><(Subscript[Ω, LT])≥-((<b>)/2)=((
(Subscript[S, ⊕]))/(2(r^3))).
3. χ = ((Subscript[ω, P]^2)/((ω^2)+ (ν^2)))+ i(((Subscript[ω, P]^2)ν)/(ω((ω^2)+ (ν^2)))); ϵ=(Subscript[ϵ, 0])(1+χ);  For ν<<ω, (Subscript[v, ϕ])=((c(1-((Subscript[ω, p]^2)/(ω^2))))^(-1/2)), (Subscript[v, g])=((c(1-((Subscript[ω, p]^2)/(ω^2))))^(1/2)); B=(((1-((Subscript[ω, p]^2)/(ω^2))))^(1/2))(E/c); 
u=(1/2)((∂_ω)((Subscript[ωϵ, r]))(E^2)+(Subscript[μ, 0]^(-1))(B^2))=(1/2)((1+((Subscript[ω, P]^2)/(ω^2)))(Subscript[ϵ, 0])(E^2)+(Subscript[μ, 0]^(-1))(B^2))=electric plus electron kinetic plus magnetic energy density=(Subscript[ϵ, 0])(E^2); S = (Subscript[ϵ, 0])EB=(Subscript[ϵ, 0])(E^2)((c(1 - ((Subscript[ω, p]^2)/(ω^2))))^(1/2))=(Subscript[uv, g]), (electron kinetic term=O((E^3))).
Heating rate = (Subscript[ωϵ, i])(E^2)=(((Subscript[ϵ, 0])(Subscript[ω, P]^2)((νE)^2))/(ω^2)); ((2ω)/π)P(∫_0^∞)dω'(1/(((ω')^2)-(ω^2)))((Subscript[ω, P]^2)/(((ω')^2)+ (ν^2)))=Im[χ] etc


4. (Subscript[j, x])=-(1/B)(∂_y)P= -(∂_y)((P/B)) -(P/(B^2))(∂_y)B. First term is gradient of dipole moment per unit volume; second is motion of center of gyration.

4. Current

Reading:

Z 281-284, 287-291, 310-312, 319-323, 346-354.
Look over these pages and note material that is new and topics you find interesting or confusing. 
Professor Church will present the Tuesday lecture. Notes posted.

Biot - Savart

(Subscript[(A^(α;β)), ;β])=-(Subscript[μ, 0])(j^α);
(∂_t)=0=>(∇^2)A=-(Subscript[μ, 0])j
=>A(r)=((Subscript[μ, 0])/(4π))∫dV((j(x))/R); where r-x=Rn;
automatically satisfies Coulomb gauge ∇·A=0
=>B=∇xA=((Subscript[μ, 0])/(4π))∫dV((j(x)xn)/(R^2))

Scalar potential for line current

dVj->Ids; Solid angle ψ=(((Subscript[μ, 0])I)/(4π))Ω

Magnetic Pinches

MHD and Astrophysical Jets

Pictor A

Massive black hole in the nucleus of giant galaxy creates outflowing relativistic jets. 



Jets created by black hole disk terminating in beam dump. Pressure in jets >> external pressure.
Suspect that jets are “long wires” confined by strong toroidal field (Subscript[B, ϕ])=(((Subscript[μ, 0]) I)/(2 πr))~5(Subscript[I, 18])(Subscript[r, kpc]^(-1))nT; return current at large cylindrical radius...
The jets appear to be stable which is surprising.

Basic Equations of Magneto-Fluid-Dynamics.

Moving (v<<c; can do v~c) conducting fluid e.g. plasma. In rest frame, E’=0=>E+vxB=0 => (∂_t)B=∇x(vxB). Induction equation; “go with flow”; see HW.
ρ((dv)/dt)≡ρ((∂_t)+v·∇)v=-∇·P+jxB => (∂_t)ρv+∇·(ρv⊗v+P+(Subscript[T, M]))=0; “Push-Pull”
Linear perturbations, variational techniques => static pinches are highly unstable - “sausage” and “kink” modes, dynamical ones may persist.

Laboratory  Z-pinch Experiments (Sandia)

Giant capacitor bank I~30 MA, V ~30MV, t~100ns dynamical radial collapse =>T~4 GK, (Subscript[L, X-ray]) ~300TW, B~1 kT, r ~ 1mm 
Surprisingly stable

When join ends of pinch in a torus have a Tokamak like ITER
Design for stability for long enough for nuclear fusion to occur ~ 1s

Bohm - Aharonov

Consider (very) long solenoid. B~constant ~Φ/A inside ~0 outside. 
However ∮ds·A= Φ =>A≠0 outside.
Induces phase shift ((eΦ)/ℏ) in Young’s slit experiment; actually measured
=> QM depends on A not B; does not move envelope consistent with classical dynamics depending on B
Quantized flux in superconductors h/2e =2 fWb mentioned in wk3
Important for topological insulators - local speciality, theoretically and experimentally.

Helmholtz coils

Single Coil

(Subscript[A, ϕ])=(((Subscript[μ, 0])I)/(4π))(∫_(-π)^π)dϕ’((cosϕ')/(((1+(r^2)-2rsinθcosϕ'))^(1/2)));

Aϕ=Integrate[Cos[ϕp](((1+(r^2)-2r Sin[θ]Cos[ϕp]))^(-1/2)),{ϕp,-π,π},Assumptions->Re[r (r+2 Sin[θ])]>-1&&Re[r (r-2 Sin[θ])]>-1]
Aϕc=Chop[FullSimplify[Aϕ/.{Sin[θ]->(((((x^2)+(y^2)))^(1/2))/((((x^2)+(y^2)+(z^2)))^(1/2))),Csc[θ]->(((((x^2)+(y^2)+(z^2)))^(1/2))/((((x^2)+(y^2)))^(1/2))),r->((((x^2)+(y^2)+(z^2)))^(1/2))}]]
Show[ContourPlot[Aϕc/.y->0,{x,.001,5},{z,.001,5}],Graphics[Disk[{1,0},.1]]]

B=∇xA=[(1/(rsinθ))(∂_θ)((Subscript[sinθA, ϕ])),-(1/r)(∂_r)((Subscript[rA, ϕ])),0];

Bs=Simplify[{(1/(r Sin[θ]))(∂_θ)(Sin[θ]Aϕ),-(1/r)(∂_r)(r Aϕ),0}]
Rcs={{(x/((((x^2)+(y^2)+(z^2)))^(1/2))),((x z)/(((((x^2)+(y^2)))^(1/2))((((x^2)+(y^2)+(z^2)))^(1/2)))),((-y)/((((x^2)+(y^2)))^(1/2)))},{(y/((((x^2)+(y^2)+(z^2)))^(1/2))),((y z)/(((((x^2)+(y^2)))^(1/2))((((x^2)+(y^2)+(z^2)))^(1/2)))),(x/((((x^2)+(y^2)))^(1/2)))},{(z/((((x^2)+(y^2)+(z^2)))^(1/2))),-(((((x^2)+(y^2)))^(1/2))/((((x^2)+(y^2)+(z^2)))^(1/2))),0}};
Bc=Chop[FullSimplify[Rcs.Bs/.{Cos[θ]->(z/((((x^2)+(y^2)+(z^2)))^(1/2))),Sin[θ]->(((((x^2)+(y^2)))^(1/2))/((((x^2)+(y^2)+(z^2)))^(1/2))),Csc[θ]->(((((x^2)+(y^2)+(z^2)))^(1/2))/((((x^2)+(y^2)))^(1/2))),Cos[2θ]->(((z^2)-(x^2)-(y^2))/(((x^2)+(y^2)+(z^2)))),r->((((x^2)+(y^2)+(z^2)))^(1/2))}]]

Show[StreamDensityPlot[{Bc[[1]],Bc[[3]]}/.y->0,{x,.001,5},{z,.001,5}],Graphics[Disk[{1,0},.1]]]

Axial Field

Bz=Simplify[Bc[[3]]/.{x->0,y->0}]

Helmholtz Coil

Two co-axial current loops separated by s. 
Create extended region of strong field

BHz=Simplify[(Bz/.{z->(s/2)+ζ} )+(Bz/.{z->(s/2)-ζ} )]
Series[BHz,{ζ,0,4}]

Second derivative vanishes when s=1.
This is the condition for an extensive strong field region.

Plot[BHz/.s->1,{ζ,-1,1}]
BHzt=Simplify[2Bc[[3]]/.{y->0,z->(1/2)}];
Plot[BHzt,{x,0,1}]

Principles very important for magnetic design - LHC, LINAC, ITER, Z-machine...

Magnetic Potential

Needs["NumericalCalculus`"]
ψ[x_,y_,z_]:=Chop[If[(x^2)+(y^2)≥1,NIntegrate[((2s z)/((((z^2)+(s^2)))^(3/2)))ArcCos[(((x^2)+(y^2)+(s^2)-1)/(2((((x^2)+(y^2)))^(1/2)) s ))],{s,((((x^2)+(y^2)))^(1/2))-1,((((x^2)+(y^2)))^(1/2))+1}],2 π  (1-z/(((1+(x^2)+(y^2)+(z^2)-2 ((((x^2)+(y^2)))^(1/2))))^(1/2)))+NIntegrate[((2s z)/((((z^2)+(s^2)))^(3/2)))ArcCos[(((x^2)+(y^2)+(s^2)-1)/(2((((x^2)+(y^2)))^(1/2)) s ))],{s,1-((((x^2)+(y^2)))^(1/2)),1+((((x^2)+(y^2)))^(1/2))}]]]
ContourPlot[ψ[x,0,z],{x,0,1},{z,0,1}]

Dynamos and Reconnection

Return to MHD and add resisitivity generally a tensor but scalar for geomagnetic field in liquid core 
E+vxB=(j/σ); adds diffusive term to induction equation (∂_t)B = ∇x(vxB) +(1/(μσ))(∇^2)B.
Field reverses on timescale -0.1-10Myr; ~10 percent over past century!
Stretch field lines non-axisymmetrically; (required by Cowling’s theorem.) 
Cooling of radioactive heating => convection, coriolis forces;  α effect  - E=α B: convert toroidal ->poloidal; 
differential rotation, poloidal -> toridal ω effect

Balanced by steady resistive diffusion and reconnection localized in space and time.

Gradients build up until reconnection rate balances flow rate. Hall effects important in plasmas.
May be explosive; e.g. coronal mass ejections.
Reconnection happens in Tokamaks too.

Homework

1. In class we described the equations of non-relativistic magneto-fluid-dynamics. In the absence of resisitivity, the first of these, the induction equation, is (∂_t)B = ∇x(vxB) . Explain why this is often described by saying that magnetic field lines are “frozen” into the flow.
(One way to approach this is to combine it with the equation of mass continuity: (∂_t)ρ+∇·ρv=0 and to compare with a simple kinematic equation that describes the evolution of a short vector joining two neighboring points that both move with the fluid. Confirm the derivation of the second dynamical equation in the form: (∂_t)ρv+∇·(ρv⊗v+P+(Subscript[T, M]))=0.

2. Explain carefully why the force on a permanent dipole m in a region where the magnetic field can be described using a scalar potential can be written in the form (Subscript[F, i])=-(Subscript[ψ, ,ij])(Subscript[m, j]).
Spin-polarized atoms can be confined within a Helmholtz “anti-coil” which is like a Helmholtz coil except that the two currents flow in opposite directions. Use the routines outlined or some you write yourself above to explore numerically how this might work. Consider, especially, the behavior of the axial field close to the axis.

3. Zangwill: Problem 11.11

Sketch Solutions

1. Combining, ((∂_t)+v·∇)(B/ρ)≡(d/dt)(B/ρ)=(B/ρ)·∇v; cf (d/dt)δl=v·∇δl. ρ((∂_t)+v·∇) v + ((∂_t)+v·∇) ((∂_t)ρ + ∇·ρv)=(∂_t)ρv+∇·ρv⊗v
2.
s=1;BAH1=Bc[[3]]/.{y->0,z->-(s/2)+ζ};BAH2=Bc[[3]]/.{y->0,z->(s/2)+ζ};
ContourPlot[BAH1-BAH2,{x,-2,2},{ζ,-2,2}]

3. Compose as three current squares. Dipoles associated with the top and the bottom squares cancel, leaving only the side square with dipole moment equal to 4(b^2)I .

5. Induction

Reading

Z 462-7, 475-9, 486-91

Electromagnetic Induction

EMF

M3=>-(OverDot[Φ])==EMF≡∮ds·E if circuit fixed in inertial frame. -(OverDot[Φ])=∮ ds·(E+vxB), if circuit moves. Often a good idea to use differential form and work in inertial frame.
Faraday Disk: B parallel to Ω. In local inertial frame with v=Ωxr, E’=0, B'=B=O((v^2)) =>V=-∫dr·vxB=-(1/2)Ω·((BR)^2)

Skin Effect

Suppose j=σE. and ignore charge and extermal current =>(∇^2)X=μσ((∂X)/(∂t))+μϵ(((∂^2)X)/(∂(t^2))); X=E,B. 
Diffusion equation with D=(((μσ))^(-1/2))when L>((((ϵ/((μσ)^2))))^(1/2)). Skin depth δ=((((2/(σωμ))))^(1/2))for oscillatory signal. X~(e^(-z/δ+i(z/δ-ωt))).  
(Subscript[J, 0]) for cylindtrical wire. 
Magnetic diffusion with same D  covered in wk4
Cu - σ=6(x10^7)(Ω^(-1))(m^(-1)). =>D=0.013(m^2)(s^(-1))=>δ=0.013(ω^(-1/2)).

Homework

Z 14.5, 14.16, 14.21. 
Weejee is working on a superconductivity problem which will be posted shortly.

6. Waves

Reading

Null

EM Waves in vacuo

Vector Potential and d’Alembertian

Basic Equations

(A^α)=[φ,A], (Subscript[F, αβ])=(Subscript[A, β,α])-(Subscript[A, α,β]); (Subscript[(A^α), ,α])=0; (Subscript[(F^(αβ)), ,β])=-(Subscript[(A^(α,β)), ,β])=(Subscript[μ, 0])(j^α); (Subscript[F, [αβ,γ]])=0.
Hence (∇^2)φ-(∂_tt)φ≡(□^2)φ=0; (□^2)A=0. Now, E=-∇φ-(∂_t)A; B=∇xA => (□^2)E=0, (□^2)B=0.
Often convenient to write A=u(t,x)s. For monochromatic wave, ((∇^2)+(ω^2))u=0 (Helmholtz); use in optics.

Plane Waves

∝(e^(iϕ)); ϕ=k·x-ωt; ω=±k; Coulomb is special choice of Lorenz, k·A=0; E=iωA;B=ikxA; E=B orthogonal and to k
General solutions (SubPlus[E])(z-t)+(SubMinus[E])(z+t) composed by Fourier integral
Superpose with different k; E=∫(((d^3)k )/(((2π))^3))(Subscript[E, k]) (e^(i(k·x-ωt))) etc

Energy

U=(Subscript[ϵ, 0])(E^2)=((B^2)/(Subscript[μ, 0])); S=U(OverHat[k]); (Subscript[Z, 0])=(Subscript[μ, 0])(E/B)=(Subscript[μ, 0])≡120π=377Ω- impedance of free space

Polarization

Elliptical

Monochromatic wave has fixed polarization; rotate axes, shift phase (Subscript[E, x])∝(e^(iϕ)), (Subscript[E, y])∝(e^(i(ϕ+δ))); δ=0 linear, =±(π/2), circular, elliptical in general.
LCP,RCP conventions are confused. LH ν has spin antiparallel to momentum  (unless overtake it) which is opposite convention from book!

ax=1.;ay=.5;
Animate[Graphics[{{Red,Arrow[{{0,0},{ax Cos[ϕ],ay Sin[ϕ]}}]},
                                {Green,  Arrow[{{0,0},{-ay Sin[ϕ],ax Cos[ϕ]}}]}},Axes->True,
AxesLabel->{Style["(Subscript[E, x])",Red]Style["(Subscript[B, x])",Green],Style["(Subscript[B, y])",Green]Style["(Subscript[E, y])",Red]}, PlotRange->{{-1,1},{-1,1}}],{ϕ,0,2π},AnimationRunning->False]

Stokes Parameters

Rotate polarization plane of linearly polarized monochromatic wave through ψ. E=E[cosψ,sinψ]cosϕ. I=(Subscript[ϵ, 0])(E^2)(cos^2)ϕ, Q = (Subscript[ϵ, 0])((Subscript[E, x]^2)-(Subscript[E, y]^2))=Icos2ψ, U=2(Subscript[ϵ, 0])(Subscript[E, x])(Subscript[E, y])=Isin2ψ =>(Q^2)+(U^2)=(I^2).
Set ψ=0 and add perpendicular oscillation in quadrature: E = [(Subscript[E, x])cosϕ,(Subscript[E, y])sinϕ]. I=(Subscript[ϵ, 0])((Subscript[E, x]^2)(cos^2)ϕ+(Subscript[E, y]^2)(sin^2)ϕ), Q = (Subscript[ϵ, 0])((Subscript[E, x]^2)(cos^2)ϕ-(Subscript[E, y]^2)(sin^2)ϕ),  V=2(Subscript[ϵ, 0])(Subscript[E, x])(Subscript[E, y])cosϕsinϕ =>(Q^2)+(V^2)=(I^2)
Combine on Poincaré Sphere: (Q^2)+(U^2)+(V^2)=(I^2). For broad band wave with partial polarization, (Q^2)+(U^2)+(V^2)<(I^2). 
Degree of linear polarization =(((((Q^2)+(U^2)))^(1/2))/I); position angle : ψ=(1/2)(tan^(-1))(U/Q); degree of circular polarization=(V/I). Note that adding π to ψ leaves the polarization unchanged, cf spinor!

Quarter - wave Plate

Two orthogonal modes of linear polarization with different n; eg calcite; thickness such that retard one mode by π/4. Incident linear at ψ=π/4 -> circular and vice versa

Waves in Uniform Media

Use D, H to get refractive index n=(((μϵ))^(1/2)), 
U=(1/2)((ϵE)^2)+(1/2)((μH)^2), S=ExH =((U(OverHat[k]))/n), Z=(E/H)=((((μ/ϵ)))^(1/2))

Reflection, Refraction and Transmission

Boundary Conditions

(Subscript[D, ⊥]),(Subscript[B, ⊥]), (Subscript[E, ||]),(Subscript[H, ||]), (Subscript[k, ||]), ω continuous at plane boundary.

Snell' s Law

(Subscript[k, ||])continuity =>nsinθ=const. for both reflected and transmitted waves;
Generalize to complex refractive index as with skin depth.

Normal Incidence

(Subscript[E, ||])->1+r=t, (Subscript[H, ||]) ->((1-r)/(SubMinus[Z]))=(t/(SubPlus[Z]))=>r=(((SubPlus[Z])-(SubMinus[Z]))/((SubPlus[Z])+(SubMinus[Z]))); t=((2(SubPlus[Z]))/((SubPlus[Z])+(SubMinus[Z])));=>|r((|)^2)(Subscript[V, g-])+|t((|)^2)(Subscript[V, g+])=(Subscript[V, g-]). energy conservation
Generalize to complex n,Z,  oblique incidence, polarization, internal reflection, Brewster angle...

Fresnel Coefficients

Consider a “TM/TE” modes incident on a plane surface (z=0) with H/E along the y direction and E/H, k lying in the x-z plane.

Solve[{(1-r)Cos[(SubMinus[θ])]==t Cos[(SubPlus[θ])],((1+r)/(SubMinus[Z]))==(t/(SubPlus[Z]))},{r,t}]
Solve[{1+r==t ,(((1-r)Cos[(SubMinus[θ])])/(SubMinus[Z]))==((t Cos[(SubPlus[θ])])/(SubPlus[Z]))},{r,t}]

Equivalent expressions using refractive indices, permittivities can be derived; Z may be complex. These are used in designing single layer and multilayer coatings for Fabry-Perot devices and X-ray mirrors

Strong Waves

Consider a single particle released from rest in a strong, monochromatic  electromagnetic wave, e.g. from a laser.
E=E(1,0,0)cosϕ, ϕ=ω(z-t), f=(qE/(mω)), α=(γ-(Subscript[u, z]))ω=-((dϕ)/(dτ))
m((dγ)/(dτ))= m(( (Subscript[du, z]))/(dτ))=(Subscript[qE, x])(Subscript[u, x])=>α=const.
m((Subscript[du, x])/(dτ)) = (γ-(Subscript[u, z]))(Subscript[E, x])=>((Subscript[du, x])/(dϕ))=-fcosϕ=>(Subscript[u, x])=(dx/(dτ))=-fsinϕ=>x=-(f/α)cosϕ
(Subscript[u, z])=(dz/(dτ))= (((ω^2) - (α^2))/(2αω))+ ((Subscript[ωu, x]^2)/(2α))=(((ω^2) - (α^2))/(2αω))+(((f^2)ω)/(2α)) (sin^2) ϕ=>z = ((((α^2)-(ω^2))/(2(α^2)ω))-(((f^2)ω)/(4(α^2))))ϕ+(((f^2)ω)/(8(α^2)))sin2ϕ
γ=(dt/(dτ))= (((ω^2) + (α^2))/(2αω))+ ((Subscript[ωu, x]^2)/(2α))=(((ω^2) + (α^2))/(2αω))+(((f^2)ω)/(2α)) (sin^2) ϕ=>t = -((((α^2)+(ω^2))/(2(α^2)ω))+(((f^2)ω)/(4(α^2))))ϕ+(((f^2)ω)/(8(α^2)))sin2ϕ
For a phase change -2π, t-z=((2π)/ω) as it should.

f=100;α=(((1+((f^2)/2)))^(1/2));
ParametricPlot[{-(f/α)Cos[ϕ],((2(α^2)-2-(f^2))/(4(α^2)))ϕ+((f^2)/(8(α^2)))Sin[2ϕ]},{ϕ,0,-4π},Frame->True,FrameLabel->{x,y},RotateLabel->False]

LCLS Undulators

LINAC Coherent Light Source (aka X-ray Laser) - accelerate electrons to 3-15 GeV (discuss later), wiggle transversely using undulator, radiate coherent X-rays, reflect into experimental hutches for spectroscopy, X-ray diffraction, high energy density physics etc...




There are 33 undulators in a 132 m hall each undulator is 3.4 m long (leaving a 0.6m gap) and contains 113 pairs of pairs of permanent magnets of alternating polarity designed to give a sinusoidal perpendicular field B=(Subscript[B, 0])sin kz where (Subscript[B, 0])=1.25T, k=((2π 113)/3.4)=209 (m^(-1)). 
Equivalently, the wavelength is 30 mm.
Here are two wavelengths ...NSNS...
                                              SNSN
The transverse equation of motion is ((Subscript[dp, x])/dt)=evB so the deflection angle is θ= ((Subscript[Δp, x])/p)=((Subscript[eB, 0])/(mckγβ)) 
Conventional to define wiggler strength parameter K=γβθ=3.5. 
The undulating electrons will radiate and we will discuss this later.
Also have focusing magnets.
The X-rays are separated from the electron beam using grazing incidence X-ray mirrors.

Metamaterials

Non-natural materials designed and fabricated to exhibit unusual electromagnetic properties
Important for GHz and THz  applications
Periodic structure < λ; effectively homogeneous

Negative Refractive Index

ENG (≡Epsilon NeGative?) materialshave (Subscript[ϵ, r])<0; MNG have (Subscript[μ, r])<0.
Can have metamaterials where ϵ, μ are both < 0. Using n(Underscript[(OverHat[k]), ])xE=μH, n(OverHat[k])xH=-ϵE, we find that n<0. E, H, k   have opposite handedness; (Subscript[V, ϕ])antiparallel to S.; refract "backwards". Cerenkov properties!!
This, in turn, can be used to create a super lens which can beat the diffraction limit of angular resolution ~λ/D. The case n=-1 can create a “perfect lens” with point focus (in principle).
http://news.stanford.edu/pr/2013/pr-meta-material-invisible-050613.html
It can also be used to create an “invisibility cloak”(also in principle).

Electromagnetic Induced Transparency

Consequence of Kramers-Kronig relations

Narrow window of transparency when (Subscript[ϵ, r ])(blue) changes rapidly.
Can also slow the group velocity of light down to ~8m (s^(-1)).
Local work in Harris, Vukevic groups.

Galactic Center Magnetar

NuSTAR

Focusing hard (6-80 keV) X-ray telescope launched in June 2012; X-ray mirrors 10 m focal length, multilayers Pt/SiC, W/Si 200m coatings!

Magnetar

http://www.astronomerstelegram.org/?read=5058
Pulsar ( neutron star moment of intertia I ~(10^38)kg (m^2),B~16GT) with spin period P=3.76s slowing down at rate (OverDot[P])=6.5x (10^(-12)) in 3Pm radius orbit around a 4 x (10^6)(Subscript[M, sun]) black hole. 
NuSTAR observed a massive flare radiating (10^33)J in a day; cf (Subscript[U, m]) ~ 4 ((πR)^3) ((B^2)/(Subscript[μ, 0]))~(10^39)J;(Subscript[U, kin])= (1/2)((IΩ)^2)~ (10^38)J
The flare launched a low frequency magnetic wave into the surrounding plasma which behaves like a perfectly conducting magnetized medium described by MHD

Alfvén Waves

Consider simple wave modes in this medium.
In particular, a transverse wave propagating along B
Linearize:  (∂_t)(Subscript[B, x])=-(∂_z)(-(Subscript[u, x])B)=>(Subscript[ωB, x])=-(Subscript[kBu, x]), (∂_t)ρ=0, ρ (∂_t)(Subscript[u, x]) =(Subscript[j, y]) B = (B/(Subscript[μ, 0]))(∂_z)(Subscript[B, x])=>(Subscript[ρωu, x])=-((Subscript[kB, x])/(Subscript[μ, 0]))B=> (ω^2)=(k^2)((B^2)/((Subscript[μ, 0])ρ))=(Subscript[V, A]^2)(k^2), (Subscript[V, A])is Alfvén speed
When k is oblique, ω=k·(Subscript[V, A]), (Subscript[V, ϕ])=(Subscript[V, A])cosθ k, (Subscript[V, g])=(Subscript[V, A]).
Energy trasnported along field direction.
We can find additional modes if we add pressure.

Transverse Wave in Plasma (cf wk 3)

The pulsar also emits radio wave pulses at the spin frequency.
To describe their propagation, linearize the fluid equation for cold plasma - just consider the electrons and treat the ions as fixed
-(Subscript[iωn, e])(Subscript[m, e])v=-(Subscript[n, e])eE => j= i(((Subscript[n, e])(e^2))/((Subscript[m, e])ω))E; imaginary conductivity
Combine with M3, 4=> (ω^2)=(Subscript[ω, p]^2)+(c^2)(k^2); where (Subscript[ω, p])=((((((Subscript[n, e])(e^2))/((Subscript[m, e])(Subscript[ϵ, 0])))))^(1/2))=56(((((Subscript[n, 0])/(1 (m^(-3))))))^(1/2))rad (s^(-1))
The refractive index is therefore n=(((1-(Subscript[ω, p]^2)/(ω^2)))^(1/2)) and (Subscript[V, ϕ])=c/n; (Subscript[V, g])=nc.
This is also valid for free electrons in a metal.
The radio pulses travel at the group velocity and are subject to a dispersive delay t=(((Subscript[ω, p]^2)L )/(2(ω^2)c)) relative to light when traversing a distance L.
For the pulses from the new magnetar, the delay is measured to be t=6.8 (Subscript[ν, 9]^(-2))s varying with the observing frequency (Subscript[ν, 9])GHz. 
This is the largest dispersion delay ever measured.

Aside on Plasma oscillation

ω=(Subscript[ω, p]). 
Add pressure - effective specific heat ratio is 3 because there is one degree of freedom.
Create waves when add electron pressure; dispersion relation: (ω^2)=(Subscript[ω, p]^2)+((3kT)/m)(k^2). Langmuir waves
Add ion inertia ->ion acoustic waves.

Magnetic Field

Combine these two approaches to consider electromagnetic waves propagating in a cold electron plasma.
The equation of motion is: -i ω (Subscript[m, e]) v = -e(E+(Subscript[vxB, 0])), where (Subscript[B, 0]) is the background field along (OverHat[z])
while Maxwell’s equations give: kx(kxE)+(ω^2)E+((i ω (Subscript[n, e])e v)/((c^2)(Subscript[ϵ, 0])))=0,
Eliminate v to obtain an eigenequation M(ω,k)·E =0 which can be solved for the dispersion relations and eigenvectors.

vk=k{Sin[θ],0,Cos[θ]}/.θ->0;vE={Ex,Ey,Ez};
dv=Simplify[vE+ ((vk\[Cross](vk\[Cross]vE))/(ω^2))+((I  n e{vx,vy,vz})/(ϵ0 ω))/.Solve [{-I ω m vx==-e(Ex+vy B),-I ω m vy==-e(Ey-vx B),-I ω m vz==-e Ez},{vx,vy,vz}]/.{B->((-m ωg )/e), n->((m ϵ0 ((ωp)^2))/(e^2))}][[1]];
dm=Simplify[Table[{dv[[i]]/.{Ex->1,Ey->0,Ez->0},dv[[i]]/.{Ex->0,Ey->1,Ez->0},dv[[i]]/.{Ex->0,Ey->0,Ez->1}},{i,1,3}]];
dr=Simplify[Eigensystem[dm]]

Homework

1. Show that the parameters f, α associated with a strong, electromagnetic wave are Lorentz invariant. 
Hence show the energy contained in a wave packet transforms as the frequency and comment upon this.
Consider a particle released from rest in circularly polarized strong wave with f=1. Show a graph of z(t) for two periods of the wave.
2. Use the Fresnel coefficients derived above to confirm that when an unpolarized wave is incident from air on a plane glass (refractive index n~1.5) interface with (SubMinus[θ])=(tan^(-1))n), the reflected wave is completely polarized.
3. Z 17.11
4. In class, we considered the propagation of electromagnetic waves in an unmagnetized electron plasma. Now add a uniform field and consider waves propagating along it. Either working from first principles or using the code written above and correcting an error, show that there are two modes with refractive index n satisfying
(n^2)=1-((Subscript[ω, p]^2)/(ω(ω±(Subscript[ω, g])))).
Describe the electric, magnetic and velocity vectors in these modes. 
There is actually a third mode. Explain its properties.
Repeat this exercise for waves propagating perpendicular to the field.
A linearly polarized wave with frequency ω >> (Subscript[ω, g]) propagates along the field. Estimate the angle through which the direction of the polarization will be rotated in traversing a distance L.
The propagation of the pulses from the magnetar can be approximated by assuming that it is parallel to a uniform field (Subscript[B, 0])=100pT.  Estimate the angle through which the plane of polarization has been rotated.

Answers

vk=k{Sin[θ],0,Cos[θ]}/.θ->0;
vE={Ex,Ey,Ez};
dv=Simplify[vE+ ((vk\[Cross](vk\[Cross]vE))/(ω^2))-((I  n e{vx,vy,vz})/(ϵ0 ω))/.Solve [{-I ω m vx==-e(Ex+vy B),-I ω m vy==-e(Ey-vx B),-I ω m vz==-e Ez},{vx,vy,vz}]/.{B->((-m ω Y)/e), n->((m ϵ0 (ω^2)X)/(e^2))}][[1]];
dm=Simplify[Table[{dv[[i]]/.{Ex->1,Ey->0,Ez->0},dv[[i]]/.{Ex->0,Ey->1,Ez->0},dv[[i]]/.{Ex->0,Ey->0,Ez->1}},{i,1,3}]]
dr=Simplify[Eigensystem[dm]];

7. Transmission

Reading

Z636-641, 667-8, 669-670, 677-682,704-705

Transmission Lines

Conducting wires etc of constant cross section; seek TEM solutions ∝(e^(i(kz-ωt))); medium may be susceptible; no longitudinal contribution to ∇.E, H so just a 2D problem. 
Boundary conditions on conductors.

Generalities

Vacuum

∇·(Subscript[B, ⊥]) = 0=>∃Φ:B=[(∂_y)Φ,-(∂_x)Φ,0](e^(i(kz-ωt))); Φ(x,y) (a stream function) is the flux per unit line length as (dy/dx)=((Subscript[B, y])/(Subscript[B, x]))=>Φ const. on field lines; also A=(0,0,Φ)
M3=>kxE=ω∇Φ(e^(i(kz-ωt))), (∇^2)Φ=0,  φ=(ω/k)Φ or more generally, (∂_t)Φ=-(∂_z)φ; M4=>kxB=((ω^2)/(kc^2))∇Φ(e^(i(kz-ωt))) =>ω=±ck, E=cB
Magnetic field lines are perpendicular to electric field (and both to axis) and are equipotentials and lie parallel to conductor surfaces; propagation along axis at c
On conductor surface, (∂_t)σ+∇·J=0.

Medium

E=((((μ/ϵ)))^(1/2))H;ω=(k/(((μϵ))^(1/2)))

Coax

Cylindrical symmetry (Subscript[H, ϕ])=((((ϵ/μ)))^(1/2))(Subscript[E, r])=(I/(2πr)); L=(μ/(2π))ln(b/a) per unit length (∂_z)V=-L(∂_t)I
Likewise, Q=2πrE per unit length or Q=CV, C=((2πϵ)/(ln(b/a)))=>(∂_t)Q = C (∂_t)V=-(∂_z)I => (∂_zz)V=LC(∂_tt)V, (∂_zz)I=LC(∂_tt)I, Telegraphers’ Equations
Follows from circuit analysis; add more capacitative and inductive units in series, parallel; add resistance (∂_z)V=-IR-L(∂_t)I etc

Parallel wires

Illustrate use of complex variables for 2D electrostatics (traditionally large part of subject, also hydro!)
In 2D φ =-(q/(2πϵ))ln(r)=-(q/(2πϵ))Re[lnz]
In general f(z)=u+iv satisfies CR =>u,v harmonic in 2D.
For two line charges, φ~ln(((z+1)/(z-1)))

φ=Log[((Sqrt[(((1+x))^2)+(y^2)])/(Sqrt[(((1-x))^2)+(y^2)]))];vE=Simplify[-{(∂_x)φ,(∂_y)φ}];vH=Simplify[-{-(∂_y)φ,(∂_x)φ}];
plt1=Show[StreamDensityPlot[{vE,φ},{x,-2,2},{y,-2,2},ColorFunction->Hue],Graphics[{PointSize[Medium],Point[{{-1,0},{1,0}}]}]];
plt2=Show[StreamDensityPlot[{vH,φ},{x,-2,2},{y,-2,2},ColorFunction->Hue],Graphics[{PointSize[Medium],Point[{{-1,0},{1,0}}]}]];
GraphicsRow[{plt1,plt2},ImageSize->500]

Waveguides

Heuristic treatment

Wireless transmission by reflection at conducting surface TEM cannot satisfy BC wo interior current, ∇xH.
Consider oblique EM plane wave modes w (Subscript[k, ⊥]^2)=((μϵω)^2)-(k^2) combine a Fourier sum over ϕ of standing modes with (Subscript[k, ⊥])=(Subscript[k, ⊥])(cosϕ,sinϕ) so as to satisfy BC
((∂_xx)+(∂_yy)+(Subscript[k, ⊥]^2))(Subscript[H, z]),(Subscript[E, z])=0 for TE,TM; can only do this for special values of (Subscript[k, ⊥])- eigenvalues with orthogonal eigenfunctions; generally fix ω, solve for k; will be cut off frequencies
BC (Subscript[E, ||])=0. TE: (Subscript[E, z])=0 plus component ∇xH parallel to surface and perpendicular to axis must vanish => (n·∇)(Subscript[H, z])=0 from M4; TM: (Subscript[E, z])=0 suffices as (Subscript[(∇xE), z])=0 
Mathematically solve scalar Helmholtz+ Dirichlet, Neuman in 2D in given geometry; cf QM energy levels
Formal proofs in Z.

Modes

Plot dispersion relation ω(k) for given mode, will be cut off ω where k=0,  and below which only evanescent propagation is possible; specifically (Subscript[ω, c])=(((μϵ))^(-1/2))(Subscript[k, ⊥])
Just like unmagnetized plasma! same algebra!; (Subscript[V, g])(Subscript[V, ϕ])=(((ϵμ))^(-1/2))  P=(Subscript[Uv, g])etc

Rectangular Waveguide

cf particle in box (Subscript[k, ⊥mn])=((π(((m^2)/(a^2))+((n^2)/(b^2))))^(1/2)).
(Subscript[TE, 10,01])(Subscript[TM, 11])are lowest  modes

Circular etc waveguides

Bessel functions... In practice have complex cross section and handle numerically

Varying cross sections

If change slowly handle using (J) WKB approximation; avoid reflection
If change abruptly, eg open/closed end, get reflections; impedance conditions

Cavities

Completely closed conducting surfaces have resonant eigenfrequencies and can be used to detect weak signals eg axion search; Tesla fields exclude axions ~2-4 μeV.

Microwave Background Telescopes

Polarization

Much effort now being expended to see signatures of inflation in the microwave background radiation polarization maps. Later.

e.g. QUaD (Church) and BICEP (Kuo) experiments; QUaD was a 2.6m telescope in a ground shield
New experiments under design


The radio waves are reflected into 31 feed horns, pass though circular waveguides and are measured using 5 mm bolometers. 
The waveguide is a high pass filter with cutoff ν>75 GHz, λ<4mm.
Copper meshes and the atmosphere act as low pass filter allowing a small bandwidth to be measured.

(Subscript[TE, 11]) Cylindrical waveguide

Choose this mode as it looks like a plane polarized EM wave
Helmholtz: ((∇^2)+(Subscript[k, ⊥]^2))(Subscript[H, z])=((∂_rr)+(1/r)(∂_r)+(1/(r^2))(∂_(ϕϕ))+(Subscript[k, ⊥]^2))(Subscript[H, z ])=0=>(Subscript[H, z])∝(Subscript[J, 1])((Subscript[k, ⊥])r)cosϕ
BC: (∂_r)(Subscript[H, z])=0 at r=R; first zero of (Subscript[J, 1])' =(((Subscript[J, 0])-(Subscript[J, 2]))/2) is 1.84 => (Subscript[k, ⊥])=((2 π)/(4mm))=1.57(x10^3)(m^(-1)) =>R=(1.84/1.57)=1.2mm
Transverse fields: ∇.B=0=>(Subscript[ikH, z])+(∂_x)(Subscript[H, x])+(∂_y)(Subscript[H, y])=0 =>ik(∂_x)(Subscript[H, z])+(∂_xx)(Subscript[H, x])+(∂_xy)(Subscript[H, y])=0; (Subscript[E, z])=0 =>ik(∂_x)(Subscript[H, z])+(∂_xx)(Subscript[H, x])+(∂_yy)(Subscript[H, x])=0; 
Helmholtz =>(Subscript[H, ⊥]) = i(k/(Subscript[k, ⊥]^2)) (Subscript[∇, ⊥])(Subscript[H, z]); M3=>E=((μω)/k)Hx(OverHat[z])

R=r/.FindRoot[BesselJ[0,r]-BesselJ[2,r]==0,{r,1.8}];
Hz=If[r<R,BesselJ[1,r]Cos[ϕ],0]/.{r->((((x^2)+(y^2)))^(1/2)),ϕ->ArcTan[x,y]};
vH={(∂_x)Hz,(∂_y)Hz};vE={vH[[2]],-vH[[1]]};
Show[StreamDensityPlot[{vH,Hz},{x,-R,R},{y,-R,R},FrameLabel->{x,y},RotateLabel->False,ColorFunction->Hue],StreamPlot[vE,{x,-R,R},{y,-R,R}],Graphics[{Thick,Circle[{0,0},R]}]]

Feed Horn

Arrange impedance at entrance so that Z=(E/H)=(Subscript[Z, 0]), matched to incoming vacuum wave.

SLAC LINAC

History

3km (SuperPlus[e])/(SuperMinus[e]) linear accelerator; waveguide operating in TM mode->(Subscript[E, z]) 
Excited by a series of klystrons - powerful microwave transmitters using up to 50 MW! (invented by Hansen, Varians!) 
Historically, beams separated into: 
SLC(50+50 GeV for 90 GeV Z physics) storage ring; =>(Subscript[E, z])~17V (m^(-1)); 
PEP-II (9+3 GeV for 5 GeV B physics)
LCLS (5-15 GeV) LCLS; just need 1km
Future linear accelerators planned for ~0.5TeV, ~ 30 (Vm^(-1))

Operation

(Subscript[V, `ϕ])>c => particles accelerated and decelerated; pre-acceleration has decelerating fields shielded in drift stage
transit time ~10μs => electron trails light by only ~(Subscript[E, GeV]^(-2))μ!; use periodic array of Cu irises to create (Subscript[V, ϕ])=c

Bloch theorem => (Subscript[E, z])(0, 0)∝(e^ikz)∑(Subscript[c, n])(e^(inκz)) wh κ=((2π)/(26 mm)); need large (Subscript[c, n]) for ω=c(k+nκ) ~ GHz.

Fiber Optics

e.g. LSST

8.4m (6.5m effective) mirror telescope being built in Chile, 10 sq deg FOV covering half sky a thousand times over 10 years.  
Prime purpose is to monitor expansion of the universe. 
SLAC leading building the camera; 3 Gpx image every 15s.
Need to communicate alerts globally in 2s; 15 TB per day on a 10 Gbps fiber down mountain.

Step Index Mode

This is geometrical optics with total internal reflection; minimize losses;  n decreases outward; ((2 x ((πa)^2) x (Subscript[πk, ⊥]^2))/(((2π))^2)) =(((a^2)(k^2))/2)(((1-((Subscript[n, 1]^2)/(Subscript[n, 2]^2))))^2)modes.
In practice concerned about evanescent modes and diffractive effects.

Graded Index Mode

Rays obey differential equation: (d/ds)n((dx)/ds)=∇n which can be derived from Fermat’s principle

Physical Optics Mode

Solve full wave optics problem with boundary conditions

Homework

1. Consider a cylindrical waveguide of radius R operating in (Subscript[TM, 32])mode in vacuo.
Adapt the treatment above for a (Subscript[TE, 11]) mode explaining the differences and relating to the boundary conditions to exhibit an image of the transverse electric and magnetic modes.
Find the cutoff frequency if R=10mm. 
2. A self-focussing graded optical fiber has a refractive index n(r)=(((Subscript[n, 0])(1-(α^2)(r^2)))^(1/2)). Working from the Helmholtz equation or otherwise, seek an axisymmetric mode with Gaussian radial profile that propagates along the fiber without spreading.
What are the phase and group velocities for this mode?
3. Z19.23

Sketch Solutions

1.

(Subscript[TM, 32]):(Subscript[H, z])=0, (Subscript[E, z]) satisfies Helmholtz ((∂_rr)+(1/r)(∂_r)+(1/(r^2))(∂_(ϕϕ))+(Subscript[k, ⊥]^2))(Subscript[E, z ])=0 with (Subscript[E, z])=0 at r=R. 
=>(Subscript[E, z])=(Subscript[J, 3])(r)cos3ϕ; R is 2nd zero of (Subscript[J, 1]);
∇·E=0 =>(Subscript[ikE, z])+(∂_x)(Subscript[E, x])+(∂_y)(Subscript[E, y])=0 =>ik (∂_x)(Subscript[E, z])+(∂_xx)(Subscript[E, x])+(∂_xy)(Subscript[E, y])=ik (∂_x)(Subscript[E, z])+(∂_xx)(Subscript[E, x])+(∂_yy)(Subscript[E, x])=0

R=N[BesselJZero[3,2]]
Ez=If[r<R,BesselJ[1,r]Sin[3ϕ],1]/.{r->((((x^2)+(y^2)))^(1/2)),ϕ->ArcTan[x,y]};
vE={(∂_x)Ez,(∂_y)Ez};vH={vE[[2]],-vE[[1]]};
Show[StreamDensityPlot[{vE,Ez},{x,-R,R},{y,-R,R},FrameLabel->{x,y},RotateLabel->False,ColorFunction->Hue],StreamPlot[vH,{x,-R,R},{y,-R,R}],Graphics[{Thick,Circle[{0,0},R]}]]

(Subscript[k, ⊥]) = ((9.76 )/(.01))= 980 (m^(-1)); (Subscript[ν, min])=((3x (10^8) x 980)/(2π))=47GHz

2.

((∂_rr)+(1/r)(∂_r)+(((n^2)(ω^2))/(c^2))-(k^2))ψ=((∂_rr)+(1/r)(∂_r)+(((Subscript[n, 0]^2)(ω^2)(1-(α^2)(r^2)))/(c^2))-(k^2))ψ=0; satisfy BC with internal reflection if ψ is (Subscript[E, ϕ])
Let ψ=(e^(-(β^2)(r^2)+ikz-ωt)); (((Subscript[αn, 0])ω)/c)=2(β^2); (((Subscript[n, 0]^2)(ω^2))/(c^2))-(k^2)= 4 (β^2)=((2(Subscript[αn, 0])ω)/c); (ω/k)=(c/(Subscript[n, 0]))((α/k)+(((1+((α^2)/(k^2))))^(1/2))); ((dω)/dk)=(c/(Subscript[n, 0]))(((1 + ((α^2)/(k^2))))^(-1/2))

ψ=(E^(-(β^2)(r^2)));Simplify[(ψ^(-1))((∂_(r,r))ψ+(r^(-1))(∂_r)ψ+(ω^2)(1-(α^2)(r^2))ψ-(k^2)ψ)]

3.

(Subscript[E, z])=sinπx ((sinπye)^(-iωt)); ω=(2^(1/2))π; B=-(2^(-1/2))i[sin πxcosπy,cosπxsinπy,0](e^(-iωt))
Poynting flux time-averages to zero; <(Subscript[T, xx])> only from ((Subscript[B, y]^2)/(2(Subscript[μ, 0]) ))  pressure which averages to a force <(Subscript[F, x])>=(ah/(16(Subscript[μ, 0]))); likewise for <(Subscript[F, y])>; no net force on ends because averaged electric tension cancels averaged magnetic pressure.

8. Radiation

Reading

Z 730-739, 752-760, 762-765, 775-780, 787-789

Relativistic radiation

Lie'nard - Wiechert Potential

Wk2,4: (∇^2)φ=-(ρ/(Subscript[ϵ, 0]))=>φ=(1/(4(Subscript[πϵ, 0])))∫(d^3)x((ρ(x))/R) ,(∇^2)A=-(Subscript[μ, 0])j=>A(r)=((Subscript[μ, 0])/(4π))∫(d^3)x((j(x))/R); where r-x=Rn;
Hence, □A=-(Subscript[μ, 0])j => A=-((Subscript[μ, 0])/(4π))∫(d^3)x((γ j)/(ξ·u)), where ξ=r-x=R(1,n) is the null separation 4-vector from the Source at x=[s,x] to the Observer at r=[t,r]; nb (Subscript[k, α]^O)(dr^α)=(Subscript[k, α]^S) (dx^α).
For a single charge, A=-(((Subscript[μ, 0])q u)/(4π ξ·u)), which is manifestly covariant; nb u =(( dx)/(dτ))=[(((1+u·u))^(1/2)),u]=γ(1,⋁)
We will also use A=((Subscript[μ, 0])/(4π))∫(dsd^3)x(( j)/(R(1-n·β)))δ(s+R-t)|((d(s+R-t))/ds)| =((Subscript[μ, 0])/(4π))∫(d^4)x(( j)/R)δ(s+R-t)

Field Tensor

ξ must remains null as x(τ) varies; consider r(τ); ξ·((dr/(dτ))-u)=0=>(Subscript[ξ, β])((dr^β)/(dτ))(∂_α)τ=(Subscript[ξ, β])(u^β)(∂_α)τ => (∂_α)τ=((Subscript[ξ, α])/(ξ·u)); nb derivatives wrt observer coordinate (r^α)and u evaluated at source.
Now (∂_α)(ξ·u)=(Subscript[ξ, β])(∂_α)(u^β)+(Subscript[u, β])(∂_α)(ξ^β)=(((Subscript[ξ, α])(Subscript[ξ, β])(a^β))/((u^β)(Subscript[ξ, β])))+(Subscript[u, β])((Subscript[(δ^β), α])-(u^β)(∂_α)τ) =(((1+ξ·a)(Subscript[ξ, α]))/(ξ·u))+(Subscript[u, α]), where (a^α)=((du^α)/(dτ)) =[a·v,a] is the acceleration, orthogonal to u 
Hence  (Subscript[A, α,β])= ((((Subscript[μ, 0])q)/(4 ((π((Subscript[ξ, γ])(u^γ)))^2)))) ((Subscript[u, α])(∂_β)((Subscript[ξ, δ])(u^δ))-(∂_β)(Subscript[u, α])(Subscript[ξ, δ])(u^δ)) =  ((((Subscript[μ, 0])q)/(4π ξ·(u^2))))((Subscript[u, α])(Subscript[u, β])-(Subscript[a, α])(Subscript[ξ, β])-(((Subscript[u, α])(Subscript[ξ, β]))/(ξ·u))(1+ξ·a)); note that (∂_α)(A^α)=0 as it must. Now, (Subscript[F, αβ])=(∂_α)(Subscript[A, β])-(∂_β)(Subscript[A, α])
Finally, F=((((Subscript[μ, o])q)/(4πξ·(u^3))))[(1+ξ·a)(ξ⊗u-u⊗ξ)-ξ·u(ξ⊗a-a⊗ξ)]

ξ={ξ0,ξ1,ξ2,ξ3};u4={u0,u1,u2,u3};a4={a0,a1,a2,a3};η={{-1,0,0,0},{0,1,0,0},{0,0,1,0},{0,0,0,1}};
ip[a_,b_]:=a.η.b;op[a_,b_]:=Outer[Times,a,b]
fuu=(1/(4π ((ip[ξ,u4])^3)))Simplify[(Outer[Times,a4,ξ]-Outer[Times,ξ,a4])ip[ξ,u4]+(Outer[Times,ξ,u4]-Outer[Times,u4,ξ])(1+ip[ξ,a4])]
fdd=η.fuu.η

Electromagnetic Field

(Subscript[E, i])= (Subscript[F, i0]), (Subscript[B, i])=(Subscript[ϵ, ijk])(F^jk); 
E=(((Subscript[μ, 0])q)/(4π))[(((1-(v^2))(n-v))/((((1-n·v))^3)(R^2)))+((n×{(n-v)×(OverDot[v])})/((((1-n·v))^3)R))]; B=(((Subscript[μ, 0])q)/(4π))n×[-(((1-(v^2))v)/((((1-n·v))^3)(R^2)))+((n×(v×(OverDot[v])}-(OverDot[v]))/((((1-n·v))^3)R))]=n×E

e1=Simplify[fuu[[1,2]]/.{ξ0->R,ξ1->R n1,ξ2->R n2,ξ3->R n3,u0->γ,u1->γ v1,u2->γ v2,u3->γ  v3,a0->(γ^4)vw,a1->(γ^4)vw β1 +(γ^2)w1,a2->(γ^4)vw v2 +(γ^2)w2,a3->(γ^4)vw v3 +(γ^2)w3}];
(1/(R^2))Simplify[Coefficient [e1,(R^(-2))]/.n1 β1+n2 v2+n3 v3->nv]+(1/R)Simplify[Coefficient [e1,(R^(-1))]/.n1 v1+n2 v2+n3 v3->nv]
b1=Simplify[fuu[[3,4]]/.{ξ0->R,ξ1->R n1,ξ2->R n2,ξ3->R n3,u0->γ,u1->γ v1,u2->γ v2,u3->γ  v3,a0->(γ^4)vw,a1->(γ^4)vw β1 +(γ^2)w1,a2->(γ^4)vw β2 +(γ^2)w2,a3->(γ^4)vw β3 +(γ^2)w3}];
(1/(R^2))Simplify[Coefficient [b1,(R^(-2))]/.n1 v1+n2 v2+n3 v3->nv]+(1/R)Simplify[Coefficient [b1,(R^(-1))]/.n1 v1+n2 v2+n3 v3->nv]

Invariants

-(((F^(αβ))(Subscript[F, αβ]))/2) =(E^2) - (B^2) =((((((Subscript[μ, 0])q)/(4π ξ·(u^3)))))^2)[((1+ξ·a)u-ξ·u ((a)·ξ])^2)=((((((Subscript[μ, 0])q)/(4π ξ·(u^2)))))^2),(( (F^(αβ))(Subscript[ϵ, αβγδ])(F^(γδ)))/8)=E.B=0

Stress-Energy Tensor

(Subscript[T, em])=(({{U, S}, {g, (Subscript[T, M])}})); (T^(αβ))=(1/(Subscript[μ, 0]))((1/4)(Subscript[F, γδ])(F^(δγ))(η^(αβ)) -(Subscript[(F^α), γ])(F^(γβ)))=>T=(((Subscript[μ, 0])(q^2))/(16(π^2)ξ·(u^6)))[(a·a ξ·(u^2)-(((1+ξ·a))^2))ξ⊗ξ-(1+ξ·a)ξ·u(ξ⊗u+u⊗ξ)+ξ·(u^2)(ξ⊗a+a⊗ξ+(1/2)η)]
We can decompose this into three parts:
(Subscript[T, near])=(((Subscript[μ, 0])(q^2))/(16(π^2)ξ·(u^6)))[-ξ⊗ξ-ξ·u(ξ⊗u+u⊗ξ)+(1/2)ξ·(u^2)η]
(Subscript[T, int])=(((Subscript[μ, 0])(q^2))/(16(π^2)ξ·(u^6)))[-2ξ·a ξ⊗ξ-ξ·u ξ·a (ξ⊗u+u⊗ξ)+ξ·(u^2)(ξ⊗a+a⊗ξ)]
(Subscript[T, far])=(((Subscript[μ, 0])(q^2))/(16(π^2)ξ·(u^6)))[(a·a ξ·(u^2)-ξ·(a^2))ξ⊗ξ]
Can verify that (∂_β)(T^(αβ))=0; in fact (∂_β)(Subscript[T, far]^(αβ))=0

Power and Radiation Reaction

Emitted Power

ξ·dξ=0, dr=0 =>  dt=ds(1-n·v)
(Subscript[S, far]^r)=(Subscript[T, far]^(0r))=(((Subscript[μ, 0])(q^2)(R^2))/(16(π^2)))(((a·a)/(ξ·(u^4)))-((ξ·(a^2))/(ξ.(u^6))))n
(Subscript[P, em])=∫((dΩR)^2)(dt/ds)(Subscript[S, far]^r)=∫((dΩR)^2)(1-n·v)(Subscript[S, far]^r)=(((Subscript[μ, 0])(q^2))/(4π))(a.a-((a.a-a·(v^2))/3))=(((Subscript[μ, 0])(q^2)a·a)/(6π))using a=[a.v,a].
(The integrals are elementary and even simpler if we recognize that the power is a Lorentz invariant and therefore can only be a function of a·a.)
(Subscript[P, em])=(((Subscript[μ, 0])(q^2)(γ^6)(((OverDot[v])^2)-vx((OverDot[v])^2)))/(6π))in terms of three velocity, acceleration
(Subscript[P, em])=(((Subscript[μ, 0])(q^4)u.F.F.u)/(6((πm)^2)))in terms of field
(Subscript[P, em])=(((Subscript[μ, 0])(q^2)a·a)/(6π)), if v⊥a; =(((Subscript[μ, 0])(q^2)a·a)/(6((πγ)^2))), if v||a

Integrate[2π Sin[θ](((1-v Cos[θ]))^(-3)),{θ,0,π},Assumptions->0<v<1];
Integrate[Sin[θ]((((Cos[θ]Cos[ψ]+Sin[θ]Sin[ψ]Cos[ϕ]-v Cos[ψ]))^2)/(((1-v Cos[θ]))^5)),{θ,0,π},{ϕ,0,2π},Assumptions->0<v<1];

Radiation Reaction

For an electron at rest with constant a, there can be no 3-momentum radiated. Therefore dp=-(Subscript[P, em])dx is satisfied and so (Subscript[P, em])is a Lorentz invariant. (nb in a general frame, ((dp)/ds)= -(Subscript[P, em])v)
This argument breaks down if a changes => dp=-(((Subscript[μ, 0])(q^2)(a·a dx+Kda))/(6π)) and K=-1 (see HW); introduce the 4-jerk j≡(da/(dτ));  resistive (or radiation reaction) 4-force is (Subscript[f, res]) = (dp/(dτ))= -(((Subscript[μ, 0])(q^2))/(6π))(a·au-j) => (Subscript[P, res]) = (Subscript[(((dp)/ds)), res])·v=(Subscript[P, em])+(((Subscript[μ, 0])(q^2))/(6π))((j^0)/γ)
If return to the same a (including zero) no extra energy radiated; temporary storage in near/int fields.

Examples

Uniform Motion

From LW

ξ·u = -γR (1 - n·v) = γr·n = -((γ((r^2) - (rxn^2)))^(1/2))=-γ ((r(1-(v^2)(sin^2)θ))^(1/2))=> A=(((Subscript[μ, 0])qu)/(4πγ ((r(1-(v^2)(sin^2)θ))^(1/2)))), F=(((Subscript[μ, 0])q)/(4((πγ)^3)(((r^3)(1-(v^2)(sin^2)θ))^(3/2))))(ξ⊗u-u⊗ξ) 
So E=(((Subscript[μ, 0])q)/(4((πγ)^3)(((r^3)(1-(v^2)(sin^2)θ))^(3/2))))γR(n-β) =(((Subscript[μ, 0])q r)/(4((πγ)^2)(((r^3)(1-(v^2)(sin^2)θ))^(3/2))))away from contemporary position of source; B=vxE

From Lorentz Transformation

Agrees as: (Subscript[E, ||])=(Subscript[E, ||]^(')); (Subscript[E, ⊥])=(Subscript[γE, ⊥]^('))=(((Subscript[μ, 0])γ q)/(4π(((x')^2)+(y^2)))); (Subscript[B, ⊥])=βxE; field lines bunch in transverse direction,

v=.9;c=.1;vE={((x(1-(v^2)))/((((x^2)+(y^2)-(v^2)(y^2)+(c^2)))^(3/2))),((y(1-(v^2)))/((((x^2)+(y^2)-(v^2)(y^2)+(c^2)))^(3/2)))};
VectorPlot[vE,{x,-1,1},{y,-1,1}]

Non-relativistic, Electric Dipole Radiation

In far field, (S^r)=(( (Subscript[μ, 0])(q^2) (axn^2))/(16(π^2)(R^2)))= (((Subscript[μ, 0])(Subscript[(Overscript[p, ..]), ⊥]^(2)))/(16(π^2)(R^2)))=((E^2)/(Subscript[μ, 0]))=> E=((Subscript[(Overscript[p, ..]), ⊥])/(4(Subscript[πϵ, 0])R)) with electric vector (polarization) evaluated at retarded time along projected dipole direction (Larmor).
More generally, (Subscript[(Overscript[p, .. ]), ⊥])->∫dV(∂_t)(Subscript[j, ⊥])
When j not harmonic, expand as ∫((dω)/(2π))(e^(-iωt))(OverTilde[j])(ω)

Movie

For a single charge with v<<1, E = (Subscript[(((Subscript[μ, 0])q)/(4π))[(n/(R^2))+ ((n×(n×(OverDot[v])))/R)], ret]) ; the first term is the Coulomb field, the second the far field
For a pure dipole comprising two opposite charges oscillating in antiphase, E = (Subscript[((Subscript[μ, 0])/(4π))[((3(p·n)n)/(R^3))-(p/(R^3))+ ((n×(n×(Overscript[p, ..])))/R)], ret])=(Subscript[((Subscript[μ, 0])/(4π))[((3(p·n)n)/(R^3))-(p/(R^3))+ (((ω^2)(Subscript[p, ⊥]))/R)], ret])

Animate[StreamPlot[{((3x y )/((((x^2)+(y^2)))^(5/2)))-((x y)/((((x^2)+(y^2)))^(3/2))),((2(y^2)-(x^2) )/((((x^2)+(y^2)))^(5/2)))+((x^2)/((((x^2)+(y^2)))^(3/2)))}Cos[t-((((x^2)+(y^2)))^(1/2))],{x,.1,4π},{y,-2π,2π}],{t,0,2π}]

Thomson scattering:

Free electron, photon energy <<m, f=(eE/(mcω))<<1
a=-((eE)/m); (Subscript[σ, T])= ((Subscript[P, em])/((Subscript[ϵ, 0])(E^2)))=((8π)/3)(Subscript[r, e]^2)=6.7(x10^(-28))(m^2); (Subscript[r, e]) =2.8fm; ((dσ)/(dΩ))=(((Subscript[r, e]^2)((Subscript[e, i])·(Subscript[e, o])))^2)=(( (Subscript[σ, T]))/2)(1+(cos^2)θ), if average over incident polarization and sum over scattered polarization

Microwave Background Polarization

Density inhomogeneity at recombination can lead to polarization. Sky pattern is E-mode. Complementary pattern is B-mode which can be produced by gravitational lensing but which may be a gravitational radiation relic of inflation. Need nK sensitivity!

Magnetic Dipole and Electric Quadrupole Radiation (outline)

When (Overscript[p, ..])=0, j->j+n·r'(∂_t)j+...->j+(1/2)(∂_t)((r'xj)xn+(r'·nj+n.jr'))
First (antisymmetric part) -> magnetic dipole radiation; second (symmetric part)->electric quadrupole radiation 
Now m=(1/2)∫dVr'xj; (Subscript[B, far]) = -(((Subscript[μ, 0 ])(Subscript[(Overscript[m, ..]), ⊥]))/(4πr)); (Subscript[P, em])=(((Subscript[μ, 0])(Subscript[(Overscript[m, ..]), ⊥]^2))/(6π))
Likewise, EQ radiation (Subscript[P, em])=(((Subscript[μ, 0])(Overscript[Q, ..]):(Overscript[Q, ..]))/(180π))in the notation of wk2 (different from Z)

Pulsars

Spinning, magnetized neutron stars (a perfect electrical conductor with M~ 3 x (10^30)kg, R~10 km) and surface magnetic flux density (Subscript[B, s]).
- Crab pulsar (Subscript[B, s])~100MT Ω~200 rad (s^(-1)).
- Millisecond pulsar (Subscript[B, s])~100 kT, Ω~3000 rad (s^(-1))
- Magnetars (Subscript[B, s])~30 GT, Ω~2 rad (s^(-1)).
Originally modeled as vacuum magnetic dipole with (Subscript[P, em])~(((Ω^4)(Subscript[B, s]^2)(R^6))/(6 π (Subscript[μ, 0])))
We now know that this model is wrong because it leads to a magnetosphere with strong electric field which would quickly fill up with plasma.

Klystrons

The Klystrons in the SLAC LINAC contain a beam of ~100 keV electrons that is bunched by an electromagnetic wave and then enters a resonant cavity with frequency ~3GHz. This excites an electromagnetic wave which is ported out through a rectangular waveguide where it accelerates the beam with an electric field ~ 17MV (m^(-1))The klystrons are arranged along the beam and phased so that the pulses of electrons see the peak accelerating electric field throughout their passage down the LINAC.

Homework

1. Work through the  treatment of magnetic dipole radiation in Z 20.7.2 and show that a permanent magnetic dipole oriented  perpendicular to the axis about which it spins with angular frequency Ω radiates a power of order (Subscript[P, em])~(((Ω^4)(Subscript[B, s]^2)(R^6))/(6 π (Subscript[μ, 0]))) where R is the radius of the star and (Subscript[B, s]) is the magnetic flux density at the stellar surface. (c=1 throughout.) 
a) Evaluate this for the three types of pulsar described above. 
b) Compute or sketch the magnetic field in the equatorial plane and contrast with the electromagnetic field radiated by a spinning electric dipole.
c) In a real pulsar the electric field surrounding the neutron star is so strong that plasma can be drawn from the surface and short out the component of electric field parallel to the magnetic field. Show that the electric charge density is ρ ~ ((ΩB)/(Subscript[μ, 0]))  and explain why the estimate of (Subscript[P, em])above remains valid.
d) The electromagnetic field also carries away angular momentum. Explain why the total torque must equal (Ω^(-1))(Subscript[P, em])
e) A spinning magnetized neutron star can also radiate linear momentum. Explain how this can happen. Hint: consider what might happen if there is magnetic quadrupole as well as a magnetic dipole.
2. Z 20.10
3. We argued in class that the resistive force acting on an emitting electron involves the 4-jerk j=(da/(dτ)). 
a) Show that (j^α)=[v·j+(((1-(v^2)))^(1/2))(a.a-(((a·v))^2)), j]
b) Explain why if (Subscript[f, res])=(dp/(dτ))=-(((Subscript[μ, 0])(q^2) )/(6π))(a.a u+Kj), then K=-1 
c) Re-express this in the form (Subscript[f, res])=(((Subscript[μ, 0])(q^2) )/(6π))(j.u u+j)

Sketch Solutions

1.

By analogy with Larmor formula P~((Subscript[μ, 0])/(6π))((Overscript[m, ..])^2)~(((Subscript[μ, 0]) )/(6π))((( ((4((πΩ)^2) (BR^3))/(Subscript[μ, 0]))))^2)[Apologies for misprint in question]
a) P~((8((πΩ)^4)(B^2)(R^6))/(3(Subscript[μ, 0])(c^3)))~ 200(Ω^4)(B^2) kW ~3 x (10^31) W (Crab), ~ (10^(30 ))W (millisecond), ~ 3 x (10^28)W (magnetar)
b) Trailing spiral (Subscript[B, ϕ]) alternating with radial wavelength ((2π)/Ω) in contrast with electric dipole radiation where we have (Subscript[E, ϕ ])
c) E ~ΩrB=>ρ~(((Subscript[ϵ, 0])E)/r); dimensional analysis or observe that transition from near to far field where r~(Ω^(-1)) ; B~(((Subscript[B, s])((R/r)))^3) and P~((B^2)/(Subscript[μ, 0]))(r^2)
d) Torque G on star does work at the rate GΩ which must equal P to conserve energy.
e) When evaluating the Poynting flux, the cross term involving the dipole and quadrupole can interfere constructively in one hemisphere and destructively in the other giveing a net radiation of linear momentum

2.

a) (Subscript[P, Ω])=(((Subscript[μ, 0])(Subscript[I, 0]^2))/(8(π^2)))(((((coskdcosθ - coskd)/(sinθ))))^2) ~(((Subscript[μ, 0])(((Subscript[I, 0]^2)(kd))^4)(sin^2)θ)/(32(π^2)))
b) p = (i/ω) ∫dzI =((2(Subscript[iI, 0]))/(k^2))(1-coskd)~(Subscript[iI, 0])(d^2) etc
c) I(0)~(Subscript[i, 0])kd => p=((2iI(0)d)/ω)etc

3.

a) u·u = -1 => u.a = 0 => (a^0) = a·v; a·a + u.j = 0 => (((a·v))^2) - a.a+((γj)^0)-γv·j=0
b) u·(dp/(dτ))=0 =>Kj·u=a·a =>K=-1
c) (Subscript[f, res])=(((Subscript[μ, 0])(q^2) )/(6π))(j-a·au)=(((Subscript[μ, 0])(q^2) )/(6π))(j·uu+j)

9. Relativity

Reading

Z 797-806, 891-898, 906-909

Synchrotron Radiation

Far Field

A = -(((Subscript[μ, 0])q u)/(4π u·ξ))=>A=(((Subscript[μ, 0])q v)/(4πR(1-n·v)))and (OverTilde[A])=∫(dte^(iωt))A=(((Subscript[μ, 0])q)/(4πR))∫((dxe)^(iωt)), using dt=ds(1-n·v).
(OverTilde[(Subscript[B, far])])=ikx(OverTilde[A])=>(OverTilde[(Subscript[E, far])])=iωnx((OverTilde[A])xn)=iω(Subscript[(OverTilde[A]), ⊥])

Gyro radiation

Particle travels in a circle of radius (κ^(-1)) with speed v and Ω=κv, observer at distance R with latitude θ
When v<<1, radiation at gyro frequency; spectrum is a δ-function; 
When v~1, get contributions from harmonics - a Fourier series. 
E=∑(Subscript[E, n])(e^(inΩt)), (Subscript[E, n])=(((Subscript[μ, 0])q Ω)/(8(π^2)R))(∫_0^((2π)/Ω))ds (e^(iω(s-n·x)))(Subscript[v, ⊥]), (Subscript[v, ⊥])=v[cos Ωs, sinΩs sinθ], r.n=(v/Ω)cosΩs cosθ =>(Subscript[E, n])~ (Subscript[J, n])(nvcosθ) and similar terms 
If there is dispersion in v, harmonics overlap; treat spectrum as continuous.

Ultrarelativistic Limit

Power

(Subscript[P, em])=(((Subscript[μ, 0])(q^2))/(6π))a·a =(((Subscript[μ, 0])(q^2))/(6π))(γ^4)(κ^2), for perpendicular acceleration with κ the curvature.

Fourier Transform

Ignoring constant phase factor, (OverTilde[E])=(((Subscript[μ, 0])q ω)/(4πR))∫ds (e^(iω(s-n·x)))(Subscript[v, ⊥])=(((Subscript[μ, 0])q ω)/(4πR))∫ds (e^(i((ωs)/2)((γ^(-2))+(θ^2)+(κ^2)(s^2)/3)))[κ s,θ], to lowest order, where θ is the observer inclination from the osculating plane, κ is the curvature at mid-point, the polarization is relative to axes parallel and perpendicular to the curvature direction, using n·x=((sinκvs cosθ)/κ) and expanding
The limits of integration can be set as ±∞ and the integral converges.

Plot[Cos[((3ξ x)/2)(1+((x^2)/2))]/.ξ->1,{x,0,10}]
NIntegrate[Cos[((3ξ x)/2)(1+((x^2)/2))]/.ξ->2,{x,0,100}]

We convert to standard integrals using s->((2(((1+(γ^2)(θ^2)))^(1/2)))/(κ γ))sinhy/3, (Subscript[ω, c])=3(γ^3)κ, x=(ω/(Subscript[ω, c])); (also expressible as Airy functions - solution of y’’=xy); nb (Subscript[ω, c])agrees with J and twice that in Z)

Simplify[ ((μ0 q ω)/(4π R)){ I κ s Sin[(ω/2) ((((1+(γ^2)(θ^2)) s)/(γ^2))+(((κ^2)(s^3))/3))],θ Cos[(ω/2) ((((1+(γ^2)(θ^2)) s)/(γ^2))+(((κ^2)(s^3))/3))]}(∂_y)(((2(((1+(γ^2)(θ^2)))^(1/2)) )/(κ γ))Sinh[(y/3)])/.{s->((2(((1+(γ^2)(θ^2)))^(1/2)))/(κ γ)) Sinh[(y/3)],ω->3(γ^3)κ x}]

Hence, (OverTilde[E])=(((Subscript[μ, 0])q x ((γ(1 + (γ^2)(θ^2)))^(1/2)))/(πR )){i(((1 + (γ^2)(θ^2)))^(1/2)) (∫_0^∞)dysin[((x(1+(γ^2)(θ^2)))^(3/2))sinhy]sinh(((2 y)/3)),γθ (∫_0^∞)dycos[x (((1+(γ^2)(θ^2)))^(3/2))sinhy]cosh((y/3))}
                 =(((3^(1/2)) (Subscript[μ, 0])q x ((γ(1 + (γ^2)(θ^2)))^(1/2)) )/(2πR)) {((i(1 + (γ^2)(θ^2)))^(1/2))(Subscript[K, 2/3])(((x(1+(γ^2)(θ^2)))^(3/2))),γθ (Subscript[K, 1/3])(((x(1+(γ^2)(θ^2)))^(3/2)))}

x=1;θm=2;Plot[{(1+(θ^2))x BesselK[(2/3),x(((1+(θ^2)))^(3/2))],θ (((1+(θ^2)))^(1/2))x BesselK[(1/3),x(((1+(θ^2)))^(3/2))]},{θ,-θm,θm}, Frame->True,FrameLabel->{"θγ","(OverTilde[E])",x,""},RotateLabel->False]

Fluence

Parseval -> (∫_(-∞)^∞)dt ((E^2)/(Subscript[μ, 0]))= (1/(R^2))(∫_0^∞)(Subscript[dωI, ωΩ]), where (Subscript[I, ωΩ])=((|(OverTilde[E])((|)^2)(R^2))/(Subscript[πμ, 0]))=((3(Subscript[μ, 0]) (q^2)(x^2)(γ^2)(1+(γ^2) (θ^2)))/(4(π^3))){(1+(γ^2) (θ^2)) (Subscript[K, 2/3]^2)(((x(1+(γ^2)(θ^2)))^(3/2)))+(γ^2)(θ^2)(Subscript[K, 1/3]^2)(((x(1+(γ^2)(θ^2)))^(3/2)))}

Angular Distribution

(Subscript[I, Ω])= ∫dω (Subscript[I, ωΩ])=(( 3(Subscript[μ, 0])(q^2) (γ^5)κ)/(4 (π^3)(((1 + (γ^2)(θ^2)))^(7/2)))))((1+(γ^2)(θ^2))(∫_0^∞)((dξξ)^2)(Subscript[K, 2/3]^2)(ξ)+(γ^2)(θ^2)(∫_0^∞)((dξξ)^2)(Subscript[K, 1/3]^2)(ξ))=((  (Subscript[μ, 0])(q^2) (γ^5)κ)/(64((π(1 + (γ^2)(θ^2)))^(7/2))))(7(1+(γ^2)(θ^2))+5(γ^2)(θ^2))

Integrate[(ξ^2)((BesselK[ν,ξ])^2),{ξ,0,∞},Assumptions->(1/3)<ν<(2/3)]

nb no 1-n·v factor unless charge moves closer to observer over averaging interval.

Plot[(((1+(θ^2)))^(-5/2))+(θ^2)(((1+(θ^2)))^(-7/2)),{θ,0,3},Frame->True,FrameLabel->{"γθ","(Subscript[I, Ω])"},RotateLabel->False]

Power Check

P=κ(∫_(-∞)^∞)dθ (Subscript[I, Ω])=((  (Subscript[μ, 0])(q^2) (γ^4)(κ^2))/(64π))(7(∫_(-∞)^∞)(dx/( (((1 + (x^2)))^(5/2))))+5(∫_∞^∞)((dxx^2)/( (((1 + (x^2)))^(7/2)))))=((  (Subscript[μ, 0])(q^2) (γ^4)(κ^2))/(6π))

Integrate[(7/(((1+(x^2)))^(5/2)))+((5(x^2))/(((1+(x^2)))^(7/2))),{x,-∞,∞}]

Spectral Power

(Subscript[P, ω ])=κ(∫_(-∞)^∞)dθ (Subscript[I, ωΩ])=(( 3 (Subscript[μ, 0])(x^2)(q^2) (γ^2)(κ^2))/(4 (π^3)))((∫_(-∞)^∞)((dθ(1+(γ^2)(θ^2)))^2)(Subscript[K, 2/3]^2)(((x(1+(γ^2)(θ^2)))^(3/2)))+(∫_(-∞)^∞)dθ (γ^2)(θ^2)(1+(γ^2)(θ^2))(Subscript[K, 1/3]^2)(((x(1+(γ^2)(θ^2)))^(3/2))))=(P/(Subscript[ω, c])) (27/(2(π^2)))(x^2) ∫dη (1 + (η^2)) ((1 + (η^2)) (Subscript[K, 2/3]^2)((((1 + (η^2)))^(3/2))x)+(η^2)(Subscript[K, 1/3]^2)((((1+(η^2)))^(3/2  ))x))
in the two polarizations; integrals can be expressed analyticaly but it is faster to compute numerically.

Iω1[x_]:=(27/(2(π^2)))(x^(2 ))NIntegrate[(((1+(η^2)))^2)((BesselK[(2/3),(((1+(η^2)))^(3/2))x])^2),{η,-∞,∞}]
Iω2[x_]:=(27/(2(π^2)))(x^(2 ))NIntegrate[(1+(η^2))(η^2)((BesselK[(1/3),(((1+(η^2)))^(3/2))x])^2),{η,-∞,∞}]
Plot[{Log[10,Iω1[(10^xx)]],Log[10,Iω2[(10^xx)]],Log[10,Iω1[(10^xx)]+Iω2[(10^xx)]]},{xx,-2,0},Frame->True,FrameLabel->{"x","(Subscript[I, ω])"},RotateLabel->False,PlotRange->{-2,.1}]

Spectrum peaks at x= 0.14, ω=0.42 (γ^3)κ; This choice of characteristic frequency is quite arbitrary.
Polarization (((Subscript[I, ω1]) - (Subscript[I, ω2]))/((Subscript[I, ω1])+(Subscript[I, ω2]))), increases from 50 to 100 percent as ω increases and the direction lies in the orbital plane; average is (3/4)

Numerical

For electrons, P=46(γ^4)(κ^(2 ))zW (≡(10^(-21))W), κ in (m^(-1)) (Subscript[ν, c])=20(γ^3)κ MHz
Magnetic acceleration:  κ =586(γ^(-1))B, (B in T)=>P=15.8(γ^2)(B^2)fW.

Cerenkov Radiation

Charged particle traveling faster than speed of light in medium, (n^(-1))=(((((Subscript[ϵ, 0])/ϵ)))^(1/2)). 
Creates Cerenkov cone (like a shock front) at angle (Subscript[θ, c])= (csc^(-1))nv. 
For an observer outside the cone there are no connectors to the source; within the cone there are two.
Modify the argument for the vector potential from wk 8 to add two equal contributions so that A=(((Subscript[μ, 0])q v)/(2π  ((r(1-(n^2)(v^2)(sin^2)θ))^(1/2))))where (r,θ)  refer to the current position of charge.
A becomes singular on the surface of the cone and E is radial, B toroidal
(Subscript[I, ωΩ])=(((Subscript[μ, 0])(q^2)n (ω^2))/(4(π^3)))|∫(dse^(iωs(1-nn·v)))(Subscript[v, ⊥])((|)^2) integrate over finite interval [-T,T] =>(Subscript[I, ωΩ=]) (((Subscript[μ, 0])(q^2)n (ω^2)(T^2)(Subscript[v, ⊥]^2))/(π^3))(sinc^2)[ωT(1-nn·v)]
Hence (Subscript[I, ω])=∫(Subscript[dΩI, ωΩ])=((2 π)/v)∫d(n·v)(Subscript[I, ωΩ])=(((Subscript[μ, 0])(q^2)ω (Subscript[v, ⊥]^2)T)/(2πv)) =>(Subscript[P, ω])=((Subscript[I, ω])/(2T))=(((Subscript[μ, 0])(q^2)ω v )/(4π))(1-(1/((n^(2 ))(v^2))))
Cerenkov cone ceases to exist at high frequency.

Integrate[(((Sin[x])^2)/(x^2)),{x,-∞,∞}];

X-ray Light Sources

SPEAR3 has 3 GeV electrons stored in a ring of 13m radius => <B>~0.8T. The current is ~0.5A
Originally a particle collider, then a simple synchrotron source, then a storage ring with larger bending magnets separated by straight sections.
The straight sections contain wiggler magnets which  deflect the particles by more than an angle ~(γ^(-1)).
The undulators in LCLS (wk 7)  deflect 10-100 fs length bunches of electrons by an angle < (γ^(-1)) strong lateral and longitudinal coherence is possible for wavelengths ~ ((Subscript[λ, w])/(2(γ^2)))corresponding to photon energies ~ 0.1-10 keV.
The emission is greatly enhanced because if N electrons are in phase the intensity ~ (N^2)

Crab Nebula

Supernova explosion contains relativistic particles and magnetic field. See ~300 MeV γ-ray synchrotron emsssion from compact region created by ~2 PeV electrons radiating in ~100 nT magnetic flux density.
The ~ few GeV electrons and positrons that stream along the magnetic field lines emanating from the poles of the central pulsar follow curving trajectories with κ~(10^(-4)) and emit coherent radio pulsars. The coherence arises from the bunching of the charges.

Uses of Cerenkov Radiation

See in light water nuclear ractors from modestly relativistic electrons.
Used in nuclear and particle experiments to measure the speed of particles and determine their nature. 
Used to detect high energy cosmic gamma rays incident upon atmosphere both through through atmospheric Cerenkov emission and on the ground using water for detecting muons
Used to detect high energy neutrinos under the Antarctic ice.

Homework

1. Using the above routines or otherwise show that the rate of photon emission in synchrotron radiation is (dN/(dϕ))=0.011γ, where ϕ is the polar angle in the plane of the orbit.
2. Z 23.16
3. A 10 TeV gamma ray is incident on the atmosphere at an altitude of 10 km where the refractive index is 1.00006. It is converted into an electron-positron pair of equal energy.
Compute the spectral flux of visible light seen on the ground. Bremsstrahlung and other interactions will create a shower of lower energy pairs. At what energy will they stop radiating?

Sketch Solutions

1. ∫dω ((Subscript[P, ω])/(ℏω κ))=0.011

(2/9) (1/137) NIntegrate[(((Iω1[x]+Iω2[x]))/x),{x,0,∞}]

2. (Subscript[I, ωΩ])=(((Subscript[μ, 0])(q^2)(ω^2))/(16(π^3)))(((βsinθ))^2)∫(dse^(iωs(1-βcosθ)))=(((Subscript[μ, 0])(q^2)(β^2)(sin^2)θ)/(16(((π^3)(1-βcosθ))^2)))
(Subscript[I, ω])=2π(((Subscript[μ, 0])(q^2)(β^2))/(16(π^3)))(∫_(-1)^1)dx(((1-(x^2)))/(((1-βx))^2))=(((Subscript[μ, 0])(q^2))/(2(π^2)))((tanh^(-1))β-1)

Integrate[((1-(x^2))/(((1-β x))^2)),{x,-1,1}]

Only up to reciprocal of initial acceleration time.
3. v~c, n~1 =>(Subscript[θ, c])~(π/2); ω~3 (x10^15) rad (s^(-1)); (Subscript[ωF, ω])~(((Subscript[μ, 0])(e^2)(ω^2)c (((n-1))^2))/(4π (H^2)))~10 aW (m^(-2))
Stop radiating when (1/(2(γ^2)))~.00006=>ℰ~50MeV

10. Fields

Reading

Z916-938

Lagrangian Formulation

Mechanics

Newton’s (vector) formulation (in terms of forces and positions) equivalent to Lagrange et al’s (scalar) formulation (in terms of energy and generalized coordinates q; L=T-V => E-L: (∂_q)L=(OverDot[(∂_(OverDot[q]))L])
Also equivalent to Hamilton’s Principle δS=δ∫dtL=0, introducing the action.
Introduced generalized momentum p=((∂L)/(∂(OverDot[q])));H=p·(OverDot[q]) - L,(OverDot[p])=(∂_q)H,(OverDot[q])=(∂_p)H,(dH/dt)=((∂H)/(∂t))=-((∂L)/(∂t))
Facilitated solution of practical problems eg celestial mechanics 
Relativistic Lagrangian L=-(m/γ)+... =>S=-m∫dτ which is a Lorentz scalar; E-L=>(d/dt)mγv=(d/dt)mu=…
Consider e.g. a planet L=-∫dx(ρ/γ)=-∫dxℒ, wh. the Lagrangian density (Subscript[ℒ, m]) =(Subscript[ρ, 0])the rest mass density in this case is also a Lorentz scalar; general property as dx≡dtdx is a Lorentz scalar

Electromagnetism

Free Fields

Generalize to fields; ODE->PDE;
Lagrangian understood by Maxwell et al as more fundamental basis for classical mechanics but inclusion of EM did not really happen until astronomer Schwarzschild in 1903
Only energy density choice for (Subscript[ℒ, em]) is (((E^2)-(B^2))/(2(Subscript[μ, 0]))); (E·B has wrong symmetry)
Constant is arbitrary unless we want to combine with mechanics in which case energy conservation will dictate this choice up to a sign.
Hence ℒ=((F:F)/(4(Subscript[μ, 0])))=((((∂_α)(Subscript[A, β])-(∂_β)(Subscript[A, α]))(η^(αδ))(η^(βγ))((∂_γ)(Subscript[A, δ])-(∂_δ)(Subscript[A, γ])))/(4(Subscript[μ, 0]))), where F:F≡(F^(αβ))(Subscript[F, βα]); stationary action subject to fixed ℒ in 3D hypersurface in spacetime
If we regard A as generalized, coordinates, E-L generalizes to ((∂ℒ)/(∂(Subscript[A, α])))=(∂_β)((∂ℒ)/(∂ ((∂_β)(Subscript[A, α]))))
Applying to our Lagrangian,(1/(Subscript[μ, 0])) (∂_β)(F^(αβ))=0 which accounts for M1,4; nb M2,3 automatically satisfied by definition of (F^(αβ ))

Interaction Lagrangian

Mechanics Lagrangians facilitate calculating interaction, eg between two oscillators; seek (Subscript[ℒ, int])  that will provide source to M1,4 AND Lorentz force for motion of charge
Only possible scalar with correct dimensions and linear in j is ∝A·j and constant defines charge.
Vary wrt (Subscript[A, α]) to identify constant as +1 st (∂_β)(F^(αβ))=(Subscript[μ, 0])(j^α)
Set ∫dxj=(qu/γ) =>L=-(m/γ) +q(A·v-φ); vary S wrt t st (d/dt)((∂L)/(∂v))=((∂L)/(∂x))=> (d/dt)(mu+qA)=q∇(A·v-φ)=q(vx∇xA+v.∇A)=>m((du)/dt)=q(-∇φ-((∂A)/(∂t))+vx∇xA)=q(E+vxB);  ·v => m((dγ)/dt)=qE·v
Hamiltonian: p=mu+qA,  H=v·p-L=((((m^2)+(p-qA)))^(1/2))+qφ; use in Schrödinger

Combined Lagrangian Density, Action

ℒ=-(Subscript[ρ, 0])+A.j + ((F:F)/(4(Subscript[μ, 0]))); S=∫dt[-(m/γ)+∫dx(A·j+((F:F)/(4(Subscript[μ, 0]))))]

Conservation Laws

Symmetry

Continuous (actually differentiable) local symmetry=> conserved current; symmetry of S or physical law not of physical system; only useful for dynamics associated with Lagrangians, not dissipative systems

Charge Current

Gauge freedom allows (Subscript[A, α])->(Subscript[A, α])+(∂_α)χ  without changing F; like a configuration space symmetry; 
If S gauge invariant, (j^α)(∂_α)χ =(∂_α)((χj)^α)-χ(∂_α)(j^α)must vanish ∀χ=>(∂_α)(j^α)=0, as surface terms vanish according to divergence th.

Coordinate Symmetry

Recall that if can transform coordinates st (Subscript[q, n]) is ignorable in L, then (Subscript[p, n ])is constant; use canonical transformation, H-J equation.
Suppose t->t+δt, q->q+δq leaves S invariant; (δt,δq are generators containing an infinitesimal); 
Hδt-p·δq , where H=((∂L)/(∂(OverDot[q])))·(OverDot[q])-L and p=((∂L)/(∂(OverDot[q]))), is conserved for each pair of generators -> conservation of energy, momentum, angular momentum...

Noether’s Theorem

Consider a single scalar field  ℒ((x^α),Φ,(∂_α)Φ), then
δℒ = ((∂_α)ℒ )((δx)^α)+((∂ℒ)/(∂Φ))δΦ+((∂ℒ)/(∂((∂_α)Φ)))δ((∂_α)Φ)=((∂_α)ℒ ) ((δx)^α)+(((∂ℒ)/(∂Φ))- (∂_α)((∂ℒ)/(∂((∂_α)Φ)))) δΦ+(∂_α)(((∂ℒ)/(∂((∂_α)Φ)))δΦ)=((∂_α)ℒ ) ((δx)^α)+(∂_α)(((∂ℒ)/(∂((∂_α)Φ)))δΦ) if satisfy E-L; also δ((d^4)x) = (d^4) x(∂_α)((δx)^α)
So δS=∫(d^4)x (∂_α)(ℒ ((δx)^α)+((∂ℒ)/(∂((∂_α)Φ)))δΦ) =>ℒ ((δx)^α)+((∂ℒ)/(∂((∂_α)Φ)))δΦ is a conserved current 
If have EM vector field Φ->(Subscript[A, β]) => (Subscript[ℒδ, β]^α)((δx)^β)+((∂ℒ)/(∂((∂_α)(Subscript[A, β]))))(Subscript[δA, β]) is conserved

Stress - Energy Tensor for Free Field

Suppose that translate in spacetime and action unchanged; in addition (Subscript[dA, β])≡(Subscript[δA, β])+((∂_γ)(Subscript[A, β]))((δx)^γ)=0 =>((Subscript[ℒδ, β]^α)-((∂ℒ)/(∂((∂_α)(Subscript[A, γ]))))((∂_β)(Subscript[A, γ])))((δx)^β)=0, ∀ ((δx)^β) 
Evaluating (Subscript[ℒδ, β]^α)-((∂ℒ)/(∂((∂_α)(Subscript[A, γ]))))((∂_β)(Subscript[A, γ]))=(1/(Subscript[μ, 0]))((1/4)F:Fη-F·F)≡T, the Stress-Energy tensor.

Cosmological Dark Energy as a Scalar Field

Important contemporary example of classical field theory
Universe has been shown to be accelerating; attributed to presence of scalar field Φ. 
The simplest ℒ is - (1/2)(η^(αβ))(∂_α)Φ(∂_β)Φ-V(Φ) 
Evaluate Stress-Energy tensor: T=∂Φ⊗∂Φ-η[(1/2)∂Φ·∂Φ+V]
If we compare with a relativistic perfect fluid, T=Pη+(P+ρ)u⊗u
We can read off: ρ=V-(1/2)∂Φ·∂Φ, P=-V-(1/2)∂Φ·∂Φ, u=((∂Φ)/(((-∂Φ·∂Φ))^(1/2)))
Manifestation of QM vacuum or emergent physics on cosmological scale.
If V,Φ const, cosmological constant: Λ=8πGV 
If Φ just varies with cosmic time, ρ=V+(1/2)((OverDot[Φ])^2), P=-V+(1/2)((OverDot[Φ])^2).
The equation of motion follows from setting the divergence of T to zero which furnishes the first law of thermodynamics: d(((ρa)^3))+Pd((a^3))=0, wh a is scale factor  ∝ size of universe 
If we ignore spatial gradients - a consequence of the initial conditions - then (Overscript[Φ, ..])+3((OverDot[a])/a) (OverDot[Φ])+(dV/(dΦ))=0
This also applies to inflation.
Another important example is general relativity where the equations are derivable from the Einstein-Hilbert action with ℒ ∝ the scalar curvature of spacetime

Final

Instructions

You have three hours to complete this exam in a single sitting unless special dispensation has been given.
You will pick it up from 207 PAB  any time after Thursday 4 June 9am and return to the same place within 24 hr before 5p Tuesday June 11
Electronic delivery is feasible if you will be out of town and make special arrangement with the TAs
You will be allowed to use one textbook, the class notes and anything you have written yourself.
Calculators and laptops will not be necessary.
You must answer all three questions which have a common three part structure:
A. Write a draft description in your own words on a topic we covered in the style of a wikipedia article and accessible to a physics undergraduate senior.
     ~100-200 words, a few equations and a sketch figure or two will suffice; cross references and bibliography are not needed.
B. Solve a simple problem
C. Make a simple application

You may need: (Subscript[m, p])=1.7 x (10^(-27))kg ≡940 MeV, (Subscript[m, e])=9.1 (x10^(-31))kg, (Subscript[k, B])= 1.4 x (10^(-23 ))J (K^(-1)), e = 1.6 x (10^(-19))C, c = 3.0 x (10^(8 ))m (s^(-1)),(Subscript[μ, 0])=4π x (10^(-7))N (A^(-2))

Exam

Q1 Electrostatics

a) Give a short, Wikipedia-style description on “Use of Bessel Functions in Classical Electromagnetism”
b) A very long conducting cylinder of unit radius about the z-axis is cut into two in the plane z=0 and the two cylinders are separated by a distance <<1.
The cylinders with z>0 is maintained at a potential +1; that with z<0  has potential -1. Show that φ =1 + (∑_(i=1)^∞)(Subscript[A, i])(Subscript[J, 0])((Subscript[k, i])r)(e^(-(Subscript[k, i ])z)) for z>0
where the constants (Subscript[A, i]), (Subscript[k, i ]) should be identified. You will find it helpful to know that ((d((Subscript[xJ, 1])(x)))/dx)= x (Subscript[J, 0])(x) and that (∫_( 0)^1)dr r (Subscript[J, 0])((Subscript[k, i])r)(Subscript[J, 0])((Subscript[k, j])r)=(1/2)(Subscript[J, 1]^2)((Subscript[k, i]))(Subscript[δ, ij]), where (Subscript[k, i]) is the i’th zero of (Subscript[J, 0])(x).
c) Sketch the equipotentials and explain qualitatively how this device can be used as a simple electrostatic lens for relativistic electrons that are initially moving parallel to the axis of the cylinder.
Without solving any more equations, outline how you would set about investigating the trajectories of the electrons in detail.

Q2 Magneto-fluid-dynamics

a) Give a short, Wikipedia-style description of “Maxwell Stress Tensor”
b) Consider a compressive, small amplitude, magnetohydrodynamical wave with wave vector perpendicular to the background magnetic field. Ignore the effects of plasma pressure and show that the wave also propagates at the Alfvén speed. Outline how the effects of plasma pressure will modify the properties of this wave.
c) A solar flare is observed and launches a hydromagnetic disturbance roughly isotropically outward in the interplanetary medium which has a magnetic flux density of 3 nT,  and a plasma density of (10^(-21))kg (m^(-3)), at the radius ~150 Gm of the Earth’s orbit. Estimate how much time will elapse after we see the flare until magnetic disturbances are registered by local spacecraft and what is the magnitude of the magnetic disturbance if the total energy released was 1ZJ (≡(10^21)J) over ~ 3hr. The interplanetary medium actually flows outward with a radial speed of 300 km (s^(-1 )) and the sound speed of the gas is ~100 km (s^(-1)). Outline qualitatively how your answer will have to be modified in practice.

Q3 Cyclotron Radiation

a) Give a short, Wikipedia-style description of “Polarization of Electromagnetic Waves”
b) Charged particles with charge q follow a circular orbit with constant curvature κ and constant speed v<<c. 
Compute the degree of circular polarization in the far field defined by C= ((<(Subscript[E, +]^2)> - <(Subscript[E, -]^2)>)/(<(Subscript[E, +]^2)> + <(Subscript[E, -]^2)>)), (where E is the electric field and +, _ refer to the two circularly polarized components in a manner which you should explain), as a function of the latitude θ measured relative to the orbital plane. Describe what happens when v ~c.
c) An early cyclotron employed closely-spaced 1 T permanent magnets arranged in a circle of radius 1m. With what kinetic energy must protons be injected to follow a circular orbit? How fast will the protons spiral inward due to radiative loss?

GOOD LUCK!

Sketch Answers

Q1 Electrostatics

a) Bessel functions arise in the solution of Laplace’s equation when there is cylindrical symmetry or when cylindrical eigenfunctions provide a natural expansion set.
Specifically, expanding in polar coordinates, r,ϕ,z,  (∇^2)φ = (∂_rr)φ + (1/r) (∂_r)φ + (1/(r^2)) (∂_(ϕϕ))φ+(∂_zz)φ=0. If we seek separable solutions of the form φ=R(r)F(ϕ)Z(z) then conventionaly we can write F''=-(m^2)F, Z''=(k^2)Z.  The boundary conditions to many problems dictate that m is an integer and k is real. In this case, the radial function R will satisfy Bessel’s equation
R''+(1/r)R'+((k^2)-((m^2)/(r^2)))R=0.
The solution that is non-singular at the origin is R =A (Subscript[J, m])(kr). Sketch of (Subscript[J, 0]).

A second solution is singular as r->0 and two more modified functions are used when k is imaginary, a circumstance that can also be encountered.
Not surprisingly, Bessel functions also arise in cylindrical solutions to the wave equation.
Modified Bessel fiunctions figure prominently in the treatment of synchrotron radiation.

b) Problem is axisymmetric and so m=0. φ(r,z)->1 for r<1 as z->∞ so k real. 
Hence φ=1+A (Subscript[J, 0])(k r) (e^(-k z))satisfies Laplace’s equation inside upper cylinder and φ(1,z>0)=1 if k is a zero of (Subscript[J, 0]).
More generally φ = 1 + (∑_i)(Subscript[A, i]) (Subscript[J, 0])((Subscript[k, i]) r) (e^(-(Subscript[k, i]) z)).
We also need φ(r<1,0)=0 =>(Subscript[A, i]) (∫_0^1)(Subscript[drrJ, 0]^2)((Subscript[k, i])r) = (1/2) (Subscript[A, i]) (Subscript[J, 1]^2)((Subscript[k, i]))=-(∫_0^1)(Subscript[drrJ, 0])((Subscript[k, i])r) = (((Subscript[J, 1])((Subscript[k, i])))/(Subscript[k, i]))=>(Subscript[A, i])=-(2/((Subscript[k, i])(Subscript[J, 1])((Subscript[k, i])))), and likewise for the lower cylinder

k=Table[N[BesselJZero[0,n]],{n,1,100}];φ=1-Sum[((2BesselJ [0,k[[n]] r](E^(-k[[n]]z)))/(k[[n]] N[BesselJ[1,k[[n]]]])),{n,1,100}];
ContourPlot[φ,{r,0,1},{z,0,1},FrameLabel->{"r","z"},RotateLabel->False]

c) Solve the equation of motion for the electrons with initial conditions as z->-∞ (Subscript[u, r])=0; (u·∇)u=-(((q(1+u.u))^(1/2))/m)∇φ, (dr/dz)=((Subscript[u, r])/(Subscript[u, z])) wh ∇≡((∂_r),(∂_z))
Observe that there will be focusing by the lower cylinder and (less) defocusing by the upper cylinder

Q2 Magneto-fluid-dynamics

a) The Maxwell stress tensor T is a three dimensional, second rank tensor that describes the stress associated with the electromagnetic field. Like all (normal) stress tensors it is symmetric and describes the force per unit are acting on the external world.

(Sometimes, it is helpful to think about this as a flux of momentum.) The forces act perpendicular and parallel to the surface. It can be expressed as the sum of an electric and a magnetic part each of which comprises an isotropic pressure ((((Subscript[ϵ, 0])(E^2))/2), ((B^2)/(2(Subscript[μ, 0]))))  I and a tension (-(Subscript[ϵ, 0]) E⊗E, -((B⊗B)/(Subscript[μ, 0]))). The form of T can be derived from the equating its divergence to the force density ρE+jxB using the Maxwell equations. An alternative approach is to regard T as a part of the special relativistic stress-energy tensor and express it in terms of products of the electromagnetic field 4-tensor (Subscript[F, αβ])=(∂_α)(Subscript[A, β])-(∂_β)(Subscript[A, α]).
b) When k ⊥ B there is only magnetic pressure not tension. Flux-freezing implies that B ∝ ρ and so the magnetic pressure has P ∝ (ρ^2). The wave is essentially a sound wave with γ=2 and the phase velocity is therefore (B/((((Subscript[μ, 0])ρ))^(1/2))). Plasma pressure will add to the restoring force and increase the wave speed.
c) v ~ ((3 x (10^(-9)))/(((4 π (10^(-7))x (10^(-21))))^(1/2)))~100 km (s^(-1))=>Δt ~1.5 Ms ~ 17d.; Wave energy density U~((10^(21-4))/(4π (((1.5 x (10^11)))^2)(10^5))) ~ (10^(-11))J (m^(-3))=>δB ~ ((((Subscript[μ, 0])U))^(1/2))~3 nT. The wave would be non-linear.
The wave will travel faster relative to the plasma on account of the thermal pressure and be convected by the plasma and so the travel time will be reduced to ~ a few days and the amplitude will be lower.

Q3 Cyclotron Radiation

a) As a transverse, vector wave, an electromagnetic wave can exhibit polarization.

Conventionally this is described by following the behavior of the oscillating electric field. If a single wave with a single frequency propagates in the z direction and the electric vector is directed along a single axis we call it linearly polarized; if the electric vector circles the z-axis we call it circularly polarized. If it is a mixture of these two case and the electric vector traces out an ellipse we call it elliptically polarized. Incoherent superpositions of waves can be partially polarized. It is then more useful to use the mean square electric field to measure the polarization. There are many conventions in use,  but a convenient one is use Stokes parameters 
I ~ < (Subscript[E, x]^2) > + < (Subscript[E, y]^2)>,  Q ~  < (Subscript[E, x]^2) > - < (Subscript[E, y]^2)>, U ~ <(Subscript[E, x])(Subscript[E, y])>, V~ <(Subscript[E, +]^2)>-<(Subscript[E, -]^2)> , where +,- refers to projections on (2^(-1/2))((OverHat[x])±i(OverHat[y]))
Waves propagating in media such as plasmas need not be transverse and can exhibit more complex polarization properties.
b) x along  Ωxn, y along nx(Ωxn), E ~ (Subscript[v, ⊥]) ~ [1, -i sinθ] where θ is the latitude. Hence C=((2sinθ)/(1+(sin^2)θ)). Note that it changes sign crossing the orbital plane.
When v~c the radiation is beamed on either side of the orbital plane and along the direction of motion. 
The two beams are strongly circularly polarized. but this will not be seen if they are combined in the detector.
c) p = eBR = 1eV/c => v~((((3 x (10^8)))^2)/(9.4 x (10^8)))~100Mm (s^(-1)); KE ~((p^2)/(2m))~50 MeV. ((dℰ)/dt) = v (dp/dt) = eBv (dR/dt)=-(((Subscript[μ, 0])(e^2)(v^4))/(6((πcR)^2))) =>(dR/dt)~ 30 μ (s^(-1))