PHY184B CORE LESSONS & OUTPUT SKILLS, Summer 2002

CORE LESSONS & OUTPUT SKILLS

PHY184B, Summer 2002

A. CONTENTS

Evaluating While Learning: a Project
Inductive Reasoning and the Game of Patterns: a Project for Three Participants
Electric Dipoles
Trajectory of a Charged Particle in a Magnetic Field, Cyclotron Orbits: a Computer Project
Magnetic Dipoles
Electrostatic Field Energy
Sourcelessness of the Magnetic Field
A Magnetic Monopole?
Magnetic Fields in Bulk Matter: Magnets
Magnetic Induction
Storage of Energy in Magnetic Fields
Self-Inductance and Inductors
The Ampere-Maxwell Equation; Displacement Current
Electric Field and Potential of Continuous Charge Distributions
Signal Velocity in a Conductor
Two-Element D.C.-Driven Series LRC Circuits
Three-Element D.C.-Driven Series LRC Circuit
Circuit Resonances
DC Conduction Along a Nerve
Electromagnetic Waves From Maxwell's Equations
Energy and Momentum in Electromagnetic Waves
Discrete Spectra
Transitions and Spectral Analysis
Colors From Spectral Distributions
Continuous Spectra: Planck's Law
Brewster's Law and Polarization
Land's Observations On Color Perception
Fraunh\"{o
Fresnel Diffraction
Diffraction Grating and X-Ray Scattering From Crystals
Optical Pumping
Laser Devices
Optical Circuits
Predicting and Specifying the Perceived Colors of Reflective Objects

B. THE SKILLS

Evaluating While Learning: a Project

Knowledge

Vocabulary: topic sentence, closure.

Inductive Reasoning and the Game of Patterns: a Project for Three Participants

Knowledge

Describe the difference between inductive and deductive reasoning.
State why induction plays an important role in the formulation of scientific theories.

Problem Solving

Draw and recognize patterns which exhibit some type of orderliness such as symmetry.

Electric Dipoles

Knowledge

Define the dipole moment of a discrete system of charges.
Define a "point dipole."

Problem Solving

Given two point charges and a distance between them, calculate the dipole moment, electric field, and electric potential they produce.
Calculate the electric field and electric potential due to a given point dipole.
Calculate the potential energy of a given dipole at a given orientation in an external electric field.
Calculate the torque on a given dipole in a given external electric field and the work done when the dipole is in the external electric field.

Trajectory of a Charged Particle in a Magnetic Field, Cyclotron Orbits: a Computer Project

Magnetic Dipoles

Knowledge

Define the magnetic dipole moment (vector) for a system of N (fictitious) magnetic monopoles.
Define the magnetic dipole moment for a magnetic dipole.
Explain the existence of magnetic dipoles in spite of the apparent non-existence of magnetic monopoles.
Write expressions for the torque, work, potential energy, and field of magnetic dipoles.

Problem Solving

Determine the torque on a magnetic dipole in an external magnetic field and the work done on it in changing its orientation.
Determine the potential energy of a given magnetic dipole or system of dipoles in an external field.
Determine the magnetic field at some given point in space due to a given magnetic dipole.

Electrostatic Field Energy

Knowledge

Derive the expression for the energy stored in the electric field of a capacitor.
State the general expression for the stored-energy-per-unit-volume associated with an electric field; derive that expression for the case of a parallel-plate capacitor.

Problem Solving

Calculate the potential energy of capacitor-type charge configurations given any two of the three quantities: capacitance, charge, and potential difference.
Calculate the forces on the plates of a capacitor or on a slab of dielectric between the plates of a capacitor, given the electrical charge state of the capacitor.

Sourcelessness of the Magnetic Field

Knowledge

Explain how this Gauss's Law for magnetic fields is a result of the physical property that magnetic "lines of force" are continuous and closed---that there are no isolated magnetic poles (or magnetic charges.)

Problem Solving

Evaluate the flux of B through surfaces (rectangular, cylindrical, spherical) directly and through the use of Gauss's Law for magnetic fields.

A Magnetic Monopole?

Knowledge

State the motivation for speculating on the existence of magnetic monopoles.
Derive Dirac's reciprocal quantization rule.
Calculate the numerical value of the fine structure constant and the Dirac magnetic charge unit.
Give an argument why, if magnetic monopoles do exist, they have so far escaped detection.

Magnetic Fields in Bulk Matter: Magnets

Knowledge

Vocabulary: ferromagnetism, magnetic domain, magnetic permeability, magnetic susceptibility, magnetization vector, magnetizing field vector.
Explain the basic mechanisms that produce ferromagnetism, including how external magnetic fields affect domains.
Starting from the definition of the magnetization vector, derive the expression that defines the magnetizing field vector.
Show that Ampere's law applied to the magnetizing field vector results in the circulation of the magnetizing field vector being the free current, whereas the circulation of the magnetic field is both the free current and the effective magnetization current.
Explain the physical effects that are responsible for the magnetization curve of ferromagnetic substances.

Magnetic Induction

Knowledge

Vocabulary: Faraday-Henry law, flux, induced current, induced potential difference, induced voltage, Lenz's law.
Write down the Faraday-Henry Law, relating the line-integral of the induced electric field to the time rate of change of the surface integral of the magnetic field across a surface bounded by the path of the line-integral, taking care to explain the sign convention. Explain in words what the physical phenomenon is that is described by this equation.

Problem Solving

Determine the voltage induced around given closed paths when a changing magnetic field in the region is specified.
Determine the voltage induced around given closed paths when the velocity of the path through a uniform static magnetic field is specified.
Use Lenz's law to determine the directions of induced voltages, currents, and magnetic fields.

Storage of Energy in Magnetic Fields

Knowledge

Derive the expression for the energy stored in the magnetic field associated with a current-carrying circuit, starting from the expression for the self-induced EMF. Explain where this energy comes from, and where it goes when the current is turned off.
Verify for some special cases where the magnetic field is confined to a limited region in space (such as within a toroid or a circuit consisting of coaxial cylindrical sheets of current), that the above expression is equivalent to B²/(2 μ₀) integrated over the volume where B exists.

Problem Solving

Given a current configuration, determine the magnetic energy density.

Self-Inductance and Inductors

Knowledge

Vocabulary: coaxial cable, henry, inductor, self-inductance.
Write down Ampere's Law and from it derive the self-inductance of a toroidal solenoid, explicitly justifying each step.
Write down Ampere's Law and from it derive the self-inductance per unit length of a coaxial cable, explicitly justifying each step.
Starting from the relation between power, voltage and current in a steady state circuit, derive the energy stored in the electric field of an inductor.
Describe the flow of energy: (a) when the current through an inductor is increased; (b) when the current through an inductor is very gradually decreased; and (c) when the current through an inductor is quickly decreased.

The Ampere-Maxwell Equation; Displacement Current

Knowledge

Demonstrate how Ampere's Law is modified when the electric field vector varies with time.

Problem Solving

Apply the Ampere-Maxwell equation to situations where even if there is no actual current in a region of space, the changing electric field induces a magnetic field whose circulation is given by the Ampere-Maxwell equation.

Electric Field and Potential of Continuous Charge Distributions

Knowledge

Vocabulary: directional derivative, equipotential surface, gradient.

Problem Solving

Given the distribution of charge in a certain region of space, calculate directly the function that gives the electric field at all points in space due to this charge distribution.
Given the function that specifies the electric potential at points in space, determine the function which describes the electric field vector at those same space points.

Signal Velocity in a Conductor

Knowledge

When a battery becomes connected to one end of a coaxial cable, describe the subsequent traveling current and voltage waves for the cases: (a) proper termination at the far "load" end; (b) proper termination only at the "input" end; and (c) proper termination at neither end. Sketch figures that illustrate the waves. %
Outline the derivation of expressions for the velocity of transmission of a switch-closing signal along a transmission line and for the line's effective initial resistance, including the case of a coaxial cable (equations need only be named and/or described). State realistic numerical values for the two output quantities.

Rule Application

Given the characteristics of a battery, a coaxial cable, and a proper-termination load resistor, plus equations for the general case and for the capacitance and inductance per unit length along a coaxial cable, determine the values of the voltage and current waves that travel down the cable after the battery is connected to it and determine the speed of the waves.

Two-Element D.C.-Driven Series LRC Circuits

Knowledge

Start with the relations for the potential drops across each of the three types of passive circuit elements and derive the relation between the current and the important circuit parameters for any two-element series DC-driven LRC circuit.
List the mechanical analogs of the circuit components and important circuit parameters for two-element DC-driven LRC circuits.
Given any two-element DC-driven LRC circuit, use analogies with the damped harmonic oscillator to sketch a graph of the time dependence of the charge on the capacitor or the current in the circuit.

Three-Element D.C.-Driven Series LRC Circuit

Knowledge

Starting from the charge-current and voltage-current relations for the three types of passive circuit element: \begin{id-subsubitems}
(a) Derive the relation between the time rate of change of charge and the circuit parameters.
(b) Given a solution for the relation, evaluate as many constants as possible without using any information about the circuit's initial state.
(c) Explain why two solution forms are necessary for the relation. \end{id-subsubitems}

Circuit Resonances

Knowledge

Show that q(t) = q_t(t) + q_s(t) is a solution for the sinusoidally-driven series LRC circuit, where q_t(t) = A e^{-γ t} sin(ω₁ t + α) and q_s(t) = B(ω) sin\left[ω t + β(ω)\right] and the driving voltage is: V₀ cos(ω t). Show that q_t(t) is a transient solution and q_s is a steady-state solution.
Given a series LRC circuit driven by a sinusoidally-varying potential, V₀ cos(ω t), and given the steady-state solution, q_s(t) = B(ω) sin[ω t + β(ω)], show or describe how one shows that: β(ω) = tan^-1[(ω₀² - ω²)L/(ω R)] and B(ω) = (V₀ cosβ)/(R ω) or equivalent.
Sketch phasor diagrams, and interpret them, to illustrate the phase relationships between voltages in the sinusoidally-driven series LRC circuit.
Show or describe how one shows that the time-average steady-state power transferred into a circuit by a sinusoidally-varying potential is: P_ave(ω) = (V²₀ R ω²/2) [L² (ω²₀ - ω²)² + R² ω²]^-1.
Sketch P_ave vs. ω in the vicinity of the resonant frequency of a sinusoidally-driven series LRC circuit both for a broad resonance and for a narrow one. Label each curve as to relative size of the resistance.

DC Conduction Along a Nerve

Knowledge

Describe the differences between the responses of an axon to stimuli that are below and above the threshold for action potential response.
Derive the voltage, current, and resistance distributions along a nerve axon due to a voltage applied at one point, where the applied potential is below the threshold for action potential response.

Electromagnetic Waves From Maxwell's Equations

Knowledge

Vocabulary: propagation (of a wave), polarization (direction of), plane-polarized (wave), monochromatic (wave).
Given Maxwell's Equations, the "curl-curl" vector identity, and the definitions of the gradient, divergence, and curl operators, derive the wave equations for electric and magnetic field vectors at chargeless currentless space-points.

Rule Application

Given the definitions of the gradient, divergence, and curl operators, verify that a given electromagnetic wave, consisting of coupled electric and magnetic waves, satisfies Maxwell's Equations.
Given the direction of polarization, direction of propagation, frequency and amplitude of a monochromatic plane-polarized electromagnetic wave, write down the electric and magnetic fields in vector form. Sketch the situation.

Energy and Momentum in Electromagnetic Waves

Knowledge

Vocabulary: electric energy current density, magnetic energy current density, energy current density, momentum current density.
Given the energy density in an electric fields, and the knowledge that the electric field energy is given by E_e = (1/8 π k_e) E² where E is the electric field, derive the expression for the time-average energy current density in the wave.

Rule Application

Relate the electric and magnetic field amplitudes to the intensity, energy density, and momentum density associated with an electromagnetic wave of a given frequency.

Discrete Spectra

Knowledge

Vocabulary: quantization, energy level, energy level diagram, ground state, excited state, ionization, hydrogen-like ion, quantum jump, principle quantum number, atomic one-electron system, atomic number, spectrometer.
State the energy level formula for atomic one-electron systems.

Rule Application

Given the atomic number Z for an atom, state the symbol for its corresponding hydrogen-like ion.

Problem Solving

Construct an energy-level diagram for any atomic one-electron system, showing energy and quantum number values for each level.
Calculate the wavelength and frequency of the photon emitted when any given atomic one-electron system makes a transition from one specified energy level to another. State the perceived color of light made up of those photons.
Given a distribution of emitted energy over a discrete set of wavelengths, use linear interpolation on a table of tri-stimulus values, plus a Chromaticity Diagram, to determine the light's chromaticity coordinates and perceived color.

Transitions and Spectral Analysis

Knowledge

Vocabulary: band spectra, continuous and line spectra, fluorescence, phosphorescence, resolving power, dispersion, spectroscope.
State the three general types of information spectroscopy yields and cite an area of application for each type.
Compare and contrast emission spectroscopy with absorption spectroscopy, including the types of light sources used.
Briefly outline a procedure for spectral analysis, including: (a) the mechanism for exciting the sample atoms or molecules; (b) how the various wavelengths are separated; (c) why the spectra usually appear as lines; (d) how the wavelengths of the spectral lines are measured; and (e) how the sample atoms or molecules are then identified.
Describe the three mechanisms by which the atoms and molecules make transitions between energy levels.

Colors From Spectral Distributions

Knowledge

Outline the procedure for combining the wavelength distribution of light energy coming from an object and the tristimulus values to produce the light's chromaticity coordinates. Justify the procedure.

Rule Application

Given the wavelength distribution of light energy coming from an object, and a table of tristimulus values, compute the light's chromaticity coordinates.

Continuous Spectra: Planck's Law

Knowledge

Vocabulary: black body or ideal absorber or emitter; black body spectrum; spectral and total emittance; Planck's Law, optical pyrometer.
Sketch the black body spectrum for a progression of temperatures, paying particular attention to the peak shift and to the general height shift.
Describe in detail how one computes the locus of black body points on the Chromaticity Diagram.
Describe how an optical pyrometer works.

Rule Application

Given the integrated form of Planck's law, determine temperature from total emittance and vice versa.

Brewster's Law and Polarization

Knowledge

Use the results obtained from Maxwell's Equations, tabulated in Eq. 26.13 of AF (see External Resouces, Item 1, below) to: \begin{id-subsubitems}
(a) Explain the phenomenon of polarization by reflection.
(b) Show how the conditions for complete polarization arise (explain how Brewster's Law comes about). \end{id-subsubitems}

Rule Application

Use Eqs. 26.13 of AF (see Item 1 of Required External Resouces, below) to calculate the intensity of reflected and transmitted radiation for some simple geometrical situations (such as normal incidence and at the polarizing angle).

Land's Observations On Color Perception

Knowledge

Describe the main points of Land's initial experiments in color perception, including: the nature of the light sources and the slides used, the results predicted by classical color mixing theory, and the results actually observed.
Given an unlabeled version of Land's color range map, state the meaning and units of both axes, and describe the colored scenes produced by combinations of stimuli from various regions demarcated on the map.
Given an unlabeled version of the coordinate graph for predicting the perceived color in an image produced by Land's experimental apparatus, state: the meaning and units of both axes, the meaning of the positively-sloped diagonal, and what colors are found on the three main regions on the map.

Fraunh\"{o

Knowledge

Derive the expression for the width of the central diffraction maximum for light of wavelength λ falling on a screen at a distance L away from a slit of width D.
State Rayleigh's definition of resolving power for a slit and for a circular aperture.
Describe the Fraunhöfer diffraction pattern due to a circular aperture.

Problem Solving

Analyze the Fraunhöfer diffraction pattern due to a single rectangular slit, finding the regions of zero intensity on the detection screen.
Use the expression for the Fraunhöfer diffraction by a circular lens aperture to determine the resolution of two distant point sources of light.

Fresnel Diffraction

Knowledge

Vocabulary: Fresnel diffraction, Fresnel zones.
Explain how Fresnel diffraction is different from Fraunhofer diffraction.
Derive the expressions for (i) the phase difference between waves from successive Fresnel zones, (ii) the amplitude of wave motion at an axial point P, and (iii) the radius of the nth Fresnel zone. Draw a sketch showing the Fresnel zones to illustrate your derivations.

Problem Solving

Given the wavelength of light incident upon a circular aperture of given radius, find the location of the points of maximum and minimum intensity along the axial line through the center of the aperture, for distances close to the aperture. Explain why these locations are points of maximum and minimum intensity.
For given points along the axial line, find the amplitude of the light arriving at these points relative to the amplitude of the incident wavefront.

Diffraction Grating and X-Ray Scattering From Crystals

Knowledge

Describe the transition from a two-slit interference pattern to one with many slits having the same slit-to-slit separation. Explain why diffraction gratings' maxima are so sharp.
For a diffraction grating, derive the expressions for the angles at which maxima are detected.
Discuss scattering of X-rays by a crystal lattice using Bragg's equation, and describe the experimental arrangement for observing Bragg scattering.

Problem Solving

Given the wavelength of light incident upon a given size grating with N lines, calculate the angles of deviation of the principle maxima.
For white light incident on a grating with N lines, calculate the angular separation for two given wavelengths of transmitted light.
Given three of the following quantities for Bragg scattering, calculate the fourth: separation of crystal lattice planes, wavelength of light, angle of incidence, order of spectrum.