Assignments and Schedule:
Questions in teal coloured fonts are supplementary to the core material for the course. Bother with them only if you can solve the rest.
Discussed dimensions and units in electromagnetism (Jackson, appendix). Find answers to the following questions:
- How can one determine the ratio k1/k2?
- Ampere's law and Biot-Savart law: make the connection between units as specified through them.
- What are possible observable consequences of the displacement current?
Read Sections 1-7 (ie, the first chapter) of "Classical Theory of Fields" by Landau and Lifschitz. Work out the following details:
- Write a four vector as a column vector with four components, and a Lorentz transformation as a matrix, L, acting on it. Write an interval as ds2=dxTGdx, where G is a 4x4 diagonal matrix with entries (c,-1,-1,-1). Check that a Lorentz transformation acts adjointly on G (ie, as LTGL) and leaves G unchanged. (See "Classical Electrodynamics" by Jackson; Chapter 11)
- What does raising and lowering of indices mean in the representation given above?
- Check that the matrices L form a group. What are the infinitesimal generators of the Lorentz group? (See "Classical Electrodynamics" by Jackson; Chapter 11)
- Prove the identities for the Levi-Civita tensor given in the footnotes in section 6.
- Work out the problems in Chapter 1 of Landau and Lifschitz.
- Work out the problems in Chapter 11 of Jackson.
- In the BaBar relativity pages, three ideas are given as the basis of relativity. Are all three needed, or can one reduce the number of basic ideas?
- If a clock were accurate to 10-12 secs, then how many times round the earth would a typical jet liner have to fly in order to test relativity? Set out the procedure to be followed in as much detail as is necessary.
- Does relativity have anything to say about the existence of rigid bodies?
- How can you observe super-luminal motion if it occurs? Are there ways of observing what seems like super-luminal motion without violating special relativity?
Follow up questions:
- Find all Lorentz tensors of rank two which are independent of the coordinates. Is it possible to have symmetric as well as anti-symmetric tensors of this kind?
- Answer the previous question for the rotation group.
- Is the Levi-Civita symbol a Lorentz tensor? If not, then is there a subgroup of the Lorentz group under which it is a tensor? Do multiple contractions of two Levi-Civita tensors form tensors? If they do, then can one prove the identities in the footnotes of section 6 of Landau and Lifschitz?
- Characterize all possible metrics on the space of spinors which remain invariant under Lorentz transformations.
Argha Banerjee and Aseem Paranjape will present a half hour talk on super-luminal motion on Monday 23 August at 5:00 PM in AG 69.
Day 3: August 16, 2004
Read chapters 2 and 3 of "Classical Theory of Fields" by Landau and Lifschitz. You may need to refer to a textbook such as "Classical Mechanics" by H. Goldstein in case you need to refresh your memory of Lagrangian mechanics, canonical momenta etc.
- What does the principle of relativity have to say about the tensor character of the laws of physics? What are Lorentz invariance and Lorentz covariance?
- For non-relativistic motion near the surface of the earth (where the gravitational field can be taken to be uniform and pointing in the z-direction) what are the allowed symmetries of motion? What does the principle of relativity then have to say about physical observations valid in this domain?
- Solve the problems in chapters 2 and 3 of Landau and Lifschitz.
Follow up questions:
- Consider a scalar field A interacting with a point particle through a scalar charge g. Write down a simple action for the particle in the external field A, and derive the equation of motion. Find the linear and angular momenta.
- Analyze the decay of a particle into 2 and 3 particles. In each case find the number of free variables. Use Kajantie's book on relativistic kinematics or the particle data book to find standard parametrizations of the free variables. In each case, write an efficient subroutine which takes as an input the 4-momentum of the initial particle and returns the 4-momenta of the final particles, after making random choices of the free variables.
- Analyze the kinematics of two particles scattering into two particles and find a standard parametrization using the above references. Write an efficient code which takes as input the two initial 4-momenta and returns as output the two final 4-momenta after making random choices of the free variables.
- Check the Jacobi Identity for the electromagnetic field strength tensor.
- Check how the components of the electromagnetic field strength tensor transform under a boost in the x-direction. Using this construct the representation of this boost acting on the three vector F=E+iH.
- Write down the pseudoscalar constructed from the Levi-Civita and two copies of the electromagnetic field strength in terms of the potentials and its derivatives.
Naren H.R. and Shayamita Ray will present a half hour talk on the horizon problem in cosmology on Thursday 26 August, at a time to be specified later. Aditya Gilra and Swapnil Patil will present a half hour talk on limits on Lorentz non-invariance in electrodynamics, and Partho Nag and Tridib Sahu will present a half hour talk on tests of Lorentz non-invariance on Thursday 26 August, at a time to be specified later.
Day 4: August 19, 2004
Technical details glossed over in the previous lectures will be discussed.
Follow up questions:
- Follow up the example of the relativistic equation of motion for a spin in an external field (Jackson chapter 11) by analyzing all possible pseudo vectors which can be constructed from the spin pseudo vector, S, the electro-magnetic field strength, F, the particle 4-velocity, u, and the 4-acceleration, w.
- For the Lie algebra of the Lorentz group, that is, the three generators for boosts, Ki, and the three generators of rotations, Si, explicitly verify the commutation relations. Why do these imply that the boosts transform as 3-vectors under rotation? Analyze Thomas precession using infinitesimal Lorentz transformations written in terms of exponentials of linear combinations of the generators. Find representations of these generators acting on spinors. Investigate Thomas precession on spinors.
- Ohm's law has the 3-dimensional rotational tensor form
ji=σijEj, where j is the
current 3-vector, E is the applied field and σ is
the conductivity tensor of the material under study. Arguments were
given for reduction in the number of components when the material
is isotropic or is an uniaxial crystal. Find the appropriate
conductivity tensor for
- the case when the material under study has a crystal structure with three axes which have no symmetry relations
- crossed electric and magnetic fields. (Why does this differ from an uniaxial crystal?)
- Define stress and strain 3-tensors in a solid. Define the elastic
coefficients through a relation between these. What forms do the
elasticity tensor take in
- an isotropic medium?
- uniaxial crystal?
- general triaxial crystal?
Day 5: August 23, 2004
Read chapter 4 of "Classical Theory of Fields" by Landau and Lifschitz. Answers to follow up questions 4 and 6 of Day 3 will need to be worked out before the lecture. You may need to refer to a textbook such as "Classical Mechanics" by H. Goldstein in case you need to refresh your memory of Lagrangian mechanics, definition of canonical momenta etc.
Follow up questions:
- Check the conditions on a Lagrangian needed to satisfy causality.
- Can the action be gauge dependent and still give rise to field equations which are gauge invariant?
- Use the electromagnetic field equations to derive the continuity equations for the electric 4-current. (By the field equations I mean the Euler-Langrange equations of the field, and not the Bianchi identity).
- Work out in detail the derivation of the electromagnetic field equations from the action. This involves manipulation of tensors that may help in later work.
- How did the choice of a constant 1/16π in the field action correspond to the choice of Gaussian units? What would correspond to other widely used units?
- Construct the 4-dimensional analogue of Stoke's theorem. Would this simplify any of the material touched upon already? If so, how? If not, where could this be applicable.
Anindya Dey, Bhargava Ram and Suryanarayan Dash will present a half hour talk on spinor representations of the Lorentz group on Monday 30 August, at 3 PM.
Day 6: August 26, 2004
Continue with chapter 4 of "Classical Theory of Fields" by Landau and Lifschitz. You may need to refer to a textbook such as "Classical Mechanics" by H. Goldstein in case you need to refresh your memory of Lagrangian mechanics, Noether's theorem etc.
The day's discussion focussed on symmetries: mainly on Noether's theorem and the topic of tensors and group representations. As follow up exercises
- Read about the field due to static charges and their multipole expansions from Landau and Lifschitz sections 40 and 41 and Jackson sections 4.1 and 4.2. The former book exhibits the tensor character in terms of multi-index objects, the latter in terms of irreducible representations (irreps) of the rotation group through the spherical harmonics.
- Learn about rotational tensors and irreps of the rotation group from the appendix on this subject in "Nuclear Physics" by De Shalit and Talmi.
Day 7: August 30, 2004
Since no student was prepared for this material on day 6, we will again attempt to continue with chapter 4 of "Classical Theory of Fields" by Landau and Lifschitz. You may need to refer to a textbook such as "Classical Mechanics" by H. Goldstein in case you need to refresh your memory of Lagrangian mechanics, Noether's theorem etc. Consider questions such as
- What makes the Lagrangian density a finite object. (For a collection of particles, in the non-relativistic limit, can you conceive of an interaction for which the Lagrangian is not linear in the number of particles?)
- For the EM field the canonical coordinates are the potentials. What are the canonical momenta? Can you use these to write the Hamiltonian of the field? How does gauge invariance show up? What do you get for the equations of motion? (This is important: see the book "The quantum theory of fields" Vol 1, by S. Weinberg, section 8.2 if you need help. You should understand the classical part of the arguments in p 344 and eqs. (8.2.7-9) on p 346. If you feel like exploring more then start from the discussion of constrained systems in Dirac's book "Quantum Mechanics")
Follow up questions
- Write the stress tensor as a sum of symmetric and antisymmetric parts. What is the contribution of the antisymmetric part to the angular momentum?
- For plane waves propagating in the +x direction, it was demonstrated that it is possible to choose a Lorentz gauge in which the components A0 and A1 of the gauge potential vanishes. Is it possible to make more components vanish?
- What condition must Lorentz transformations satisfy if they are to diagonalize the stress tensor for plane wave solutions of the Maxwell equations? What Lorentz transformations, if any, satisfy these conditions?
- Construct possible actions, the equations of motion, consider the role of
symmetries, the stress tensor and plane wave solutions for
- a real scalar field φ
- a complex scalar field ψ=ψR+iψI, such that the action is invariant under changes of phase, ie, for ψ'=exp(iα)ψ, where α is a constant, independent of the position.
- M different scalar fields, φ1 ... φM such that the action is invariant under linear combinations of field components which are the same at each space-time point.
We continue with plane wave solutions. Read up on polarization, the Stokes parameters, and Fourier analysis from Landau and Lifschitz, "Classical Theory of Fields" (chapter 6).
Follow up questions:
- Find the little groups of various classes of 4-vectors.
- Check whether circularly polarized waves carry angular momentum. Does its projection on the direction of motion have anything to do with the polarization?
- For a monochromatic wave, the polarization 4-vector, ε, is a new vector in the problem (the wave vector k is the other). Why should the stress tensor contain only the term kμkν? Why are terms such as εμkν or εμεν absent? (See Jackson's chapter on plane waves for the construction of polarization vectors.)
- How is the polarization vector related to the angular momentum of the plane wave? (Does the discussion on day 4 of spin precession due to an external force help?)
- Prove that the Stokes parameters ξ2 and ξ12 + ξ32 are Lorentz invariant.
- Evaluate the angular integrals obtained in Fourier transforming two dimensional functions which transform as irreducible tensors under rotations.
- Write down the angular integrals which would be obtained in Fourier transforming three dimensional functions which transform as irreducible tensors under 3-dimensional rotations. Do these reduce to the integrals you have already performed above? Why?
- Check the definition of a linear vector space. Do square integrable functions form a linear vector space?
- Write down the Fourier transformation in 1 dimension as a matrix (make a lattice if that helps to keep define things). Check that this is unitary, and that the inverse matrix yields the correct definition of the inverse Fourier transformation. Check that the delta functions provide an obvious basis for functions on the original space.
- Construct the infinitesimal generator of rotations in a plane and translations along a line. Are these equal? What is the difference between the two groups?
Write ups of the first round of seminars are due by Sunday, September 5. Send either a ps or a pdf file by mail. The submission is acceptable if the received date stamp on the mail is September 5 or earlier, and the file can be opened and navigated by ghostview or acrobat reader.
Day 9: September 6, 2004
We re-examine the notion of causality. We use causality to define Green's functions for the wave equation. Use the books by Jackson, Schwinger or Griffiths to prepare. Alternatives are older field theory texts such as that by Bjorken and Drell.
Follow up the lecture with these questions:
- Can you regulate the propagator of the wave equation without taking principal parts? Use a different regulator and check whether the solutions are the same.
- Write the function 1/x in an integral with a regular test function as the sum of the prinicipal part of 1/x and a delta function. (Be careful about the coefficient of the delta function.)
- For the harmonic oscillator the advanced and retarded propagators were defined. Check that a given solution of the inhomogeneous equation can be written using either propagator, but then they differ by a solution of the homogeneous equation. Use the result of the previous problem.
- Complete the construction of the propagator (Green's function) for the electromagnetic wave equation.
- Simplify the expression for the EM propagator when the inhomogeneous terms (currents) have cylindrical symmetry.
- Simplify the expression for the EM propagator when the currents have spherical symmetry.
- Prove the three equivalences set up in Titchmarsh's theorem.
- Work out in detail the construction of the model for absorption of light in Jackson (chapter on waves). What do the dispersion (Kramers Kronig) relations for the propagator have to say? (This is important)
- Suppose the imaginary parts of the conductivity function σ(ω) for two different materials differ by exp(-tω)P(ω), where P(ω) is a polynomial of order n. Then can one derive bounds on the difference of the DC conductivities? (This is important)
- Reverse the question. If the DC conductivities of two materials are known to differ, and they absorb light at the same frequencies, then in what ways can the absorptive (imaginary) part of σ(ω) differ? (This is important)
- If there is a piece of glass which is transparent to light at all wavelengths, then what can you say in general about its complex conductivity? (This is important)
- The EM wave equation is Lorentz covariant. How does this extra structure affect the dispersion relations for the propagator?
Day 10: September 9, 2004
We examine the boundary conditions for fields at a conductor. We find solutions for waveguides and cavities, using symmetries to diagonalize the field equations. Use the book by Jackson to prepare for this material.
Follow up questions:
- What are the definitions of globally elliptic, parabolic and hyperbolic second order linear partial differential equations in two dimensions? What are the canonical forms of these equations? How does one solve globally hyperbolic equations by the method of characteristics?
- Construct globally elliptic, parabolic and hyperbolic second order linear partial differentual equations in two dimensions in which the coefficients of the second derivative terms are not constant.
- Construct a linear second order partial differential equation in two dimensions which is not in one of these classes.
- Find two different orthonormal and complete bases on all square integrable functions in one dimension. Prove the completeness of both these bases.
- Show that the boundary conditions Ez=0 and dn Bz=0 at the conducting surface of a waveguide implement exactly the boundary conditions that the normal component of B and the tangential component of E vanish.
- Explicitly solve the equations for a waveguide of square cross section and use Mathematica to draw graphs of the field intensities of the 3 lowest modes for any ω above ωmin. Compute the values of ωmin for these three modes.
- Explicitly solve the equations for a waveguide of circular cross section and use Mathematica to draw graphs of the field intensities of the 3 lowest modes for any ω above ωmin. Compute the values of ωmin for these three modes.
The Mathematica notebooks that you create to solve the last two equations should be submitted by mail by midnight of September 16, 2004. As usual, the submission will be considered complete only if the received time stamp is before the last date, and the program successfully runs.
Day 11: September 13, 2004
We move on to the problem of accelerated charges. Use Landau and Lifschitz or Jackson to read up on retarded potentials, antennas, synchrotron radiation.
Follow up the lecture with the following problems:
- What happens at the cutoff frequency of waveguides which leads to an infinite phase velocity?
- What makes the product of the phase and group velocities a constant independent of frequency?
- Can you compute the Q value of a classical hydrogen atom? How fast does the electron lose energy when starting from the lowest possible quantum level? From the next level? In general?
- If a charged particle in an uniform magnetic field radiates, then what is the Q value? Does it depend on the field? Does the radiation have a preferred polarization? Does
- How does a charged particle radiate when it undegoes uniform, rectilinear acceleration? Spectrum, polarization etc.
The course uptil now dealt with properties of the electromagnetic field: the symmetries and field equations, specific solutions of the field equations- mainly wave solutions and the generation of fields by moving charges. We now put this material to use and begin the study of matter.
Prepare for this lecture by reading the following sections from Landau and Lifschitz, "Electrodynamics of Continuous Media". In class we will deal with specific sections of chapters 1 and 2, described below. Section 1 is basic material that we have used before. Section 5 is an application of the stress tensor. Read sections 6, 7 and 9 keeping in mind the notion of causality. Section 10 applies notions of the energy and momentum densities of fields to thermodynamics. Section 13 is a simple application of finite groups and their irreducible tensors to the study of in-medium electrodynamics.
Follow up questions:
- We discussed coarse graining of microscopic equations in order to obtain effective theories of materials considered as continuous media. The microscopic equations such as the EM field equations or Newton's equations for the motion of particles contain no dissipative terms. How do such terms (viscosity in fluid dynamics, electrical conductivity of materials) arise due to coarse graining?
- Study sections 2 and 5 of Landau and Lifschitz "Electrodynamics of Continuous Media" to understand the forces and torques on conductors due to applied electromagnetic fields.
- Complete the construction of coarse grained equations for static magnetic properties of materials (Landau and Lifschitz, "Electrodynamics of Continuous Media, sections 29 and 30).
- Do magnetic fields produce forces and torques on media? Study the question quantitatively using appropriate generalizations of the material from question 2.
- On day 10 of the course we came across the notion of a skin depth for the penetration of fields inside a conductor (Jackson, section 8.1). What is the physics of this?
Day 13: September 20, 2004
We continue from the last lecture with the first two chapters from Landau and Lifschitz, "Electrodynamics of Continuous Media". In class we will deal with specific sections of chapters 1 and 2, described below. The main emphasis now will be the thermodynamics of materials in electromagnetic fields. Re-read sections 2 and 5, and the section 10 till the end of chapter 2.
- For a system of conductors show that the derivative of the energy with respect to the potential gives the charge and vice versa.
- Show that the mutual capacitance of two conductors is negative.
- Derive a bound on the off-diagonal components of the mutual conductance in terms of the self capacitances of a system of conductors on a field.
- In an inhomogeneous external field, the gradient of the field provides a direction. In which direction does a conductor move: along or opposite to the field?
- A function of one variable (a curve) is given by an equation y=y(x). Define
the slope, p=y'(x). Then the curve can also be specified by saying how y can
be reconstructed as p varies along the curve. This is a Legendre transform.
(Example: The energy U of a system is a function of S, then T=dU/dS. A Legendre
transform to the free energy, F, expresses the same relationship in the same
system by giving F in terms of the variable T.) Find Legendre transforms of
the following curves:
- y=a x
- y=Log x
- Find the contribution to the entropy of a homogeneous and isotropic dielectric when the dielectric constant of a material is temperature dependent.
- In a crystal the dielectric constant becomes a tensor of rank 2. This tensor can always be diagonalized. Is there any special simplification for either these eigenvalues or the corresponding eigenvectors if the crystal structure admits a spontaneous polarization?
- What is the rank of a tensor which connects the elastic stress tensor of a material with the electric field? What are the symmetries of this new tensor?
Day 14: September 23, 2004
Using complex analytic methods for solving potential problems in a plane: Tridip Sadhu. Causality and the solutions of linear second order PDE's: Aditay Gilra.
Day 15: September 24, 2004
Basic fourier analysis and its connection with the translation group: Bhargava Ram. Causality and the theorem of Titchmarsh: H.R. Naren.
Day 16: September 25, 2004
Causality and the Kramers-Kr\"onig relations: Aseem Paranjape. The irreducible representations of finite groups: Swapnil Patil. Lie groups and their irreducible representations: Partha Nag.
Day 17: September 27, 2004
Fourier analysis and the solutions of PDE's as exercises in finding representations of Lie groups: Shamayita Ray. Lattices, their symmetries and Fourier analysis: Argha Banerjee.
Day 18: September 28, 2004
Representations and properties of the delta function: Suryanarayan Dash.
Day 19: October 1, 2004
Multipole fields and representations of the rotation group: Anindya Dey.
Day 20: October 4, 2004
A second look at all the material covered in the course.
We will study the motion of charged particles in electromagnetic fields. As preparation for the class, study the appropriate chapter and all the solved examples from Landau and Lifschitz "Classical Theory of Fields".
Follow up questions:
- Consider a charge accelerated by an uniform static external electric field (Case 1). What kind of radiation does the charge emit: polarization, power spectrum, angular distribution etc? Does it lose energy by radiation faster or slower than the energy it gains from the accelerating field? Does this depend on factors such as the particle's current momentum? Is it possible for such an accelerated particle to have a limiting momentum less than c? What do these considerations have to do with designing linear accelerators?
- Consider a charge accelerated by an uniform magnetic field (Case 2). Construct dimensionless combinations of system parameters and initial conditions and check what their values have to do with the dynamics. Interpret these dimensionless quantities in terms of ratios of dynamical quantities of equal dimension. How many different time scales can you construct from these variables (initial conditions and system parameters)? Neglect radiation from the charge for this part of the analysis.
- Characterize the radiation emitted from a charge in a constant uniform external magnetic field (polarization, power spectrum, angular distribution) assuming that the rate of energy loss is small. What dimensionless parameter does one have to examine to check that this approximation is valid? The leading term assumes that the gyromagnetic frequency ωg is constant. Check whether the first correction term is linear in (dωg/dt)/ωg2, or whether it involves a higher power or a higher derivative. Can you argue for your result?
- Can one deduce the magnetic field at the surface of a distant star by looking at the radiation coming from it, without making untestable assumptions about the origin of the radiation?
- Can you look at a bubble chamber or emulsion picture of elementary particle interactions, and deduce the value of the magnetic field strength used in these detectors? Conversely, can you identify the particles that made these tracks?
- For a charge in uniform static external magnetic and electric fields (Case 3), construct all possible dimensionless combinations of system parameters and initial conditions. Interpret these in terms of ratios of physical quantities with equal dimension which allow one to say something about the physics of the problem.
- In a certain limit of these dimensionless quantities (which?), the motion can be analysed as a small perturbation of case 2, yielding a drift velocity, vd. In general, there is no such time-independent drift velocity, because the center of the gyro-orbit accelerates. Can one write the corrections in a power series of an appropriate dimensionless quantity? Can one find the first term beyond the drift approximation?
- In the opposite limit of the value of the same dimensionless quantity, the motion should be analyzable as a small perturbation of case 1. What is the nature of the leading perturbation?
- In materials where Ohm's law holds, the drift approximation must be exact for case 3. Clearly then, the charge under investigation must lose the energy that it gains from the field through collisions with other particles in the material. What properties of the material must be measured in order to deduce that Ohm's law must follow?
- What initial conditions can be varied for particles to escape from a magnetic trap? Can you draw the regions of phase space in which particles are trapped? Assume the minimum information that you need to assume.
We will study plasmas, that is to say, materials with mobile charge carriers. We will show that such materials are characterized by Debye screening and plasma oscillations, and that there is a dimensionless variable that says whether or not the coarse grained picture of this material (at length scales larger than the Debye screening length) is fluidlike. As preparation for the class, study the appropriate chapters of Sturrok's book.
Day 23: October 25, 2004
We will continue the study of plasmas by extending the dimensional arguments to relativistic and quantum plasmas. We shall also study plasmas with overall non-zero charge, and see whether they differ from ``one-component plasmas'' which are charge neutral (ie, plasmas which are charge neutral but only one charge is mobile).
Follow up for lectures on days 22 and 23
- Write a Mathematica notebook to compute the gradient operator and the Laplacian in any system of coordinates in three dimensions, given the expressions for the cartesian coordinates in terms of the new ones. (Deadline: midnight of Oct 26; evaluation criteria: correctness and speed of execution)
- For a non-neutral plasma in a Penning trap,
- evaluate the Green's functions G(
- estimate the correction to the confinement criterion coming from the mechanical part of Lz.
- check whether Debye screening occurs.
- compute the plasma oscillation frequency.
- evaluate the Green's functions G(
- Find out the number densities of charge carriers and temperatures of the following plasmas- salt solution in water at room temperature, a metal, the terrestrial ionosphere, the jovian ionosphere, the solar corona, the solar core, the interior of a tubelight, current tokamak designs, interstellar gas, the universe just before decoupling of matter and radiation. For each of these, check whether quantum and relativistic effects are important. Tabulate the Debye length and plasma frequency for each of these plasmas.
- We have performed a dimensional analysis of the Debye screening length in terms of the microscopic parameters of the plasma and found that λD(e,m,n,T)=n-1/3f(e,Λ), in a relativistic quantum theory, where Λ is the plasma parameter, and e is the dimensionless charge, Investigate the scaling of &lambdaD in a non-relativistic quantum theory. This would describe a metal at room temperature. Check in any solid state physics book (for example, Ashcroft and Mermin) whether the form you derive is valid.
- For a non-relativistic classical plasma we found that λD(e,m,n,T)=n-1/3f(Λ), where f(Λ) was found to be linear in the plasma parameter Λ. Is this an approximation, valid only for large Λ? Can you find the next terms in the expansion, or the full function f(Λ)?
Day 24: October 28, 2004
We will examine the propagation of small frequency electromagnetic waves through a plasma by writing down a dielectric tensor. Restrictions due to causality will be investigated. Read appropriate sections of "Electromagnetic Theory" by Jackson, and "Electrodynamics of Continuous Media" by Landau and Lifschitz.
Follow up questions:
- Construct the Argand diagram for the impedance, Z(ω) of a simple LCR circuit, ie, one with only one resonance. How does the form change as the damping term vanishes, ie, the pole in the response function comes closer to the real axis? (To be used in later lectures)
- Construct a formal proof that the resistivity of a material is positive.
- A linear circuit is usually constructed by putting together resistors, capacitors and inductors of given ratings in any combination whatsoever. However, these are coefficients of the real and imaginary parts of the Z(ω). Since Z(ω) is causal, its real and imaginary parts must be related by a Kramers-Kronig relation. Why does this not constrain the values of the circuit elements which can go together? (In other words, how can circuits be built with modular elements?)
- In the limit of large ω, the dielectric constant, ε of a material goes to unity. Thus one cannot write a Kramers Kronig relation for ε. Show that one can write such a relation for ε-1. (Hint: What quantity other than D must satisfy a causality relation with respect to E?)
- For an isotropic medium introduce the dielectric constant ε(k,ω) and the magnetic permeability μ(k,ω). How are these related to the two functions εL and εT which were constructed from the tensor ε?
- When changing the definition of the tensor ε by adding to it a piece in ΠT, one has to change the definition of H in terms of B. Can such a redefinition be used to get the relation between the pair ε and μ and the pair εL and εT?
- Construct three orthogonal projection operators to be used in the wave equation in the presence of an external magentic field.
- For a single particle in an external electric and magnetic field, write the equations of motion for the Fourier components of the velocity, v(ω). Trade the value of the external magnetic field by the gyro-frequency ωg. Solve the resulting equations. (To be used in later lectures).
Day 25: November 1, 2004
We will continue to examine the propagation of small frequency electromagnetic waves through a plasma by writing down a dielectric tensor. Read appropriate sections of "Electromagnetic Theory" by Jackson, and "Electrodynamics of Continuous Media" by Landau and Lifschitz.
Follow up questions:
- Construct the Argand diagrams for the response functions of the LCR circuit and a damped harmonic oscillator. Are they exactly the same? If not why not? (The response function is defined as the ratio of the output signal and input force in the Fourier domain)
- If the response of a system at time t depends on the applied force over the duration t-δ≤t'≤t only, then can the quantity δ be extracted from a knowledge of the response function Z(ω) for all values of ω? How?
- With the definition γ=2b/m, (where 2b is the coefficient of the damping term in the equation of motion of the charge in an electric field E and m is the mass), find the dispersion relations for the longitudinal and transverse modes in a plasma. For the transverse modes make the assumption that γ is much smaller than ωp.
- Assuming that γ=0, and that a magnetic field B=Bz is applied
to the plasma, write down the following quantities in terms of ω, k,
ωp and ωg:
- the longitudinal, transverse and Hall conductivities, σL, σT and σH.
- the dielectric tensor ε.
- dispersion relations for the three independent modes in the two cases when k is in the z-directions and the y-direction.
- For the last part of the previous problem find all the zeroes and divergences of the dispersion relations k2(ω2). Recall that every zero of this function is a cutoff, and every divergence can be a resonance if the function changes sign at the resonance. Thus, this question asks you to find all the cutoffs and resonances. Use Mathematica to help you.
- Find the phase and group velocities for each of these modes in both the cases (k and B parallel or orthogonal). Plot them. Use Mathematica to make your job easier.
Day 26: November 4, 2004
We will examine the passage of electromagnetic waves through a plasma in the presence of an external magnetic field. Read chapter 6 of Sturrock.
Follow up questions:
- Complete the analysis of the Argand diagram plot for the damped harmonic oscillator. As the damping coefficient is changed, how does this change? Can the change in the peak of the response function be seen in the Argand diagram? Can the change in the width of the peak of the response be seen?
- The damping of the plasmon mode (plasma oscillations) due to thermal effects is called Landau damping. Is the wave damped or the motion of electron damped? In other words, how can you decide which way the energy is flowing: from electrons to waves or waves to electrons.
- When γ=k(T/M)1/2 is much smaller than ωp, how are the transverse modes damped? Is there any special behaviour in the damping at λD?
- Examine Faraday rotation in detail. What information can you obtain about physical conditions in a plasma by examining Faraday rotation of waves passing through it. Can you use this to probe the earth's ionosphere? (How does one produce polarized radio waves?) Can you design an experiment to see this in a salt solution in the lab?
- For use in the remaining problems you will need to build a Mathematica notebook which will take a dispersion relation and use this to plot the time development of a wavepacket. It would be useful to play with Gaussian and square wavepackets. You should be able to plot or listen to the wavepacket.
- Examine the propagation of the helicon mode: phase velocity, group velocity, the broadening of a wavepacket. Examine the propagation of a wavepacket. Stay away from resonances so that you can neglect damping.
- Examine the propagation of the Alfven wave: phase velocity, group velocity, the broadening of a wavepacket. Examine the propagation of a wavepacket. Stay away from resonances so that you can neglect damping.
- Examine the propagation of a whistler: phase velocity, group velocity, the broadening of a wavepacket. Examine the propagation of a wavepacket: listen to it. Work out the frequencies of a whistler in the atmosphere (you need to know physical conditions in the ionosphere and the strength of the earth's magnetic field). Procure a radio and listen to whistlers: which band do you need to listen to? Does this sound like your program?
- Examine the propagation of light through a plasma when the wave vector is orthogonal to the applied magnetic field. When B is very large, what needs to be added to the analysis to get the resonances at multiples of ωg? Do you need to understand a cyclotron better to understand these resonances?
Day 27: November 18, 2004
We will study elements of Magneto-Hydro-Dynamics (MHD). See Landau and Lifschitz "Electrodynamics of Continuous Media", sections 63-69. In particular we will study dissipationless MHD (infinite conductivity, zero viscosity). In this limit magnetic lines of force are frozen into the fluid (Alfven's theorem), magnetohydrostatics is simple, and the linearized hydromagnetic equations give rise to Alfven and magneto-acoustic waves. (See an obit of Alfven and his Nobel page)
Follow up with
- Is MHD valid for the plasma in a tubelight? Estimate the Debye screening length for the answer. In the same vein, is MHD a valid approximation for droplets of mercury flowing on a tabletop?
- Write down the Euler equation for MHD using elementary arguments about forces on current carrying wires, and cross check this argument starting with the expression for the stress tensor.
- Find relations between the pressure, density and magnetic field changes for the Alfven wave. (In class we found the relation between the magnetic field and velocity.) How do fluid elements move in an Alfven wave? (Plot trajectories of fluid elements.)
- Find the solutions for the speeds of the fast and slow magneto-acoustic waves. Is there a difference in the motion of the fluid elements for these two modes?
- Find relations between the pressure, density and magnetic field changes for the magneto-acoustic waves. How do these differ from the same quantities for the Alfven wave?
- In the limit of an incompressible fluid, how do the fluid elements move for the two polarizations of the Alfven wave?
- For a liquid metal such as Mercury, what are the speeds of the Alfven and magneto-acoustic waves in the magnetic field of the earth? Can these be observed? By increasing the DC magnetic field H0, can one make it easier to see these?
- Compute the speeds of Alfven and magneto-acoustic waves in as many different plasmas as you can think of.
Days 28 and 29: November 22 and 25, 2004
These contact hours will be devoted to questions on previous material.
- Estimate the Debye screening length of the plasma inside a tube light. Is it likely that MHD is applicable to this plasma?
- By estimating the voltage drop across a tube light, find the conductivity of the plasma inside the tube.
- Find the skin depth of EM waves at the line frequency, ie, 50 Hz, for the plasma in the tube. Is it likely that the ideal MHD equations are applicable to this plasma? If not, then assuming this conductivity to be independent of frequency, at what frequency could this plasma be considered to be an ideal magneto-fluid? What is the phase velocity of Alfven waves at these frequencies?
- At frequencies such that the plasma inside a tube light can be considered to be described by the ideal MHD equations, expand these equations to second order in small deviations from equilibrium and thereby find the maximum field (oscillating) field strength such that the linearized equations apply.
- Repeat the above estimates for the plasma in the earth's ionosphere. (Estimate the density of the atmosphere at the appropriate height by considering the atmosphere to be a compressible fluid in hydrostatic equilibrium, with the temperature dependent only on the height. Landau and Lifschitz "Fluid Mechanics" may be of use. Cross check your answer with measurements of the pressure at the ionosphere.)
- Repeat the same exercise for the sun's corona and photosphere.
- Repeat the exercise for a saturated solution of common salt at normal room temperature.
Questions in teal coloured fonts are supplementary to the core material for the course. Bother with them only if you can solve the rest.
© Sourendu Gupta. Created on Jun 26, 2004. Last updated on Nov 23, 2004.