Teaching

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Electricity and Magnetism

This topic outline is incomplete in many respects. Updates are being posted periodically.

These lecture notes are for the advanced undergraduate course PHY331 and for the graduate courses PHY530/PHY630.

In PHY331 the goal is cover parts of the first 14 modules and some of the simpler exercises.

In PHY530 parts of most earlier modules serve as review materials. The goal is to cover the first 14 modules in more depth and focus on the more advanced applications/exercises.

In PHY630 students are expected to be familiar with the materials of at least the first 11 modules. Nevertheless, some key concepts from the early modules will be briefly reviewed. The later modules are more topical in nature and need not be covered in any particular sequence.

Depending on the level of preparedness of the students in class, adjustments will be made in each course.

A different set of lecture notes and slides for a one-semester introductory course on electricity and magnetism (PHY204) is available elsewhere:
http://www.phys.uri.edu/gerhard/PHY204/lecslif14.html

1. Electrostatics I [lln5] 

• Coulomb force between point charges
• Coulomb force mediated by electric field
• Charge densities
• Electric potential
• Gauss’s law for the electric field
• Electric flux
• Electrostatic field at surfaces and interfaces
• Electrostatic field determined via Gauss’s law
• Electrostatic energy
• Differential relations versus integral relations
• Electrostatic force on extended charged object
• Interaction energy and self-energy for extended charged objects
• Multipole expansion
• Torque and force on electric dipole

Exercises:

• Electric field of a charged rod I [lex1] 
• Electric field of a charged ring I [lex2] 
• Electric field of a charged spherical shell [lex3]  
• Electric field of a charged ring II [lex4] 
• Electric field of a charged rod II [lex5]  
• Electric field of a charged rod III [lex6]  
• Electric field of V-shaped line charge [lex7]  
• Electric field of parabolic line charge [lex8] 
• Electric dipole field [lex9] 
• Electrostatic field of two point charges [lex39]
• From electric potential to electric field and back [lex40] 
• Electric flux through cube and net charge inside [lex41] 
• Electric potential of charged rod [lex42] 
• Electric field of uniformly charged concentric spheres [lex46] 
• Electric field of uniformly charged coaxial cylinders [lex47] 
• Electrostatic interaction energy of point charges [lex120]
• Electric field generated by linear arrays of point charges [lex128]
• Electrostatic force between two point charges [lex187]
• Oppositely charged semicircles [lex188] 
• Electric field of planar charge distributions [lex189]
• Electric potential of charged ring I [lex192] 

Additional Materials:

• Self-energy and interaction energy of point charges [lam8]
• Surface electric dipole layers [lam9]
• Green’s theorem [lam11] 
• Mathematica syntax items for E&M [lam21.nb]
• Vector analysis [gmd1] 
• Generalized functions [gmd3] 

2. Electrostatics II [lln6] 

• Electrical insulators and conductors
• Electrostatic properties of conductors
• Specifications of conductors at equilibrium
• Electric field at the surface of a conductor
• Electrostatic pressure
• Existence and uniqueness of electrostatic equilibrium
• Boundary value problem
• Electric field in empty cavity
• Capacitor
• Method of images for induced charges
• Solutions in search of a problem
• Applications with planar, spherical, or cylindrical symmetry

Exercises:

• Parallel-plate capacitor [lex10]
• Cylindrical capacitor [lex11]
• Spherical capacitor [lex12]
• Point charge near plane conducting surface [lex13]
• Electric dipole near plane conducting surface I [lex14]
• Electric dipole near plane conducting surface II [lex15]
• Point charge near perpendicular plane conducting surfaces [lex16] 
• Conducting sphere in uniform electric field [lex17]
• Line charge near plane conducting surface [lex18]
• Conducting cylinder in uniform electric field [lex19] 
• Conducting hyperbolic trough [lex43]
• Conducting sphere between point charges [lex124]

Additional Materials:

• Electric potential of point charge near conducting sphere [lam10]
• Coordinate systems [gmd2]

3. Electrostatics III [lln7]

• Uniqueness theorem for Laplace equation
• Separable solutions in Cartesian coordinates
• Separable solutions in spherical coordinates
• Separable solutions in cylindrical coordinates
• Solutions from conjugate functions
• Application to charged plane surface
• Application to charged cylindrical surface

Exercises:

• Square pipe made of conducting walls at different potential [lex20]
• Conducting rectangular pipe with end-plate potential [lex21]
• Conducting hemispherical shells at different electric potential [lex22]
• Oppositely charged hemispherical shells [lex23]
• Series expansion of off-center Coulomb potential [lex24]
• Conducting half-cylindrical shells at different electric potential [lex25]
• Conducting plates intersecting at right angle [lex26]
• Electric potential and field at edge of large conducting plate [lex27]
• Fringe electric potential and field of parallel plates [lex28]
• Conducting hemispheres at opposite electric potential [lex125]
• Electrostatic potential inside rectangular box I [lex126]
• Electrostatic potential inside rectangular box II [lex127] 

Additional Materials:

• Meaning of Orthogonality [lam15] 
• Expansions in orthogonal functions [lam12]
• Laplace equation in spherical coordinates [lam13] 
• Laplace equation in cylindrical coordinates [lam14] 
• Legendre polynomials and functions [lam16]
• Spherical Harmonics [lam17]
• Bessel functions [lam18]
• Multipole expansion generalized [lam19]

4. Dielectrics I [lln9] 

• Induced electric dipole moments
• Polar molecules
• Polarization and bound charge
• Displacement field
• Boundary conditions involving dielectrics
• Linear dielectrics
• Poisson equation for uniform linear dielectric
• Clausius-Mossotti model
• Energy density in dielectric
• Capacitance with dielectric
• Impact of dielectric added to capacitor

Exercises:

• Uniformly polarized dielectric sphere [lex29] 
• Dielectric sphere polarized by uniform electric field [lex30]
• Lateral force on dielectric slab between parallel plates [lex31] 
• Point charge near plane surface of dielectric I [lex32] 
• Capacitor with stacked dielectrics I [lex44] 
• Capacitor with stacked dielectrics II [lex45]
• Point charge near plane surface of dielectric II [lex48] 
• Insertion of conducting slab between capacitor plates I [lex53]
• Insertion of conducting slab between capacitor plates II [lex54]

Additional Materials:

• Mesoscopic fields and sources [lam20]

5. Electric Currents I [lln11]

• Electric current density
• Electric current
• Charge conservation
• Ohm’s law
• Device with capacitance and resistance
• Electric field driving steady current
• Classical model of conductivity
• Power dissipation in resistive materials
• Electric current in vacuum tube
• Resistor circuits in steady state
• Capacitor circuits at equilibrium
• RC circuits

Exercises:

• Charge accumulated in conductor with steady current [lex33]
• Capacitor circuit I [lex34] 
• Resistor circuit I [lex35]
• Resistor circuit II [lex49] 
• Capacitor circuit II [lex55] 
• Capacitor circuit III [lex56]
• RC circuit I at equilibrium [lex57]
• RC circuit I with currents flowing [lex58] 
• RC circuit I: transfer of energy between devices [lex61]

6. Magnetostatics I [lln12]

• Lorentz force
• Magnetic force on a charged particle
• Magnetic force on current-carrying conductor
• Biot-Savart law
• Magnetic flux
• Gauss’s law for the magnetic field
• Ampère’s law restricted to steady states
• Steady-state magnetism and electrostatics
• Consistency of the Biot-Savart law with Ampère’s law
• Vector potential
• Magnetic dipole moment
• Torque and force on magnetic dipole

Exercises:

• Magnetic dipole field [lex36]
• Motion in crossed electric and magnetic fields [lex50]
• Magnetic field of straight current segment I [lex51]
• Magnetic field of straight current segment II [lex52]
• Magnetic field of a circular current I [lex59]
• Magnetic field of a square-shaped wire [lex60]
• Velocity selector [lex62]
• Magnetic moment of rotating charged disk [lex63]
• Magnetic moment of rotating charged spherical shell [lex64]
• Magnetic field along the axis of a solenoid [lex65] 
• Mass spectrometer [lex71]
• Magnetic moment of rotating charged solid sphere [lex78] 
• Helmholtz coil and beyond [lex129] 
• Magnetic dipole interaction I [lex130]
• Magnetic dipole interaction II [lex131]
• Magnetic dipole interaction III [lex133]
• Magnetic dipole interaction IV [lex148]
• Current density and magnetic dipole moment [lex190] 

Additional Materials:

• Principle of relativity [lam27]

7. Magnetostatics II [lln13]

• Magnetic dipole moments of elementary particles
• Electric and magnetic dipoles — commonalites and differences
• Magnetization and bound currents
• Magnetic field H and magnetic induction B
• Scalar magnetic potential
• Boundary conditions
• Diamagnetism
• Paramagnetism
• Ferromagnetism

Exercises:

• Current-carrying magnetic slab [lex66]
• Vector potential of uniformly magnetized sphere [lex67]
• Magnetic field of uniformly magnetized sphere [lex68]
• Magnetic material between coaxial cylinders with current [lex69]
• Magnetized material with cavities of different shapes [lex70]
• Solid sphere placed in a uniform magnetic field [lex72] 
• Magnetic shielding inside a magnetizable spherical shell [lex73]
• Dynamic response of electric and magnetic moment to torque [lex122]
• Vector potential generated by magnetic material [lex191]
• Electric dipole near long electrically charged rod [lex195]
• Magnetic dipole near long straight current-carrying wire [lex196]

8. Electrodynamics I [lln14]

• Faraday’s law of electromagnetic induction
• Motional EMF
• Consistency of motional EMF with Faraday’s law
• Faraday disk generator
• Eddy currents and magnetic attenuation
• Alternating current generator
• Cyclotron
• Betatron
• Inductance
• Self-induction
• Energy stored in inductor
• Mutual induction
• Magnetic field energy density
• RL circuits
• Electromagnetic oscillator (LC circuit)
• LC circuit with two modes
• RLC circuit

Exercises:

• Energy dissipation in Faraday disk generator [lex74]
• Faraday wheel I [lex75]
• Free fall attenuated by eddy current [lex76]
• Inductance of a toroid [lex77]
• RL circuit I [lex79]
• RL circuit turning into LC circuit [lex80]
• Motional EMF I [lex81] 
• Mutual inductance of solenoid and ring [lex86]
• RC circuit turning into LC circuit [lex87]
• Circuit breaker contest I [lex88]
• Conducting ring moving into region of magnetic field I [lex137]
• Conducting ring moving into region of magnetic field II [lex138] 
• Circuit breaker contest II [lex139]
• AC circuit with RLC devices in series [lex193] 
• AC circuits I [lex203]
• AC circuits II [lex204]
• AC circuits III [lex205]

Additional Materials:

• Alternating current circuits [lam28]

9. Electrodynamics II [lln15]

• Maxwell’s equations in vacuum
• Scalar potential and vector potential
• Gauge invariance
• Maxwell’s equations in matter
• Path from restricted scenarios toward generality
• Poynting theorem
• Electromagnetic waves in vacuum
• General plane-wave solution
• Energy and momentum densities
• Spherical wave solution

Exercises:

• Conduction current versus displacement current I [lex82]
• Maxwell’s equations for scalar and vector potentials [lex83]
• Vector and scalar potentials of spherical wave [lex84] 
• Electric and magnetic fields of spherical wave [lex85]
• Energy delivery into current-carrying wire [lex89] 
• Longitudinal and transverse current densities [lex134]
• Scalar and vector potentials in the Coulomb gauge [lex135]
• Conduction current versus displacement current II [lex140] 
• Circularly polarized plane electromagnetic wave [lex141] 
• Field energy between capacitor plates I [lex142]
• Field energy inside solenoid I [lex143]
• Maxwell stress tensor for linearly polarized plane wave [lex206]

10. Relativity I [lln16]

• Postulates of special relativity
• Spacetime coordinate transformations
• Flash of light expanding
• Paradigm shift
• Einstein’s signature clock
• Time dilation
• Length contraction
• Relativity of simultaneity
• Time dilation paradox resolved
• Length contraction paradox resolved
• Addition of velocities
• Events and worldlines
• Mass and energy
• Relativistic momentum
• Relativistic energy

Exercises:

• Exchange of light signals [lex90]
• Internal clock of pions [lex92] 
• Jet propulsion [lex93]
• Photon absorption and photon emission [lex94]
• Moving calendars [lex144] 
• Who passes more quickly? [lex145]  
• Interstellar travel [lex160]
• Time on the fly [lex161]
• Interstellar speed control [lex162]
• Exchange of light signals II [lex163]
• Two events, two views, one common ground [lex164]
• K meson decay [lex165]
• Lorentz transformation I [lex166]
• Lorentz transformation II [lex167]
• Observing a rod in transverse motion [lex168]
• Lorentz invariance of momentum conservation [lex169]
• Relativistic mass from momentum conservation [lex170]

Additional Materials:

• Doppler effect for sound and light [lam22]
• Skater paradox [lam23]
• Minkowski diagram [lam24]
• Twin paradox [lam25]
• Rapidities [lam26]

11. Relativity II [lln25]

• Spacetime
• Lorentz transformation matrix
• Lorentz invariance of 4-vectors
• Kinematics
• Energy and momentum
• Dynamics
• Lorentz force and electromagnetic field tensor
• Lorentz transformation of electromagnetic field tensor
• Partial derivatives Lorentz transformed
• Current 4-vector and continuity equation
• Maxwell’s equations in covariant form
• Lorentz invariance of Maxwell equations
• 4-vector potential
• Summary list of Lorentz transformations
• Electromagnetic wave observed in moving frames
• Energy-momentum flux tensor

Exercises:

• Velocity addition rule from principle of relativity [lex91]
• Fields between moving capacitor plates I [lex95] 
• Point charge in uniform motion I [lex96]
• Fields between moving capacitor plates II [lex146]
• Lorentz invariance of Maxwell’s equations [lex194] 

12. Optics I [lln17]

• Electromagnetic wave in dielectric
• Reflection and refraction at plane dielectric interface
• Total internal reflection
• Normal-incidence reflectivity
• Nonreflecting surface via dielectric coating
• Plane-wave incidence at an angle
• Reflection and refraction of TE plane wave
• Reflection and refraction of TM plane wave
• Brewster angle
• Energy conservation
• Electromagnetic waves in a conductor
• Reflection from conductor
• Dispersion
• Dispersion in a dielectric
• Dispersion in a plasma

Exercises:

• Anti-reflection coating [lex97]
• Fresnel equation for TE wave [lex98]
• Fresnel equation for TM wave [lex99]
• Electromagnetic wave in a conductor [lex100]
• Reflection of electromagnetic wave from a conductor [lex101]
• Driven harmonic oscillator: steady-state solution [lex102]
• Dispersion and absorption in a dielectric [lex103]
• …

13. Optics II [lln18]

• Electromagnetic wave between parallel conducting plates
• Transverse electric and magnetic (TEM) wave
• Transverse electric (TE) wave
• Energy transport in TE wave
• Geometrical interpretation of phase/group velocities
• Transverse magnetic (TM) wave
• Energy transport in TM wave
• Rectangular wave guide
• TE modes in rectangular wave guide
• TM modes in rectangular wave guide
• Wave guide with cross section of arbitrary shape
• TE modes in wave guides of arbitrary cross section
• TM modes in wave guides of arbitrary cross section
• Conditions for TEM modes in wave guides of arbitrary cross section
• TEM mode in coaxial cable

Exercises:

• TE mode in rectangular wave guide [lex104]
• Surface charge and current in rectangular wave guide I [lex105]
• Surface charge and current in rectangular wave guide II [lex106]
• Helmholtz equation for wave guide I: TE modes [lex107]
• Helmholtz potential for wave guide II: TM modes [lex108]
• TEM mode in coaxial cable I: electric and magnetic fields [lex109]
• TEM mode in coaxial cable II: impedance [lex110]
• Phase velocity and group velocity [lex172] 
• …

14. Radiation I [lln19]

• Retarded potentials
• Radiation from electric dipole
• Hertzian dipole (special case)
• Radiation from magnetic dipole
• Electric dipole radiation at arbitrary distance
• Half-wave linear antenna
• Radiation from an accelerated charged particle
• Light scattering from bound charged particle

Exercises:

• Green’s function of ∇2+k2  [lex111] 
• Planar surface current abruptly established [lex112]
• Electric dipole moment from charge density or current density [lex113]
• Magnetic radiation field of electric dipole [lex114]
• Electric radiation field of electric dipole [lex115]
• Poynting vector and radiation power of electric dipole [lex116]
• Antenna resistance of Hertzian dipole [lex117]
• Radiation fields an Poynting vector for half-wave linear antenna [lex118]
• Larmor formula for radiation of accelerated charged particle [lex119]
• …

15. Magnetism I [lln22]

• Angular momentum of electrons
• Two-electron wave function
• Atomic electrons in a magnetic field
• Diamagnetism
• Langevin and Brillouin paramagnetism
• Van Vleck paramagnetism
• Fine structure
• Hund’s rule
• Russell-Saunders coupling
• Landé g-factor
• Nuclear spins
• Hyperfine structure

Exercises:

• Diamagnetic response of electron in circular orbit [lex121]
• Dynamic response of electric and magnetic moment to torque [lex122]
• Hyperfine-coupling Hamiltonian [lex123] 
• Magnetic ground state of some Lanthanide ions [lex173]
• …

16. Magnetism II [lln23]

• Crystal field
• Low-field and high-field regimes
• Orbital quenching
• Jahn-Teller effect
• Nuclear magnetic resonance
• Bloch equations
• Electron spin resonance
• Mössbauer spectroscopy
• Muon-spin rotation
• Exchange interaction
• Mediated exchange interaction

Exercises:

• Level populations and rate of energy absorption in NMR [lex171]
• Bloch equations I [lex174]
• Bloch equations II [lex175]
• Exchange integral [lex176]
• …

17. Quantum Optics I [lln24]

• Levels of quantization
• Blackbody radiation
• Einstein coefficients
• Radiative transition rates
• Linewidth and lineshape
• Photon statistics
• Light with Poisson statistics
• Light with Pascal statistics
• Light with super-Poisson statistics
• Light with sub-Poisson statistics
• Statistics of photon number states
• …

Exercises:

• Mean and variance of Poisson distribution [lex149]
• Mean and variance of Pascal distribution [lex150]
• Erlang distributions I [lex151]
• Erlang distributions II [lex152]
• Photon statistics from Planck’s radiation law [lex153]
• Super-Poisson interpolation statistics I [lex177]
• SuperPoisson intepolation statistics II [lex178]
• Differential recursion relation or Poisson and Pascal statistics [lex179]
• Photon detection events from thermal radiation [lex180] 
• Poisson distribution from maximizing uncertainty [lex181]
• …

18. Quantum Optics II [lln26]

• Photon bunching and antibunching
• Light modeled as harmonic oscillations
• Phasor diagram and field quadratures
• Coherent states of light
• Squeezed states of light
• Criteria for the classification of light
• Photonic number states
• Photonic coherent states
• Photonic thermal states
• Phase-space representations of photonic states
• Intensity correlations of photonic number states
• …

Exercises:

• Coherent state of quantum harmonic oscillator I [lex154]
• Coherent state of quantum harmonic oscillator II [lex155]
• Position distribution in harmonic oscillator [lex182]
• Quantum uncertainty in ground state of harmonic oscillator [lex183] 
• Phase-space distribution in harmonic oscillator [lex184]
• Coherent state of quantum harmonic oscillator III [lex185]
• Mixed versus pure photonic states [lex186]
• …

Additional Materials:

• Density operator [lam1]
• First- and second-order correlation functions [lam2]
• Quantization of the electromagnetic field [lam3]
• Uncertainty [lam4] 
• Quantum harmonic oscillator [lam5] 
• …

19. Quantum Optics III [lln27]

• Resonant light-atom interactions
• Pure versus mixed quantum states
• Bloch sphere
• Schrödinger equation of interacting system
• Einstein coefficients in the weak-field limit
• Rabi oscillations in the strong-field limit
• Optical cavities
• Two-level atom in optical cavity
• Weak-coupling regime
• Purcell effect
• Strong-coupling regime
• Cold atoms from laser cooling
• Optical molasses
• Doppler limit for laser cooling
• …

Exercises:

• Rabi oscillations off resonance [lex156]
• Einstein coefficients for hydrogen 1s →2p transition [lex157]
• Einstein coefficients for hydrogen 2s →3p transition [lex158]
• Jaynes Cummings model I [lex159]
• …

Additional Materials:

• Quantum time evolution and measurement [lam6]
• …

Some Relevant Textbooks and Monographs:

• R. H. Good: Classical Electromagnetism. Saunders, 1999.
• D. J. Griffiths: Introduction to Electrodynamics. Prentice Hall,  1999.
• J. D. Jackson: Classical Electrodynamics (3rd Ed.) Wiley, New York 1999.
• G. L. Pollack and D. R. Stump: Electromagnetism. Addison Wesley, San Francisco 2002.
• J. R. Reitz, F. J. Milford, and R. W. Christy: Foundations of Electromagnetic Theory. Addison Wesley, 1993
• S. Blundell: Magnetism in Condensed Matter. Oxford University Press 2011.
• M. Fox: Quantum Optics. Oxford University Press 2014.
• P. Lambropoulos and D. Petrosyan: Fundamentals of Quantum Optics and Quantum Information. Springer, 2007
• R. Wald: Advanced Classical Electromagnetism. Princeton University Press 2022.
• …

Last updated 08/31/23