Modeling and Simulating with Electromagnetism. Tim Thirion COMP 259 Physically-based Modeling, Simulation and Animation April 13, 2006. Before We Begin ….
Physically-based Modeling, Simulation and Animation
April 13, 2006
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?
The gravitational force is
The Coulomb force is
The ratio is
Electromagnetism is the prevailing force on a huge range of physical scale …
On the smallest scales, EM dominates where nuclear forces drop off.
From 1 nm = 10 Å to 1 cm, we can begin modeling nanomolecules, organic molecules, and microdevices.
On the scale of everyday experience, we again see multiple applications
On the scale of the earth, geo* applications
At higher scales, gravity dominates. However, EM still plays a role as light…
Coulomb’s Law gives the force between two charged particles at rest:
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?
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.
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.
What force will a positive “test” charge feel if placed into an electric field?
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 outwardflow or flux is:
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
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:
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:
The circulation of the magnetic field around a closed loop is proportional to the net current flowing through it.
From vector calculus, Stokes’ Theorem says
Apply this, and make the surface infinitesimally small:
Differential form of 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:
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 …
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.
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!
Suppose we have a surface S with a curve boundary C, then
In the language of vector calculus
As we did for Gauss’ Law, shrink S to an infinitesimally small surface to get the differential form:
Faraday’s Law of Induction:
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?
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:
Faraday’s Law of Induction
Analog of Gauss’ Law for Magnetism
Ampere’s Law with Maxwell’s Extension
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.
“LIC emulates the effect of a strong wind blowing a fine sand.”
LIC improves on DDA (digital differential analyzer).
DDA used straight line approximations in the vector field.
To generate streamlines:
The final convolution step:
k(w) is the convolution kernel.
Earth’s magnetosphere is caused primarily by two effects:
The strength of earth’s magnetic field decays exponentially; half-life 1400 years, reversals every 250,000 years (500,000 years overdue)
Electromagnetic phenomena are incredibly diverse.
Theory and methods are relatively simple.
Phenomena can be incredibly complex.
There’s plenty of room at the bottom!