# Plasma Processes. m v = ee. (2)

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Plasma Processes In the preceding few lectures, we ve focused on specific microphysical processes. In doing so, we have ignored the effect of other matter. In fact, we ve implicitly or explicitly assumed propagation through a vacuum for most applications. It s time to take on matter! When we introduced Maxwell s equations, we very solemnly defined D = ǫe and H = B/µ to include the effects of matter, where ǫ is the dielectric constant and µ is the magnetic permeability. Is this necessary? No! Remember that Maxwell s equations explicitly include sources, in the form of ρ and j. If we do this consistently, for all charges and currents (whether or not they are in a medium), then Maxwell s equations for E and B alone work just fine. Thus, Maxwell s equations for a vacuum work fine in that case, as long as both free and bound charges are included. In that case, we can again think about the propagation of radiation, this time more generally. Again let s assume a space and time variation of the form exp i(k r ωt). Then we get ik E = 4πρ, ik B =, (1) ik E = i(ω/c)b, ik B = (4π/c)j i(ω/c)e. Now, in general this could be pretty tough, if the medium is something arbitrary (air, water, glass). In our case, though, we re specifically interested in a plasma, which can be loosely defined as an ionized gas that is globally neutral. That means that all charges are mobile in principle. However, as we ve done before, we ll assume that the ions are basically stationary for our purposes, and mainly serve to keep the plasma neutral. Another important simplifying assumption is that the plasma is isotropic. Ask class: what does that imply about the magnetic field? It implies that there is no significant external magnetic field, because that would break isotropy. Let s consider nonrelativistic electrons. A given electron follows Newton s law m v = ee. (2) There can be an internal magnetic field, just not an ordered one. Ask class: why, then, have we neglected the magnetic component of the force? It s because that term is of order v/c, which is small if the motion is nonrelativistic. Given our assumption about the space and time variations of quantities, this means The current density is j = nev, meaning that we get v = ee/(iωm). (3) j = σe, (4)

2 where the conductivity is σ = ine 2 /(ωm). This is Ohm s law; the current responds directly to the electric field. Note, however, that this statement requires isotropy. Ask class: can they think of a specific example in which anisotropy will lead to a different response to an applied electric field? If there is a strong magnetic field, charges and currents move along the field a lot better than across, so no longer is there this linear relation. In general, in fact, the conductivity is a tensor. However, for our situation we can treat it as a scalar due to isotropy. so that From charge conservation, our expi(k r ωt) assumption gives If we define the dielectric constant by (note that this is real, since σ has an i in it), we get iωρ + ik j = (5) ρ = ω 1 k j = σω 1 k E. (6) ǫ 1 4πσ/(iω) (7) ik ǫe =, ik B =, ik E = i(ω/c)b, ik B = i(ω/c)ǫe. Well, lookee here. This looks just like the source-free vacuum equations we had before, except for ǫ. Indeed, arguing as before, we find that k, E, and B are mutually perpendicular. However, we find that c 2 k 2 = ǫω 2. (9) Since ǫ depends on ω, we no longer have the simple vacuum situation in which all frequencies travel at the same rate, the phase velocity is the same as the group velocity, and so on. Thus, the presence of a plasma introduces dispersion, where wave packets spread and there is effectively an index of refraction. If we substitute in expressions, we can rewrite the dielectric constant as ǫ = 1 (ω p /ω) 2, (1) where ω p = 4πne 2 /m is called the plasma frequency. Numerically, ω p = n 1/2 s 1 if n is in cm 3. Ask class: from these definitions and the dispersion relation, what does this tell us about propagation when ω < ω p? It means that k is imaginary: k = (i/c) ω 2 p ω 2. Ask class: what does that mean about the propagation of radiation below ω p? It means that there is an exponential cutoff in the amplitude, with a distance scale of order 2πc/ω p. Therefore, effectively, frequencies below ω p can t propagate in a plasma. Side note 1: one way to get quick intuition in a number of astrophysical situations is to have a number of characteristic quantities in mind. The plasma frequency is an (8)

3 example: if you have a plasma of a given number density and are considering propagation of electromagnetic waves, you should compare the frequency of the wave with the plasma frequency. If a magnetic field is involved, think of the cyclotron frequency. If a high density plasma is relevant, think of the Fermi energy. Stuff like that. It helps you decide quickly what regime you re in and what processes are likely to be relevant. Side note 2: since σ is purely imaginary, Ohm s law means that there is a 9 phase shift between j and E. Therefore, in a time-averaged sense, j E = and there is no net work done by the field in an isotropic plasma. That also means there is no dissipation, so below the plasma frequency you have a pure reflection. Thus, you can probe the ionosphere of the Earth by finding out when a wave of a given frequency is completely reflected. You can also communicate intercontinentally by bouncing low-frequency waves off of the ionosphere. Now back to our regular program. Electromagnetic radiation travels at a velocity different from c, due to the presence of matter. The phase velocity, v ph ω/k, is greater than the speed of light. However, the physically relevant speed is the group velocity v gr = c 1 ωp/ω 2 2, which is less than c (this is the speed at which wave energy or information travel). The frequency dependence means that there is dispersion in the propagation of light over a variety of frequencies. One especially useful application is to pulsars. Suppose a pulsar is a distance d away. Ask class: how long does it take for light of a given frequency to reach us? The time is t p (ω) = d ds/v g (ω). (11) Here s measures the line of sight distance to us. Plasma frequencies in the ISM are really low, usually 1 3 Hz or so, so we can assume ω ω p and therefore ( ) Therefore, the propagation time is v 1 g 1 c ωp 2 ω 2. (12) t p d d c + (2cω2 ) 1 ωp 2 ds. (13) This is the vacuum time (d/c) plus some extra. There is therefore a gradient in the time as a function of frequency, dt p /dω = 4πe2 cmω D, (14) 3 nds is the dispersion measure. In principle this can be used to find the distance to a pulsar, given an estimate of the average number density of plasma in the ISM. In practice, the errors are pretty large! where D d Let s complicate things a bit by assuming we have a fixed external magnetic field B in the plasma. Ask class: qualitatively, what does that do? It means that the plasma and

4 propagation in it are no longer isotropic, since the magnetic field introduces a preferred direction. It also means that not all polarizations are equal in their propagation properties. We can make some progress by thinking about the special case of propagation along the direction of the field, and by considering only cold plasma (i.e., nonrelativistic motion). Also, let s assume that the magnitude of the external field is a lot greater than the magnitude of the fields of the propagating wave, so that the equation of motion becomes mdv/dt = ee e c v B. (15) For propagation along a fixed direction we only have to consider two polarization modes. Let s choose circular modes: E(t) = Ee iωt (ǫ 1 iǫ 2 ), (16) where - gives right circular and + gives left circular polarization. If the wave propagates along B = B ǫ 3, then by substituting we find that the steady-state velocity has the form v(t) = ie E(t), (17) m(ω ± ω B ) where ω B = eb /mc is the cyclotron frequency for nonrelativistic particles. Ask class: what are the implications of this, as the wave goes through the medium? Since the speeds of the different polarizations are different, there will be a net rotation of the polarization vector as the wave propagates through the plasma. One can rewrite this as an expression for the dielectric constant: ωp 2 ǫ R,L = 1 ω(ω ± ω B ). (18) In general, an electric field vector with wavenumber k traveling a distance d will rotate in phase by k d. If the wavenumber is not constant along the path, this must be integrated. Before getting the formula for Faraday rotation (the angle of rotation after some distance), let s think about some limits. Ask class: what do they expect the rotation to be if the field strength is very small? It should also be small. Ask class: does the equation satisfy this constraint? Yes, because small B means small ω B, and hence a small difference between the polarizations. Ask class: from the formula, what happens at extremely large B? There, also, the difference is small, because the ω B in the denominator means that ǫ 1 when B is big. As can be verified easily (see book), in the common astrophysical limits that ω ω p and ω ω B we have for our angle of rotation Here B is the component of B parallel to the line of sight. θ = 2πe3 d nb m 2 c 2 ω 2 ds. (19)

5 Now let s think about ways in which this might be applied in practice. First, Ask class: if the magnetic field is uniform (no change in direction or magnitude in the region), what happens to the degree of polarization due to Faraday rotation? Nothing; the angle simply rotates. Okay, then, Ask class: what if the region has a tangled magnetic field and you observe a region that involves many tangles? Then, different parts are rotated by different amounts, so the net result is a decrease in the degree of polarization. This is sometimes used in observations of active galactic nuclei. Many times parts of the spectrum are thought to be due to synchrotron radiation, which you remember produces highly polarized light. However, observed with low angular resolution, there is little polarization. Observations of higher angular resolution do give net polarization, so by comparing the two one can estimate the line of sight integrated magnetic field. Similar methods are used to estimate the magnetic field strength in the interstellar medium or molecular clouds, when more direct spectroscopic information is unavailable. Finally, we can just mention in passing one other effect of plasmas. Since the speed of electromagnetic waves is less than the vacuum speed of light, it becomes possible for particles to travel faster than the local speed of electromagnetic waves. This produces effects similar to shocks in the atmosphere, and generates Cerenkov radiation, which is bluish (i.e., it has a spectrum tipped towards high frequency, and would actually look blue to the eye). Only particles traveling faster than c/n emit radiation, which has been used to detect neutrinos (if they scatter electrons in water, the electrons can move faster than c/n) and estimate the energies of cosmic rays by using materials with different indices of refraction.

### Set 5: Classical E&M and Plasma Processes

Set 5: Classical E&M and Plasma Processes Maxwell Equations Classical E&M defined by the Maxwell Equations (fields sourced by matter) and the Lorentz force (matter moved by fields) In cgs (gaussian) units

### Typical anisotropies introduced by geometry (not everything is spherically symmetric) temperature gradients magnetic fields electrical fields

Lecture 6: Polarimetry 1 Outline 1 Polarized Light in the Universe 2 Fundamentals of Polarized Light 3 Descriptions of Polarized Light Polarized Light in the Universe Polarization indicates anisotropy

### Particle acceleration and generation of high-energy photons

Particle acceleration and generation of high-energy photons For acceleration, see Chapter 21 of Longair Ask class: suppose we observe a photon with an energy of 1 TeV. How could it have been produced?

### Neutrinos, nonzero rest mass particles, and production of high energy photons Particle interactions

Neutrinos, nonzero rest mass particles, and production of high energy photons Particle interactions Previously we considered interactions from the standpoint of photons: a photon travels along, what happens

### A Propagating Wave Packet The Group Velocity

Lecture 7 A Propagating Wave Pacet The Group Velocity Phys 375 Overview and Motivation: Last time we looed at a solution to the Schrödinger equation (SE) with an initial condition (,) that corresponds

### PHYS4210 Electromagnetic Theory Quiz 1 Feb 2010

PHYS4210 Electromagnetic Theory Quiz 1 Feb 2010 1. An electric dipole is formed from two charges ±q separated by a distance b. For large distances r b from the dipole, the electric potential falls like

### 1. Why photons? 2. Photons in a vacuum

1 Photons 1. Why photons? Ask class: most of our information about the universe comes from photons. What are the reasons for this? Let s compare them with other possible messengers, specifically massive

### Overview in Images. S. Lin et al, Nature, vol. 394, p , (1998) T.Thio et al., Optics Letters 26, (2001).

Overview in Images 5 nm K.S. Min et al. PhD Thesis K.V. Vahala et al, Phys. Rev. Lett, 85, p.74 (000) J. D. Joannopoulos, et al, Nature, vol.386, p.143-9 (1997) T.Thio et al., Optics Letters 6, 197-1974

### Clusters: Context and Background

Clusters: Context and Background We re about to embark on a subject rather different from what we ve treated before, so it is useful to step back and think again about what we want to accomplish in this

### Macroscopic dielectric theory

Macroscopic dielectric theory Maxwellʼs equations E = 1 c E =4πρ B t B = 4π c J + 1 c B = E t In a medium it is convenient to explicitly introduce induced charges and currents E = 1 B c t D =4πρ H = 4π

### 9 The conservation theorems: Lecture 23

9 The conservation theorems: Lecture 23 9.1 Energy Conservation (a) For energy to be conserved we expect that the total energy density (energy per volume ) u tot to obey a conservation law t u tot + i

### David J. Starling Penn State Hazleton PHYS 214

All the fifty years of conscious brooding have brought me no closer to answer the question, What are light quanta? Of course today every rascal thinks he knows the answer, but he is deluding himself. -Albert

### Class XII Chapter 8 Electromagnetic Waves Physics

Question 8.1: Figure 8.6 shows a capacitor made of two circular plates each of radius 12 cm, and separated by 5.0 cm. The capacitor is being charged by an external source (not shown in the figure). The

### MCQs E M WAVES. Physics Without Fear.

MCQs E M WAVES Physics Without Fear Electromagnetic Waves At A Glance Ampere s law B. dl = μ 0 I relates magnetic fields due to current sources. Maxwell argued that this law is incomplete as it does not

### Electromagnetic fields and waves

Electromagnetic fields and waves Maxwell s rainbow Outline Maxwell s equations Plane waves Pulses and group velocity Polarization of light Transmission and reflection at an interface Macroscopic Maxwell

### Exam in TFY4240 Electromagnetic Theory Wednesday Dec 9, :00 13:00

NTNU Page 1 of 5 Institutt for fysikk Contact during the exam: Paul Anton Letnes Telephone: Office: 735 93 648, Mobile: 98 62 08 26 Exam in TFY4240 Electromagnetic Theory Wednesday Dec 9, 2009 09:00 13:00

### Chapter 6. Maxwell s Equations, Macroscopic Electromagnetism, Conservation Laws

PHYS 532 Lecture 7 Page 1 Chapter 6. Maxwell s Equations, Macroscopic Electromagnetism, Conservation Laws 6.1 Maxwell Displacement Current and Maxwell Equations Differential equations for calculating fields

### Theoretische Physik 2: Elektrodynamik (Prof. A-S. Smith) Home assignment 9

WiSe 202 20.2.202 Prof. Dr. A-S. Smith Dipl.-Phys. Ellen Fischermeier Dipl.-Phys. Matthias Saba am Lehrstuhl für Theoretische Physik I Department für Physik Friedrich-Alexander-Universität Erlangen-Nürnberg

### Chapter 33. Electromagnetic Waves

Chapter 33 Electromagnetic Waves Today s information age is based almost entirely on the physics of electromagnetic waves. The connection between electric and magnetic fields to produce light is own of

### xkcd.com It IS about physics. It ALL is.

xkcd.com It IS about physics. It ALL is. Introduction to Space Plasmas! The Plasma State What is a plasma? Basic plasma properties: Qualitative & Quantitative Examples of plasmas! Single particle motion

### Chapter 9. Reflection, Refraction and Polarization

Reflection, Refraction and Polarization Introduction When you solved Problem 5.2 using the standing-wave approach, you found a rather curious behavior as the wave propagates and meets the boundary. A new

### Maxwell s equations and EM waves. From previous Lecture Time dependent fields and Faraday s Law

Maxwell s equations and EM waves This Lecture More on Motional EMF and Faraday s law Displacement currents Maxwell s equations EM Waves From previous Lecture Time dependent fields and Faraday s Law 1 Radar

### 4. Energy, Power, and Photons

4. Energy, Power, and Photons Energy in a light wave Why we can often neglect the magnetic field Poynting vector and irradiance The quantum nature of light Photon energy and photon momentum An electromagnetic

### H ( E) E ( H) = H B t

Chapter 5 Energy and Momentum The equations established so far describe the behavior of electric and magnetic fields. They are a direct consequence of Maxwell s equations and the properties of matter.

### 1. Why photons? 2. Photons in a vacuum

Photons and Other Messengers 1. Why photons? Ask class: most of our information about the universe comes from photons. What are the reasons for this? Let s compare them with other possible messengers,

### Lecture 4. 1 de Broglie wavelength and Galilean transformations 1. 2 Phase and Group Velocities 4. 3 Choosing the wavefunction for a free particle 6

Lecture 4 B. Zwiebach February 18, 2016 Contents 1 de Broglie wavelength and Galilean transformations 1 2 Phase and Group Velocities 4 3 Choosing the wavefunction for a free particle 6 1 de Broglie wavelength

### Electromagnetic optics!

1 EM theory Electromagnetic optics! EM waves Monochromatic light 2 Electromagnetic optics! Electromagnetic theory of light Electromagnetic waves in dielectric media Monochromatic light References: Fundamentals

### PH 222-2C Fall Electromagnetic Waves Lectures Chapter 33 (Halliday/Resnick/Walker, Fundamentals of Physics 8 th edition)

PH 222-2C Fall 2012 Electromagnetic Waves Lectures 21-22 Chapter 33 (Halliday/Resnick/Walker, Fundamentals of Physics 8 th edition) 1 Chapter 33 Electromagnetic Waves Today s information age is based almost

### PHYS 110B - HW #5 Fall 2005, Solutions by David Pace Equations referenced equations are from Griffiths Problem statements are paraphrased

PHYS 0B - HW #5 Fall 005, Solutions by David Pace Equations referenced equations are from Griffiths Problem statements are paraphrased [.] Imagine a prism made of lucite (n.5) whose cross-section is a

### PHYS 532 Lecture 10 Page Derivation of the Equations of Macroscopic Electromagnetism. Parameter Microscopic Macroscopic

PHYS 532 Lecture 10 Page 1 6.6 Derivation of the Equations of Macroscopic Electromagnetism Parameter Microscopic Macroscopic Electric Field e E = Magnetic Field b B = Charge Density η ρ = Current

### EXAMINATION QUESTION PAPER

Faculty of Science and Technology EXAMINATION QUESTION PAPER Exam in: Fys-2009 Introduction to Plasma Physics Date: 20161202 Time: 09.00-13.00 Place: Åsgårdvegen 9 Approved aids: Karl Rottmann: Matematisk

### n=0 l (cos θ) (3) C l a lm 2 (4)

Cosmic Concordance What does the power spectrum of the CMB tell us about the universe? For that matter, what is a power spectrum? In this lecture we will examine the current data and show that we now have

### Polarization Mode Dispersion

Unit-7: Polarization Mode Dispersion https://sites.google.com/a/faculty.muet.edu.pk/abdullatif Department of Telecommunication, MUET UET Jamshoro 1 Goos Hänchen Shift The Goos-Hänchen effect is a phenomenon

### Report submitted to Prof. P. Shipman for Math 540, Fall 2009

Dynamics at the Horsetooth Volume 1, 009. Three-Wave Interactions of Spin Waves Aaron Hagerstrom Department of Physics Colorado State University aaronhag@rams.colostate.edu Report submitted to Prof. P.

### The Wave Function. Chapter The Harmonic Wave Function

Chapter 3 The Wave Function On the basis of the assumption that the de Broglie relations give the frequency and wavelength of some kind of wave to be associated with a particle, plus the assumption that

### Cold plasma waves. Waves in non-magnetized plasma Cold plasma dispersion equation Cold plasma wave modes

Cold plasma waves Waves in non-magnetized plasma Cold plasma dispersion equation Cold plasma wave modes EM wave propagation through and interaction with plasmas belong to central issues of plasma physics.

### POLARIZATION OF LIGHT

POLARIZATION OF LIGHT OVERALL GOALS The Polarization of Light lab strongly emphasizes connecting mathematical formalism with measurable results. It is not your job to understand every aspect of the theory,

### Physics 505 Homework No. 8 Solutions S Spinor rotations. Somewhat based on a problem in Schwabl.

Physics 505 Homework No 8 s S8- Spinor rotations Somewhat based on a problem in Schwabl a) Suppose n is a unit vector We are interested in n σ Show that n σ) = I where I is the identity matrix We will

### Theory of Electromagnetic Fields

Theory of Electromagnetic Fields Andrzej Wolski University of Liverpool, and the Cockcroft Institute, UK Abstract We discuss the theory of electromagnetic fields, with an emphasis on aspects relevant to

### The atom cont. +Investigating EM radiation

The atom cont. +Investigating EM radiation Announcements: First midterm is 7:30pm on Sept 26, 2013 Will post a past midterm exam from 2011 today. We are covering Chapter 3 today. (Started on Wednesday)

### ˆd = 1 2π. d(t)e iωt dt. (1)

Bremsstrahlung Initial questions: How does the hot gas in galaxy clusters cool? What should we see in their inner portions, where the density is high? As in the last lecture, we re going to take a more

### Explanations of quantum animations Sohrab Ismail-Beigi April 22, 2009

Explanations of quantum animations Sohrab Ismail-Beigi April 22, 2009 I ve produced a set of animations showing the time evolution of various wave functions in various potentials according to the Schrödinger

### Magnetohydrodynamic Waves

Magnetohydrodynamic Waves Nick Murphy Harvard-Smithsonian Center for Astrophysics Astronomy 253: Plasma Astrophysics February 17, 2016 These slides are largely based off of 4.5 and 4.8 of The Physics of

### Lecture 11: Polarized Light. Fundamentals of Polarized Light. Descriptions of Polarized Light. Scattering Polarization. Zeeman Effect.

Lecture 11: Polarized Light Outline 1 Fundamentals of Polarized Light 2 Descriptions of Polarized Light 3 Scattering Polarization 4 Zeeman Effect 5 Hanle Effect Fundamentals of Polarized Light Electromagnetic

### Time dependent perturbation theory 1 D. E. Soper 2 University of Oregon 11 May 2012

Time dependent perturbation theory D. E. Soper University of Oregon May 0 offer here some background for Chapter 5 of J. J. Sakurai, Modern Quantum Mechanics. The problem Let the hamiltonian for a system

### REFLECTION AND REFRACTION OF PLANE EM WAVES

REFLECTION AND REFRACTION OF PLANE EM WAVES When an electromagnetic wave hits a boundary between different materials, some of the wave s energy is reflected back while the rest continues on through the

### Appendix A2. Particle Accelerators and Detectors The Large Hadron Collider (LHC) in Geneva, Switzerland on the Border of France.

Appendix A. Particle Accelerators and Detectors The Large Hadron Collider (LHC) in Geneva, Switzerland on the Border of France. Prepared by: Arash Akbari-Sharbaf Why Build Accelerators? Probe deeper From

### Characterization of Left-Handed Materials

Characterization of Left-Handed Materials Massachusetts Institute of Technology 6.635 lecture notes 1 Introduction 1. How are they realized? 2. Why the denomination Left-Handed? 3. What are their properties?

### Principles of Mobile Communications

Communication Networks 1 Principles of Mobile Communications University Duisburg-Essen WS 2003/2004 Page 1 N e v e r s t o p t h i n k i n g. Wave Propagation Single- and Multipath Propagation Overview:

### 2/8/16 Dispersive Media, Lecture 5 - Thomas Johnson 1. Waves in plasmas. T. Johnson

2/8/16 Dispersive Media, Lecture 5 - Thomas Johnson 1 Waves in plasmas T. Johnson Introduction to plasma physics Magneto-Hydro Dynamics, MHD Plasmas without magnetic fields Cold plasmas Transverse waves

### Overview - Previous lecture 1/2

Overview - Previous lecture 1/2 Derived the wave equation with solutions of the form We found that the polarization of the material affects wave propagation, and found the dispersion relation ω(k) with

### Wave Phenomena Physics 15c. Lecture 11 Dispersion

Wave Phenomena Physics 15c Lecture 11 Dispersion What We Did Last Time Defined Fourier transform f (t) = F(ω)e iωt dω F(ω) = 1 2π f(t) and F(w) represent a function in time and frequency domains Analyzed

### Lecture notes 5: Diffraction

Lecture notes 5: Diffraction Let us now consider how light reacts to being confined to a given aperture. The resolution of an aperture is restricted due to the wave nature of light: as light passes through

### BASIC WAVE CONCEPTS. Reading: Main 9.0, 9.1, 9.3 GEM 9.1.1, Giancoli?

1 BASIC WAVE CONCEPTS Reading: Main 9.0, 9.1, 9.3 GEM 9.1.1, 9.1.2 Giancoli? REVIEW SINGLE OSCILLATOR: The oscillation functions you re used to describe how one quantity (position, charge, electric field,

### Electromagnetism Physics 15b

Electromagnetism Physics 15b Lecture #18 Maxwell s Equations Electromagnetic Waves Purcell 9.1 9.4 What We Did Last Time Impedance of R,, and L Z R = V R = R Z I = V = 1 Z R I iω L = V L = iωl I L Generally

### and the radiation from source 2 has the form. The vector r points from the origin to the point P. What will the net electric field be at point P?

Physics 3 Interference and Interferometry Page 1 of 6 Interference Imagine that we have two or more waves that interact at a single point. At that point, we are concerned with the interaction of those

### E E D E=0 2 E 2 E (3.1)

Chapter 3 Constitutive Relations Maxwell s equations define the fields that are generated by currents and charges. However, they do not describe how these currents and charges are generated. Thus, to find

### Wave Phenomena Physics 15c. Lecture 15 Reflection and Refraction

Wave Phenomena Physics 15c Lecture 15 Reflection and Refraction What We (OK, Brian) Did Last Time Discussed EM waves in vacuum and in matter Maxwell s equations Wave equation Plane waves E t = c E B t

### Complex number review

Midterm Review Problems Physics 8B Fall 009 Complex number review AC circuits are usually handled with one of two techniques: phasors and complex numbers. We ll be using the complex number approach, so

### Ref: J. D. Jackson: Classical Electrodynamics; A. Sommerfeld: Electrodynamics.

2 Fresnel Equations Contents 2. Laws of reflection and refraction 2.2 Electric field parallel to the plane of incidence 2.3 Electric field perpendicular to the plane of incidence Keywords: Snell s law,

### Solving with Absolute Value

Solving with Absolute Value Who knew two little lines could cause so much trouble? Ask someone to solve the equation 3x 2 = 7 and they ll say No problem! Add just two little lines, and ask them to solve

### Lecture 1 Notes, Electromagnetic Theory II Dr. Christopher S. Baird, faculty.uml.edu/cbaird University of Massachusetts Lowell

Lecture 1 Notes, Electromagnetic Theory II Dr. Christopher S. Baird, faculty.uml.edu/cbaird University of Massachusetts Lowell 1. Overview of the Course - Last semester we covered electrostatics, magnetostatics,

### Electromagnetic waves in free space

Waveguide notes 018 Electromagnetic waves in free space We start with Maxwell s equations for an LIH medum in the case that the source terms are both zero. = =0 =0 = = Take the curl of Faraday s law, then

### Physical Processes in Astrophysics

Physical Processes in Astrophysics Huirong Yan Uni Potsdam & Desy Email: hyan@mail.desy.de 1 Reference Books: Plasma Physics for Astrophysics, Russell M. Kulsrud (2005) The Physics of Astrophysics, Frank

### Ideal Magnetohydrodynamics (MHD)

Ideal Magnetohydrodynamics (MHD) Nick Murphy Harvard-Smithsonian Center for Astrophysics Astronomy 253: Plasma Astrophysics February 1, 2016 These lecture notes are largely based on Lectures in Magnetohydrodynamics

### Line Broadening. φ(ν) = Γ/4π 2 (ν ν 0 ) 2 + (Γ/4π) 2, (3) where now Γ = γ +2ν col includes contributions from both natural broadening and collisions.

Line Broadening Spectral lines are not arbitrarily sharp. There are a variety of mechanisms that give them finite width, and some of those mechanisms contain significant information. We ll consider a few

### Yell if you have any questions

Class 36: Outline Hour 1: Concept Review / Overview PRS Questions Possible Exam Questions Hour : Sample Exam Yell if you have any questions P36-1 Before Starting All of your grades should now be posted

### Physics 221B Spring 2018 Notes 34 The Photoelectric Effect

Copyright c 2018 by Robert G. Littlejohn Physics 221B Spring 2018 Notes 34 The Photoelectric Effect 1. Introduction In these notes we consider the ejection of an atomic electron by an incident photon,

### Keywords: Normal dispersion, Anomalous dispersion, Absorption. Ref: M. Born and E. Wolf: Principles of Optics; R.S. Longhurst: Geometrical

28 Dispersion Contents 28.1 Normal dispersion 28.2 Anomalous Dispersion 28.3 Elementary theory of dispersion Keywords: Normal dispersion, Anomalous dispersion, Absorption. Ref: M. Born and E. Wolf: Principles

Electromagnetic Waves Properties. The electric and the magnetic field, associated with an electromagnetic wave, propagating along the z=axis. Can be represented by E = E kˆ, = iˆ E = E ˆj, = ˆj b) E =

### Radiation processes and mechanisms in astrophysics I. R Subrahmanyan Notes on ATA lectures at UWA, Perth 18 May 2009

Radiation processes and mechanisms in astrophysics I R Subrahmanyan Notes on ATA lectures at UWA, Perth 18 May 009 Light of the night sky We learn of the universe around us from EM radiation, neutrinos,

### de = j ν dvdωdtdν. (1)

Transfer Equation and Blackbodies Initial questions: There are sources in the centers of some galaxies that are extraordinarily bright in microwaves. What s going on? The brightest galaxies in the universe

### Observational Parameters

Observational Parameters Classical cosmology reduces the universe to a few basic parameters. Modern cosmology adds a few more, but the fundamental idea is still the same: the fate and geometry of the universe

### EM waves: energy, resonators. Scalar wave equation Maxwell equations to the EM wave equation A simple linear resonator Energy in EM waves 3D waves

EM waves: energy, resonators Scalar wave equation Maxwell equations to the EM wave equation A simple linear resonator Energy in EM waves 3D waves Simple scalar wave equation 2 nd order PDE 2 z 2 ψ (z,t)

### Edwin Soeryadjaya Problem Theoretical 3: Mirage

The refractive index of the air varies with temperature. Cold air is denser than warm air and has therefore a greater refractive index. Thus a temperature gradient in the atmosphere is always associated

### SW103: Lecture 2. Magnetohydrodynamics and MHD models

SW103: Lecture 2 Magnetohydrodynamics and MHD models Scale sizes in the Solar Terrestrial System: or why we use MagnetoHydroDynamics Sun-Earth distance = 1 Astronomical Unit (AU) 200 R Sun 20,000 R E 1

### Electromagnetic Waves & Polarization

Course Instructor Dr. Raymond C. Rumpf Office: A 337 Phone: (915) 747 6958 E Mail: rcrumpf@utep.edu EE 4347 Applied Electromagnetics Topic 3a Electromagnetic Waves & Polarization Electromagnetic These

### Lecture 38: Equations of Rigid-Body Motion

Lecture 38: Equations of Rigid-Body Motion It s going to be easiest to find the equations of motion for the object in the body frame i.e., the frame where the axes are principal axes In general, we can

### Constructive vs. destructive interference; Coherent vs. incoherent interference

Constructive vs. destructive interference; Coherent vs. incoherent interference Waves that combine in phase add up to relatively high irradiance. = Constructive interference (coherent) Waves that combine

### Sound and Light. Light

Sound and Light Light What do you think? Read the two statements below and decide whether you agree or disagree with them. Place an A in the Before column if you agree with the statement or a D if you

### 2 u 1-D: 3-D: x + 2 u

c 2013 C.S. Casari - Politecnico di Milano - Introduction to Nanoscience 2013-14 Onde 1 1 Waves 1.1 wave propagation 1.1.1 field Field: a physical quantity (measurable, at least in principle) function

### Electricity & Magnetism Study Questions for the Spring 2018 Department Exam December 4, 2017

Electricity & Magnetism Study Questions for the Spring 2018 Department Exam December 4, 2017 1. a. Find the capacitance of a spherical capacitor with inner radius l i and outer radius l 0 filled with dielectric

16. More About Polarization Polarization control Wave plates Circular polarizers Reflection & polarization Scattering & polarization Birefringent materials have more than one refractive index A special

### The Strange Physics of Nonabelian Plasmas

The Strange Physics of Nonabelian Plasmas Guy Moore, McGill University Review: What does Nonabelian mean? Instability of a Uniform magnetic field Radiation and the LPM effect Plasma instabilities My units:

### Atomic cross sections

Chapter 12 Atomic cross sections The probability that an absorber (atom of a given species in a given excitation state and ionziation level) will interact with an incident photon of wavelength λ is quantified

### Today in Physics 218: Fresnel s equations

Today in Physics 8: Fresnel s equations Transmission and reflection with E parallel to the incidence plane The Fresnel equations Total internal reflection Polarization on reflection nterference R 08 06

### Electrodynamics HW Problems 06 EM Waves

Electrodynamics HW Problems 06 EM Waves 1. Energy in a wave on a string 2. Traveling wave on a string 3. Standing wave 4. Spherical traveling wave 5. Traveling EM wave 6. 3- D electromagnetic plane wave

### Uniform Plane Waves Page 1. Uniform Plane Waves. 1 The Helmholtz Wave Equation

Uniform Plane Waves Page 1 Uniform Plane Waves 1 The Helmholtz Wave Equation Let s rewrite Maxwell s equations in terms of E and H exclusively. Let s assume the medium is lossless (σ = 0). Let s also assume

### Electromagnetic Theory

Summary: Electromagnetic Theory Maxwell s equations EM Potentials Equations of motion of particles in electromagnetic fields Green s functions Lienard-Weichert potentials Spectral distribution of electromagnetic

### Chapter 17: Magnetism

Chapter 17: Magnetism Section 17.1: The Magnetic Interaction Things You Already Know Magnets can attract or repel Magnets stick to some things, but not all things Magnets are dipoles: north and south Labels

### PH575 Spring Lecture #26 & 27 Phonons: Kittel Ch. 4 & 5

PH575 Spring 2009 Lecture #26 & 27 Phonons: Kittel Ch. 4 & 5 PH575 Spring 2009 POP QUIZ Phonons are: A. Fermions B. Bosons C. Lattice vibrations D. Light/matter interactions PH575 Spring 2009 POP QUIZ

### Chapter 11. Special Relativity

Chapter 11 Special Relativity Note: Please also consult the fifth) problem list associated with this chapter In this chapter, Latin indices are used for space coordinates only eg, i = 1,2,3, etc), while

### Introduction to Polarization

Phone: Ext 659, E-mail: hcchui@mail.ncku.edu.tw Fall/007 Introduction to Polarization Text Book: A Yariv and P Yeh, Photonics, Oxford (007) 1.6 Polarization States and Representations (Stokes Parameters

### MITOCW 8. Electromagnetic Waves in a Vacuum

MITOCW 8. Electromagnetic Waves in a Vacuum The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high-quality educational resources

### Concepts in Theoretical Physics

Concepts in Theoretical Physics Lecture 1: The Principle of Least Action David Tong Newtonian Mechanics You've all done a course on Newtonian mechanics so you know how to calculate the way things move.

### Physics 11b Lecture #10

Physics 11b Lecture #10 Magnetic Fields S&J Chapter 29 What We Did Last Time Electromotive forces (emfs) atteries are made of an emf and an internal resistance Resistor arithmetic R = R + R + R + + R series

### PHYS 390 Lecture 23 - Photon gas 23-1

PHYS 39 Lecture 23 - Photon gas 23-1 Lecture 23 - Photon gas What's Important: radiative intensity and pressure stellar opacity Text: Carroll and Ostlie, Secs. 9.1 and 9.2 The temperature required to drive