Modeling and Simulating with Electromagnetism

Tim ThirionCOMP 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. 12. 10¹3. 10²4. 10³

SolutionThe gravitational force is

The Coulomb force isThe 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 Intel’s 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: Coulomb’s Law

Coulomb’s Law gives the force between two charged particles at rest:

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Coulomb’s Law

The Law of Superposition holds

Why doesn’t an electron collide with the positively charged protons in a nucleus?

Does an electron act on itself?

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Vector FieldsVector 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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FluxSuppose 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 Maxwell’s Equations:

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CirculationAs 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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Ampere’s Law

From vector calculus, Stokes’ Theorem says

Apply this, and make the surface infinitesimally small:

Differential form of Ampere’s Law:

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Ampere’s 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 Ampere’s Law:

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Problem

Coulomb’s Law holds for static charge configurations.

Moving charges generate magnetic fields.

How do magnetic fields affect the motion of charged particles?

Coulomb’s 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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ApplicationModeling 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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Faraday’s Law

As we did for Gauss’ Law, shrink S to an infinitesimally small surface to get the differential form:

Faraday’s 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 Maxwell’s Equations:

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The Maxwell Equations

Gauss’ Law

Faraday’s Law of Induction

Analog of Gauss’ Law for Magnetism

Ampere’s Law with Maxwell’s Extension

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Visualizing Vector FieldsThere 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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LICThe final convolution step:

k(w) is the convolution kernel.

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LIC Results

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Modeling the Magnetosphere

Earth’s magnetosphere is caused primarily by two effects:

• The convection of ionized liquid metals in the Earth’s outer core

• The solar winds: a vast flow of plasma (a stream of free ions)

The strength of earth’s magnetic field decays exponentially; half-life 1400 years, reversals every 250,000 years (500,000 years overdue)

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Visualizationshttp://

svs.gsfc.nasa/gov/search/Keyword/Magnetosphere.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.

There’s plenty of room at the bottom!

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Questions?

thirion@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/802TEAL3D/index.html