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Time Dependent Wave Equation 1D One Dimensional Optical Resonator Modes
Gas Physics Introduction
Gas Physics Introduction Chapter 1:Energy Distribution Evolution Energy Diffusion from Mono-energetic of a Three Dimensional Gas Rotating Gas in a 3D Toroid Gas Velocity Equilibration 2D Gas Velocity Equilibration 3D Circulating Gas in a Wind Tunnel Density Modulated Rotating Gas in a 3D Toroid Position Diffusion (Mixing) of a Three Dimensional Gas Energy Distribution 3D with MathJax Equations and Embedded Canvases Gas Energy Distributions 2D Rigid Rotors Energy Distribution 2D 1D Gas Energy and Momentum Distribution Rotor Energies 2D Rotor Energies 3D H2O Energies 3D CO2 Energies 3D 2D Rigid Arrays with Rotation and Translation 3D Rigid Arrays with Rotation and Translation Gas Energy Distributions 1D, 2D, 3D Chapter 2:Particle Scattering Effects
Inelastic Collision Conversion to Internal Energy Brownian Motion Animation 2D Using Hard Sphere Scattering Large Disc Kinetics 2D Condensation of a Gas on a 2D Solid Maxwell's Demon Energy Seperator 2D Minimize Final Energy 2D and 3D Gas Diffusion 2D Gas Expansion 2D Linear Gas Expansion 2D Constant Pressure Gas Expansion 2D Circular Gas Explosion 2D Circular MagnetoCaloric Effect 2D Convection Loops in a Numerical Gas Chapter 3: Gas Energy Equipartition Energy Equipartition of a Two Dimensional Gas with Two Different Masses Energy Equipartition of a Three Dimensional Gas with Two Different Masses Gas Energy Equipartition 3D
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Photoelectric Effect Alpha Particle Emission from a Large Nucleus 2D Black Body Emission from Charged Harmonic Oscillators Quantum Animation of Energy Band Gap 1D Animation of Quantum Wave Packets in a pn Junction 1D Quantum Mirror Focusing by Iterating the Schrodinger Equation Quantum Lens Focusing by Iterating the Schrodinger Equation Quantum Two Mirror Resonator by Iterating the Schrodinger Equation Entangled Quantum Object Propagation in a Parabolic Potential Classic and Quantum Object Propagation in a Parabolic Potential Accurate Wave Packet Propagation in a Parabolic Potential Comparison of Quantum to Classical Physics 1D-Time Dependent Accurate Wave Packet Propagation in a Swaged Potential Series Solution of Wave Packet Propagation in a Infinite Square Potential Propagation of a Wave Packet in a Parabolic Potential Propagation of a 2D Wave Packet in a 2D Parabolic Potential Propagation of an Asymmetric Shaped Wave Packet in a 2D Parabolic Potential Two Slit Diffraction by Iterating the Schrodinger Equation Two Potential Slit Diffraction by Iterating the Schrodinger Equation Propagation of an Electron Wave Packet in an Infinite Square Well (Series Solution) Propagation of a Free Wave Packet (Series Solution) Series and Algebraic Solutions for Propagation of a Wave Packet in a Parabolic Potential Comparison of Quantum to Classical Physics 1D-Time Independent Mexican Hat Stationary Quantum States of a Finite Square Potential Well Various Wave Packet Motion in a Finite Square Potential Well Various Wave Packet Motion in a Swaged Potential Well Comparison of Quantum to Classical Physics 1D-Time Independent Cosine/Parabola Electromagnetic Modes of a 2D Cavity Black Body Radiation Experiment and Theory Emission of the Inner Walls of a Black Body Cavity Multiple Slit Diffraction Statistics of Indistinguishable Dice or Particles Quantum Field Collapse Visualization Matrix Operations and Eigenmodes
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Chapter 7
Chapter 8
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Tumbling Block on Inclined Plane New Roller on Inclined Plane Sliding Block on Inclined Plane Classic Objects Motion 1D Generation of Curve for Fastest Time between Two Points Curve for Fastest Time between Two Points Bead Sliding On a Stiff Wire Sound Wave in Two Solid Media Mechanical Properties of a 2D Numerically Modeled Lattice Vibrating Reed Numerical Model 2D Laminar Flow of a Fluid in a 2D Bearing Animation of a Fountain 2D Gravity Leveling of a 2D Liquid Longitudinal and Shear Wave Sound Speed in a 2D Lattice Mechanical Properties of a 2D Numerically Modeled Lattice Vibration Frequencies of a 2D Numerically Modeled Lattice Damped Harmonic Oscillator Motion and Modes of a Particle in a Mexican Hat Potential Forced Oscillator Roller Bearing Cam and Roller Follower Simple Hybrid Gear Train Ratchet Animation Child's Swing Height Increase Mechanics Roller Chain Animation Elasticity of 3D Gas Driving a Piston Free to Forced Harmonic Oscillator Transition Physics of Mounting a Tire on a Rim Physics of a Pendulum Clock Escapement Physics of a Gas Pressure Gauge
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Physics of a Gas Pressure Gauge Ion Drift in Neutral Particle Sea Diffusion Equation Solved by Finite Difference Method Linear Particle Diffusion Heat Flow 1D with Time Dependent Boundary Solid Cube in Liquid Column Drag on a Large Sphere Due to a Digital Gas Animation of a Gas Centrifuge Laminar Flow of a Gas in an Annular Space Laminar Flow of a Fluid in a 2D Bearing Incompressible Fluid Flow through a Constriction Fluid Flow Due To Pressure Gradients
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Parallel Resistors Gas Discharge Animation Oscilloscope Animation Permanent Magnet in Solenoid Cross Product of Two Vectors Magnetic Braking with Cylindrical Geometry Stopping a Falling Magnet by Eddy Currents Magnetic Brake Magnetic Field of a Long Solenoid Coil Light Propagation in a Transparent Dielectric Material Dipole Array Animations Angular Distribution of Emission of Arrays of Excited Dipoles Geometric Derivation of Snell's Refraction Angle Law Electromagnetic (EM) Wave Response to a Row of Dipoles Electromagnetic (EM) Wave Response to a Single Dipole Optics of a Transparent Plate by Iterating the Maxwell Wave Equation Diffraction by Iterating the Maxwell or Schrodinger Wave Equations Lens Focusing by Iterating the Maxwell or Schrodinger Wave Equations Optical Wave Packet Propagation by Finite Difference Matrix 1D Potential(x,y) Using Laplace's Equation Div(gradV)=0 with Variable Boundary Geometry Potential(x,y) Using Laplace's Equation Div(epsilon*gradV)=0 Laplace's Equation Div(gradV)=0 with Mixed BCs Potential(x,y) for Surface Electrodes with Dielectric Slab Dipolar Media Field Plots LCR Oscillator Animation Beam Trajectory of a Moving Laser Beam Trajectory in a Square Ring Laser Doppler Effect Converting Magnetic Forces to Electric Forces Converting Permanent Magnet Orbital Moments to Solenoid Currents Electron Beam Collimation by Axial Magnetic Field Electron Buncher for Klystron Charged Particle Spatial and Speed Bunching Electrical Conduction-Particle Picture Electromagnetic Bell Triode Vacuum Tube
Chapter 4
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E=mc^2 and Mexican Jumping Bean Radar Pulse Return Time Interval Experiment to Determine Gamma Current Loop with a Rotating Test Particle Moving Fabry-Perot Interferometer Space Time Invariant Intervals of a Moving Clock Relativity Transformations by Requiring Simulataneity Accelerated Space Trip Clock Times and Rates Minguzzi Invariant Inertial Frame Time Accelerated Space Trip with Light Pulse Signaling
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Chapter 4
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Semiconductor Electron Energies and Band Gap 1D Semiconductor Electron Energies and Band Gap 2D Simple Fermi Surface p-nJunctionDiode Depletion Width Temperature Dependence of 2D Fermi-Dirac Particles
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Assembly of 3D Parts Using Javascript Web Graphics Library 3D Compared to Javascript 3D Graphics 3D Graphics Tutorial Beginnings Proof of Pythagoras a^2+b^2=c^2
Chapter 4
Chapter 5
Chapter 6
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Multiple Slider Animation Template for Plotting with HTML Enhanced Sliders or Ranges Collapsible Dropdown Menu Example Arrow Functions Fast Fourier Transform (FFT) Demonstration
Animation of a Progress Bar
Template for Scientific Plots
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Time Dependent Wave Equation 1D One Dimensional Optical Resonator Modes
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Probability of Birthday Match Random Creation of Probability Function Profiles Two Dies Probability Distribution Three Dies Probability Distribution
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
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Light Rays in a Transparent Isosceles Prism Light Ray in a Series of Wedges of Various Index of Refraction Snells Law of Refraction Refraction Due to Dipoles in Two Media Refraction and Reflection Two Media Animation of Concave Mirror Reflection Animation of Convex Lens Refraction Prism Refraction-Wave Picture Thick Lens Ray Trace Monochromator (Czerny-Turner Type) Wavelets from Concave Mirror> Thick Lens Focusing
Chapter 4
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Sagnac Effect Fiber Optical Gyro (FOG) Ring Laser Gyro (RLG)
Chapter 4
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Paticle Motion in a Venturi: Wall Reflections Paticle Motion in a Venturi Potential Elliptical Ion Trap Using Boundary Potential Distribution Prediction of Hard Disc Collision Time and Position Rigid Rotor Collisions
Chapter 4
Chapter 5
Chapter 6
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Acoustic Wave with Built-in Pressure Acoustic Waves in 2D Numerically Modeled Solid Wave 1D on a Linear Array Waves on Discrete Model of Vibrating String Linear Acoustic Wave in a 3D Gas Spherical Acoustic Wave in a 3D Gas Laterally Perturbed Two Dimensional Gas with Lennard Jones Repelling Forces Longitudinally Perturbed Two Dimensional Gas with Lennard Jones Repelling Forces
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Bicep and Forearm Animation Blood Pressure Measurement Multiple Dose Residual Medicine Vs Time The Eye and its Resolution Breathing Animation Scoliosis Animation Muscle Animation
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Chapter 5
Chapter 6
Chapter 7
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Scattering from a Crystalline Target 2D Scattering from a Crystalline Target 3D
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Energy Conduction in a 2D Numerically Modeled Solid Lattice Unbalanced Gas Dynamics:Energy Flow in a Gas
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12

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Energy Distribution of 3D Charged Rigid Arrays

Energy Distribution of 3D Charged Rigid Arrays In this animation, we will examine the energy distributions of 3D rigid arrays that interact with each other via potentials between the their elements and elements of other arrays. As such, these arrays will be subject to both repelling forces on their centers of mass (CM) as well as torques that tend to rotate the array about its CM. Therefore, for the 3 dimension case that we sill study here, we will have 6 degrees of freedom. These are position in the x, y and z directions as well as rotation about the (x,y,z) axes with repect to the CM. In order to compute the rotations we first compute the moments of inertia, Ix, Iy, and Iz about the three principle axes. Then we will have the equations for changes of the spin rates about these axes

`I_xdomega_x=(bbrxxbbF)bbx, I_y*domega_y=(bbxxbbF)bby, I_z*domega_z=(bbrxbbF)bbz`

where the vector cross product `(bbrxbbF)`is the torque due to force `bbF` at particle vector position `bbr` of the array. To keep the equations simple we will choose the particle locations so that the array is symmetric about its center of mass (CM). Then the expressions for moments of inertia are

`I_x=sum_im_ix_i`

, etc. Having computed `domega_x`, etc., we then need to increment `omega_x`, etc., by that amount and then increment the `x` angle of the `jth` array, `Phix_j`, of the array about the `x` axis by the angle `dPhi_x=omega_x`. Then we need to compute the positions of all masses with respect to the center of mass by these equations. This is done using the (x,y,z) rotation matrices for the angle increments `(omega_x,omega_y,omega_z)` at each increment of time where the time increments are video frames. It is important to preserve the order of rotation of the `(x,y,z)` angles. Since there are 6 degrees of freedom, we should therefore expect that the energy distributions would be

`D(E)=E^2*exp(-E/E_c)`

where `E_c` is some constant that is related to the average energy, E_(avg), and determined by the value of the ratio of the following integrals

`E_(avg)=(intED(E)dE)/(intD(E)dE)`.

It turns out the above ratio is `E_(avg)=3E_c` which means that `E_c=E_(avg)/3` and then `D(E)` becomes

`D_p(E)=CE^2exp(-(3E)/E_(avg))`

where C can be chosen so that `D(E)` represents either a probability (and integrates to 1.0) or, more conveniently for plotting, `C` can cause the peak value of `D(E)` to equal 1. To make `D(E)` equal 1 at its peak value, we differentiate the original expression with respect to E and find that the peak value of `D(E)` is at `Ep=E_c/2`. Then the value of `D(E)` at its peak is

`D_(max)(Ep)=E_c^2exp(-2E_c/E_c)`

Then we choose

`C=exp(2)/(4Ec*Ec)`

wich causes the peak value of Dp(E) to equal 1. So

`D_p(E)=exp(2)(E^2/(4Ec^2))exp(-E/Ec)=exp(2)(9E^2/(4E_(avg)^2))exp(-3E/E_(avg))`

The plots show the distributions of the total kinetic energy (black), the translational kinetic energy (green), the rotational kinetic energy (blue), and the expression above (black) which is a smooth curve whereas the others are stepped since they are histograms. Because it takes a long time to compute any single plot, I have chosen to present some Final Distribution Pictures (Dist i..) which the learner is free to click on and view. Just close the picture window after viewing to return to the application.

In this case, I have had great difficulty with the usual `1/r^2` (where r is the distance between the elements) forces between elements. I have instead used the expression `F=q^2exp(-r/r_C)` where `r_C` is a constant chosen by the learner and q represents the charge on each element.

Over limited range of the variable parameters this force value is stable but cannot be expected to be stable for all parameters. If an unstable set of parameters is chosen, at the left middle of the page a message 'Maximum Energy Exceeded...' will appear and you should change at least parameter and press Start again.

Since we are dealing with 3D here, I have chosen to give some indication of the relative depths (z) into the screen of particular array pairs. I have change the color from black to red if the z depth difference between two Arrays is less than their diameter. When two arrays are red and they also are within a diameter of each other in the (x,y) plane, that means that they are strongly interacting. Two black arrays or a black and a red do not interact strongly.

Dist 1 Dist 2 Dist 3 Dist 4 Dist 5 Dist 6 Dist 7 Dist 8 Dist 9 Dist 10