Modeling and Simulating with Electromagnetism

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Modeling and Simulating with Electromagnetism

• Tim Thirion
• COMP 259
• Physically-based Modeling, Simulation and
  Animation
• April 13, 2006

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Before We Begin

• A question If I place a proton at the North pole
  and another at the South pole, what is the
  approximate ratio of the strength of the
  electrostatic force to the gravitational?
• 1
• 10¹?
• 10²?
• 10³?

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Solution

• The gravitational force is
• The Coulomb force is
• The ratio is
• Relevant constants

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Why is gravity so weak?

• The Four Physical Forces
• Strong Nuclear (binds nucleons)
• Weak Nuclear (some forms of nuclear decay)
• Electromagnetic
• Gravitational
• The first three have been shown to be
  indistinguishable in certain (Big Bang-like)
  conditions
• Uniting the four forces is the greatest
  outstanding problem in physics (String Theory,
  etc.)

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Outline

• Why should a computer scientist care about
  electromagnetism (EM)?
• The Fundamentals Statics and Dynamics
• Visualizing Vector Fields using LIC
• Application Modeling the Magnetosphere
• FEMs, Materials Science and Nanoscience
• Questions and (Hopefully) Answers

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Orders of Magnitude

• Electromagnetism is the prevailing force on a
  huge range of physical scale
• On the smallest scales, EM dominates where
  nuclear forces drop off.
• Scale 10 pm (average atom radius) 10 nm
• Must use QEM
• Fundamental particles, origin of the universe
• Molecule formation (chemistry)
• Smallest feature of Intels chips (65 nm, as of
  2006)

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Orders of Magnitude

• From 1 nm 10 Å to 1 cm, we can begin modeling
  nanomolecules, organic molecules, and
  microdevices.
• 1 nm is the radius of a carbon nanotube
• 2 nm is the diameter of a DNA helix
• Nanoscience and materials science simulation
  would occur mostly at this scale
• Electrostatic effects are prevalent

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Orders of Magnitude

• On the scale of everyday experience, we again see
  multiple applications
• 1 cm 1,000 km 1 Mm
• Approximations of the interaction of light and
  matter (rendering)
• Modeling of solids, crystals, x-ray diffraction
  simulations
• On the scale of the earth, geo applications
• The ionosphere and magnetosphere
• Lightning and weather systems

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And Beyond

• At higher scales, gravity dominates. However, EM
  still plays a role as light
• Star formation (QM, gravity, fluids, and light
  propagation)
• Galaxial modeling, supernovae (models needed to
  predict release of energy and particles)
• Cosmic background radiation models
• And so on

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Electrostatics Coulombs Law

• Coulombs Law gives the force between two charged
  particles at rest

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Coulombs Law

• The Law of Superposition holds
• Why doesnt an electron collide with the
  positively charged protons in a nucleus?
• Does an electron act on itself?

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Vector Fields

• Vector fields associate a vector with each point
  in space.
• The curl of a vector field gives the circulation
  within a volume.
• The divergence of a vector field gives the
  outward flow from a volume.

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Fields

• All of electromagnetism is concerned with
  deriving and utilizing the magnetic and electric
  fields.
• Both are functions of space and time
• As we shall see, they are deeply interconnected.
• In fact, they are essentially different aspects
  of the same phenomenon.

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Electric Fields

• What force will a positive test charge feel if
  placed into an electric field?
• More concisely

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Flux

• Suppose we have a closed surface.
• In the case of a fluid, we can ask, are we losing
  or gaining fluid in the enclosed volume?
• The net outward flow or flux is

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Electric Flux

• Electric fields do not flow because they are
  not the velocity of anything.
• We can still compute the flux using E.
• It turns out that
• Or

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Gauss Law

• A result from vector calculus, Gauss Theorem,
  says
• Using a charge density
• Taking the limit as V goes to zero
• The first of Maxwells Equations

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Circulation

• As with flux, we can define the amount of
  circulation present in a field.
• Draw a closed curve, how quickly does the fluid
  inside travel around this curve?
• The circulation is

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Circulation with the Magnetic Field

• The circulation of the magnetic field around a
  closed loop is proportional to the net current
  flowing through it.

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Amperes Law

• From vector calculus, Stokes Theorem says
• Apply this, and make the surface infinitesimally
  small
• Differential form of Amperes Law

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Amperes Law

• This is not fully general. Also must consider
  electric flux through S
• Using techniques from vector calculus, we arrive
  at the general differential form of Amperes Law

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Problem

• Coulombs Law holds for static charge
  configurations.
• Moving charges generate magnetic fields.
• How do magnetic fields affect the motion of
  charged particles?
• Coulombs Law is no longer the full story

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The Lorentz Force

• The total force on a charged particle due to
  electric and magnetic fields is
• Note the presence of the cross product and the
  dependency on velocity, not acceleration.

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Application

• Modeling the dynamics of charged particles
  immersed in E and B fields.
• Simply need to balance quantities, and use your
  favorite integrator with the Lorentz force!
• See http//www.levitated.net/p5/chamber/

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Circulation of the Electric Field

• Suppose we have a surface S with a curve boundary
  C, then
• In the language of vector calculus

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Faradays Law

• As we did for Gauss Law, shrink S to an
  infinitesimally small surface to get the
  differential form
• Faradays Law of Induction

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The Last Equation

• Recall Gauss Law
• Is there a similar analog for magnetism?
• That is, can we encapsulate magnetic charges in
  a surface, and measure the magnetic flux?

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The Last Equation

• There is no (as yet observed) magnetic charge or
  monopole.
• The magnetic field is divergence free, there is
  no inward or outward flow, to or from a point.
• The last of Maxwells Equations

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The Maxwell Equations

• Gauss Law
• Faradays Law of Induction
• Analog of Gauss Law for Magnetism
• Amperes Law with Maxwells Extension

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Visualizing Vector Fields

• There are many techniques available for
  determining and rendering field lines.
• We can trace particles through the field, use
  stream lines, or use icons. That is, place a
  relevant symbol along regular sample points
  (arrows, ellipsoids, etc.)
• Some methods use Gaussian linear solvers,
  conjugate gradient methods, spot noise, reaction
  diffusion textures, etc.
• One of the most interesting is Line Integral
  Convolution.

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Line Integral Convolution

• LIC emulates the effect of a strong wind blowing
  a fine sand.
• Idea
• For each sample in the vector field
• Compute a stream line starting at a cell, moving
  forward and backward a determined distance
• Use the points covered to index a white noise
  texture
• Convolve the texture points to determine the
  corresponding pixel color for the cell.

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Visual LIC

• LIC improves on DDA (digital differential
  analyzer).
• DDA used straight line approximations in the
  vector field.

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Visual LIC

• To generate streamlines

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LIC

• The final convolution step
• k(w) is the convolution kernel.

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LIC Results
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Modeling the Magnetosphere

• Earths magnetosphere is caused primarily by two
  effects
• The convection of ionized liquid metals in the
  Earths outer core
• The solar winds a vast flow of plasma (a stream
  of free ions)
• The strength of earths magnetic field decays
  exponentially half-life 1400 years, reversals
  every 250,000 years (500,000 years overdue)

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Visualizations

• http//svs.gsfc.nasa/gov/search/Keyword/Magnetosph
  ere.html

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Finite Element Methods (FEMs)

• As we have seen, FEMs begin with discretization
  (tetrahedra, cubes, )
• Nearly every computational physics problem can be
  represented by matrices
• Highly specialized, dense
• A Finite Element Computation of the
  Gravitational Radiation emitted by a Point-like
  object orbiting a Non-rotating Black Hole
• Advanced Finite Element Method for
  Nano-Resonators
• An Algorithm for Constructing Polynomial Systems
  Whose Solution Space Characterizes Quantum
  Circuits

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Computational Materials Science

• Already becoming an important new topic in
  physical simulation
• Current topics
• Deformation of metals (bouncing metal balls?)
• Micromagnetic modeling (with mesoscale physics)
• Phase Field Modeling (applied solidification)
• Discovering/Designing effective Hamiltonians
• Quantum dots, quantum information,
  superconductors
• Surfaces and interfaces

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Final Thoughts

• Electromagnetic phenomena are incredibly diverse.
• Theory and methods are relatively simple.
• Phenomena can be incredibly complex.
• Theres plenty of room at the bottom!

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Questions?

• thirion_at_cs.unc.edu

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Bibliography

• Classical Electrodynamics, J.D. Jackson, John
  Wiley Sons, Inc., 2001
• The Feynman Lectures on Physics, R.P. Feynman,
  R.B. Leighton, and M. Sands, Addison Wesley
  Publishing Company, Inc., 1963
• Fundamentals of Physics, D. Halliday, R. Resnick,
  J. Walker, John Wiley Sons, Inc., 2003
• A Dynamical Theory of the Electromagnetic Field,
  J.C. Maxwell, Scottish Academic Press, Ltd., 1982
• The Nature of Solids, A. Holden, Dover
  Publications, Inc., 1965

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Bibliography

• Finite Element Method for Electromagnetics, J.L.
  Volakis, A. Chatterjee, and L.C. Kempel, IEEE
  Press, 1998
• Imaging Vector Fields using Line Integral
  Convolution, B. Cabral and L. Leedom, Proceedings
  of ACM SIGGRAPH 1993
• Computational Physics Lecture Notes, A.
  MacKinnon, available on the internet (please
  e-mail me)

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Links

• Center for Theoretical and Computational
  Materials Science http//www.ctcms.nist.gov/
• TEAL at MIT http//web.mit.edu/8.02t/www/802TEAL3
  D/index.html