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Time Dependent Wave Equation 1D One Dimensional Optical Resonator Modes
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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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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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Solar Power Generation Planet Formation Illumination of Planets Change of Orbit due to Asteroid Impact Earth Time Zones Day Lengths Astronomical Aberration Kepler Planetary and Sun Orbits Model of Ocean Tides Gravity Course Iteration Spacecraft Speed Assistance by Gravity Hohmann Transfer from One Circcular Orbit to Another Star Distances Using the Parallax Method
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
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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
Chapter 7
Chapter 8
Chapter 9
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Chapter 11
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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
Chapter 11
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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
Chapter 5
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Sagnac Effect Fiber Optical Gyro (FOG) Ring Laser Gyro (RLG)
Chapter 4
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Chapter 6
Chapter 7
Chapter 8
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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
Chapter 7
Chapter 8
Chapter 9
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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
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
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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
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
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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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Light Propagation in a Transparent Dielectric Material

The goal here is the demonstrate electromechanically how a wave behaves when it enters a dielectric material after exiting a vacuum. Asking why a light wave slows down when it enters a dielectric of permittivity `epsilon` is the same as asking why Maxwell's wave equation contains the parameter `n^2=epsilon/epsilon_0` where where `epsilon_0` is the permittivity of vacuum and `n>1` is the index of refraction. The use of `n` in Maxwell's wave equation results in the wave phase speed being reduced by factor `1/n` in the dielectric. That also results in the energy antinodes in the dielectric being spaced at distance 1/n closer together than in the vacuum.

Calculation of the wave energy density in the dielectric

An important fact is that in free space (vacuum) the relation between the electric intensity, `E_0`, and magnetic intensity, `H_0`, is

`E_0/H_0=sqrt(mu_0/epsilon_0)`

where `mu_0` is the magnetic permeability of vacuum and `epsilon_0` is the permittivity of vacuum. Now, when the wave enters a dielectric medium, the magnetic intensity does not change but the electric intensity,`E`, follows the same equation:

`E/H_0=sqrt(mu_0/epsilon)`

where `epsilon` is the permittivity of the medium. For a high frequency wave we can write

`epsilon=n^2epsilon_0`

where `n` is the index of refraction of the dielectric medium. Therefore we can write:

`E/H_0=sqrt(mu_0/(n^2epsilon_0))=(E_0/H_0)/n`

so `E` becomes

`E=E_0/n`

In the case of a radiation wave, the electric field is "loaded" or reduced by the dielectric material.

Wave Energy Conservation

The rate of energy flow in medium `epsilon` has to be the same as in medium `epsilon_0`. Energy flow of a plane wave must be measured as `"Joule"sm^(-2)sec^(-1)`. The Poynting vector magnitude is `S=EH="Watts"m^(-2)="Joules"m^(-2)sec^(-1)`. In `epsilon_0` `S_0=E_0H_0` while in `epsilon` we have `S=EH_0=(E_0/n)H_0` which is less than `S_0`. In fact `S=S_0/n`. This is the "bare field" power per unit area. But we must also consider the energy associated with the motion of the electron clouds of the atoms of the dielectric. Since `S` has units watts`meter^(-2)` and the energy density has units Joules`m^(-3)`, we must divide `S` by a speed `msec^(-1)` to obtain wave energy density. An appropriate equation for the total energy density associated with the wave is

`U=(EH_0)/v=(E_0H_0)/(nv)=(E_0H_0)/c`

from which we find that

`nv=c`
`v=c/n`

This shows that the rate of wave energy flow is constant if `v=c/n`. and this applies to both vacuum and dielectric material. The total energy density induced by the wave also includes the mechanical energy of the moving dipoles.

Origin of `epsilon_0`, the Permittivity of Vacuum

In order to calculate what would be the speed of light in vacuum, Maxwell needed to have independent values for the parameters `epsilon_0` and `mu_0`. He could obtain `epsilon_0` from the Coulomb force between two charged bodies and that is the measurement I will discuss here. By Coulomb's Law the repelling force between two charges, `q_1` and `q_2` (coulombs) is measured to be

`bbF=bbhatr(q_1q_2)/(4piepsilon_0r^2)`      

where `r` is the distance between the charge centers and `bbhatr` is a unit vector in the `r` direction.

Derivation of the Force Equation

For the derivation we will make one of `Q` much larger than the other `q` i.e.

`bbF=bbhatr(Qq)/(4piepsilon_0r^2)`      

Now we invoke Gauss's law for the electric field, `bbE`, of `Q`.

`epsilon_0intbbE*dbbs=Q`      

where the integral is over the surface of a sphere surrounding `Q` and `dbbs` is an outward vector surface increment of the sphere and the centered `*` dot represents the dot product of `bbE` with `dbbs`. This equation for `bbE` is also valid in a dipolar medium where `epsilon` can replace `epsilon_0`. Obviously, if `bbE` is parallel to `dbbs` everywhere the the integral is just `4pir^2|bbE|`, where `r` is the distance from the sphere center, so the equation becomes:

`epsilon_04pir^2|bbE|=Q`      

and

`bbE=Qbbhatr/(4piepsilon_0r^2)`      

The force on `q` due to this electric field is `qbbE` and becomes:

`F=bbhatr(Qq)/(4piepsilon_0r^2)`      

Measuring `epsilon_0` in Vacuum

After measuring `|bbF|` we can compute the value of `epsilon_0` as

`epsilon_0=(qQ)/(4pi|bbF|r^2)`      

To measure `bbF` due to charges, one might resort to placing known changes `q` on each of two spheres just as was done for measuring the gravitational constant `G`. A big problem with this measurement is being sure of the value of `q`. Charging the spheres is done by providing a known current for a known time. But other parts of the apparatus may have charge too.

Explanation of the Controls

Dielectric Constant Slider

For this animation the dielectric constant `epsilon` is the square of the index of refraction, `n`. Increasing `epsilon` using its slider has 3 effects:

1:It decreases the amplitude by the factor 1/`sqrt(epsilon)` of the wave in the dielectric in Canvas 1.

2: It decreases the spacing between waves and wave crests shown in Canvases 1 and 2 by the factor 1/`sqrt(epsilon)`.

3:It decreases the wave and wave crest speed by the factor 1/`sqrt(epsilon)` of the wave in the dielectric in Canvases 1 and 2.

Note that in both Canvases, when a wave or wave crest is reaches the boundary in the vacuum/dielectric, a new wave or wave crest always is born in the dielectric/vacuum. That is much easier to see in Canvas 2 than Canvas 1 because of the amplitude differences in Canvas 1.

Number of Random Dipoles Slider

This slider changes the number of randomly placed dipoles that react to the moving wave in Canvas 1. If you decrease the number of dipoles to, say, 200 and check the Single Step box and then press Start successively while watching a single dipole, you can see that the dipoles' rotation angles are such that they tend to cancel the electric field of the wave. This canceling effect reduces the wave's electric field by the factor `1/sqrt(epsilon)`.

Explanation of Graphics

Canvas 1 shows the relative motion and amplitude of the wave in both vacuum and dielectric material. To avoid the complexity of having wave reflections at the interfaces between vacuum and dielectric I have chosen to have applied a 1/4th wave antireflection coating (see darker layers at both of these interfaces). The reflections that might have occurred bring into play the Fresnel equations which are an entirely different subject. Thus the graphics are explained more than adequately under "Explanation of the Controls"