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Time Dependent Wave Equation 1D One Dimensional Optical Resonator Modes
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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
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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
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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
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Sagnac Effect Fiber Optical Gyro (FOG) Ring Laser Gyro (RLG)
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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
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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
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Chapter 12

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Electromagnetic Modes of a Rectangular Cavity

This page is a prelude to another page on the subject of radiation from a black body cavity. It is that subject that led Max Planck to conclude that energy (modes) could be added to the cavity only in increments proportional to the frequency of the electromagnetic (EM) radiation to be added. The constant of proportionality was the celebrated Planck's constant. h. This page shows a few of the lower electromagnetic (EM) modes of a rectangular cavity that has zero tangential field values at its inner boundary. The contours are color-coded with values corresponding to the color bar. The viewer may view various (nX,nY) by clicking the Start Random button.

Each of the random patterns seen corresponds to a different mode of the cavity. Larger numbers of antinodes, of course, correspond to shorter wavelength and therefore higher frequencies and higher photon energies. From the patterns it is clear that the vector fields are not zero at the inner boundaries of the rectangle. The learner is reminded that only the tangential component of E needs to be zero at a perfectly conducting wall. The normal component will generally be non-zero as is the case here. The magnitude of the wave vector, k, is

`k=sqrt(k_x^2+k_y^2)`

The most important lesson to learn from these graphics is the following: For larger `k vector magnitudes it is obvious that the number of possible modes (field maxima) per unit change of k,

`(deltan)/(deltak)=2pik`

which is the area of a circular ring which has a radius of `k` and a width of `deltak`. You can see that the number of lobes contained inside the ring shown is much larger when `n_X` and `n_Y` are larger. The number of modes inside the ring width shown are plotted versus ring radius. Note that this number increases approximately linearly with ring radius. This will be the case for the two dimensional cavity shown here because the area of the ring is `pi/2rdr` where `r` the radius and `dr` is the ring width.

For a 3D cavity, we should expect that the number of modes inside a spherical shell of thickness `dr` and of radius `r` is `4pir^2dr`.

As instructed above the contour plot, the learner should click on one of the green circles which repreasent the mode numbers, `(n_X,n_Y)`, to see the particular EM mode associated with that mode number. In a macroscopic black body that Planck analyzed the mode numbers in in the 10s of thousands and it is not feasible to show their patterns here. For simplicity and to save screen space, only one quadrant of the possible mode numbers is shown but the other three quadrants are equally likely.

As always, if, with the Chrome browser, the learner wants to view the code for this animation they should just click F12 and select Sources and then press F5 to choose one of the Javascript files.