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| ===A Computational Model=== | | ===A Computational Model=== |
| !DOCTYPE html
| | https://www.glowscript.org/#/user/sboebel3012/folder/MyPrograms/program/EMWaveVisualization |
| <html>
| |
| <div id="glowscript" class="glowscript">
| |
| <meta http-equiv="Content-Type" content="text/html; charset=utf-8">
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| <link type="text/css" href="https://www.glowscript.org/css/redmond/2.1/jquery-ui.custom.css" rel="stylesheet" />
| |
| <link type="text/css" href="https://www.glowscript.org/css/ide.css" rel="stylesheet" />
| |
| <script type="text/javascript" src="https://www.glowscript.org/lib/jquery/2.1/jquery.min.js"></script>
| |
| <script type="text/javascript" src="https://www.glowscript.org/lib/jquery/2.1/jquery-ui.custom.min.js"></script>
| |
| <script type="text/javascript" src="https://www.glowscript.org/package/glow.3.2.min.js"></script>
| |
| <script type="text/javascript" src="https://www.glowscript.org/package/RSrun.3.2.min.js"></script>
| |
| <script type="text/javascript"><!--//--><![CDATA[//><!--
| |
| | |
| // START JAVASCRIPT
| |
| ;(function() {;
| |
| var ρσ_modules = {};
| |
| ρσ_modules.pythonize = {};
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| | |
| (function(){
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| function strings() {
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| var string_funcs, exclude, name;
| |
| string_funcs = set("capitalize strip lstrip rstrip islower isupper isspace lower upper swapcase center count endswith startswith find rfind index rindex format join ljust rjust partition rpartition replace split rsplit splitlines zfill".split(" "));
| |
| if (!arguments.length) {
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| exclude = (function(){
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| var s = ρσ_set();
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| s.jsset.add("split");
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| s.jsset.add("replace");
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| return s;
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| })();
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| } else if (arguments[0]) {
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| exclude = Array.prototype.slice.call(arguments);
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| } else {
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| exclude = null;
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| }
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| if (exclude) {
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| string_funcs = string_funcs.difference(set(exclude));
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| }
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| var ρσ_Iter0 = string_funcs;
| |
| ρσ_Iter0 = ((typeof ρσ_Iter0[Symbol.iterator] === "function") ? (ρσ_Iter0 instanceof Map ? ρσ_Iter0.keys() : ρσ_Iter0) : Object.keys(ρσ_Iter0));
| |
| for (var ρσ_Index0 of ρσ_Iter0) {
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| name = ρσ_Index0;
| |
| (ρσ_expr_temp = String.prototype)[(typeof name === "number" && name < 0) ? ρσ_expr_temp.length + name : name] = (ρσ_expr_temp = ρσ_str.prototype)[(typeof name === "number" && name < 0) ? ρσ_expr_temp.length + name : name];
| |
| }
| |
| };
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| if (!strings.__module__) Object.defineProperties(strings, {
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| __module__ : {value: "pythonize"}
| |
| });
| |
| | |
| ρσ_modules.pythonize.strings = strings;
| |
| })();
| |
| async function __main__() {
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| "use strict";
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| var display = canvas;
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| var scene = canvas();
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| | |
| function round(num, n=0) {return Number(num.toFixed(n))}
| |
| | |
| var version, print, arange, __name__, type, ρσ_ls, scene, amplitude, speed_of_light, wave_length, frequency, omega, k, length_of_wave, E_arrows, B_arrows, i, t, dt, z;
| |
| version = ρσ_list_decorate([ "3.2", "glowscript" ]);
| |
| Array.prototype['+'] = function(r) {return this.concat(r)}
| |
| Array.prototype['*'] = function(r) {return __array_times_number(this, r)}
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| window.__GSlang = "vpython";
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| print = GSprint;
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| arange = range;
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| __name__ = "__main__";
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| type = pytype;
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| var strings = ρσ_modules.pythonize.strings;
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| | |
| strings();
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| "5";
| |
| scene = ρσ_interpolate_kwargs.call(this, canvas, [ρσ_desugar_kwargs({title: "Electromagnetic Wave Simulation", width: 800, height: 600, center: vector(5, 0, 5), background: color.black, forward: vector(1["-u"]()["*"](.5), 1["-u"]()["*"](.5), 1["-u"]()["*"](.5))})]);
| |
| "7";
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| amplitude = 1;
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| "8";
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| speed_of_light = 1;
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| "9";
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| wave_length = 5;
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| "10";
| |
| frequency = speed_of_light["/"](wave_length);
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| "11";
| |
| omega = 2["*"](pi)["*"](frequency);
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| "12";
| |
| k = 2["*"](pi)["/"](wave_length);
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| "13";
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| length_of_wave = 20;
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| "16";
| |
| E_arrows = ρσ_list_decorate([]);
| |
| "17";
| |
| B_arrows = ρσ_list_decorate([]);
| |
| "18";
| |
| for (var ρσ_Index1 = 0; ρσ_Index1["<"](length_of_wave["+"](1)); ρσ_Index1++) {
| |
| i = ρσ_Index1;
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| "19";
| |
| E_arrows.append(ρσ_interpolate_kwargs.call(this, arrow, [ρσ_desugar_kwargs({color: color.red, axis: vector(0, amplitude, 0), shaftwidth: .1, pos: vector(0, 0, i)})]));
| |
| "20";
| |
| B_arrows.append(ρσ_interpolate_kwargs.call(this, arrow, [ρσ_desugar_kwargs({color: color.cyan, axis: vector(amplitude, 0, 0), shaftwidth: .1, pos: vector(0, 0, i)})]));
| |
| }
| |
| "23";
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| t = 0;
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| "24";
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| dt = .01;
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| "25";
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| while (true) {
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| "26";
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| (await rate(100));
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| "29";
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| var ρσ_Iter2 = range(len(E_arrows));
| |
| ρσ_Iter2 = ((typeof ρσ_Iter2[Symbol.iterator] === "function") ? (ρσ_Iter2 instanceof Map ? ρσ_Iter2.keys() : ρσ_Iter2) : Object.keys(ρσ_Iter2));
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| for (var ρσ_Index2 of ρσ_Iter2) {
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| i = ρσ_Index2;
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| "30";
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| z = ρσ_getitem(E_arrows, i).pos.z["-"](1["*"](speed_of_light)["*"](t));
| |
| "31";
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| ρσ_getitem(E_arrows, i).axis.y = amplitude["*"](sin(k["*"](z)["-"](1["*"](omega)["*"](t))));
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| "32";
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| ρσ_getitem(B_arrows, i).axis.x = amplitude["*"](cos(k["*"](z)["-"](1["*"](omega)["*"](t))));
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| }
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| "34";
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| t=t["+"](dt);
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| }
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| };
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| if (!__main__.__module__) Object.defineProperties(__main__, {
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| __module__ : {value: null}
| |
| });
| |
| | |
| ;$(function(){ window.__context = { glowscript_container: $("#glowscript").removeAttr("id") }; __main__() })})()
| |
| // END JAVASCRIPT
| |
| | |
| //--><!]]></script> | |
| </div>
| |
| </html>
| |
|
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|
| ==Examples== | | ==Examples== |
Spencer Boebel '24
Electromagnetic Waves
The Main Idea
An electromagnetic wave is what happens when we put all four of Maxwell's equations together. A changing electric field creates a magnetic field, which changes and creates an electric field, starting it all over again. This is the basic premise of an electric wave, and we shall derive other properties presently.
A Mathematical Model
The mathematical equations that allow us to model this topic are Maxwell's equations; but a simpler way to figure it out is to consider a notion from General Relativity. According to General Relativity, the speed of information is [math]\displaystyle{ c }[/math]. Now consider an electron moving at a velocity [math]\displaystyle{ \vec{v}(t) }[/math]. Now consider an observation point a location [math]\displaystyle{ \vec{r} }[/math] from the electron. Let us say the electron is oscillating, so we have that [math]\displaystyle{ \vec{v}=\cos{t} \vec{k} }[/math]. Then since the information that the electron is moving takes a while to get to the position [math]\displaystyle{ \vec{r} }[/math], so instead of getting the Coulomb field [math]\displaystyle{ \mathbf{E}=\frac{kq_e}{|\vec{r}|^2}\hat{r} }[/math] we get the electric field [math]\displaystyle{ \mathbf{E} = -\frac{q}{4\pi\epsilon_0} \left[ \frac{\mathbf{e}_{r'}}{r'^2} + \frac{r'}{c} \frac{d}{dt} \left(\frac{\mathbf{e}_{r'}}{r'^2}\right) + \frac{1}{c^2} \frac{d^2}{dt^2} \mathbf{e}_{r'} \right] }[/math], where [math]\displaystyle{ \vec{r}' }[/math] is the direction that the electron appears to be according to the observer, but is not actually. The magnetic field can then be written as [math]\displaystyle{ \mathbf{B} = -\frac{\mathbf{e}_{r'} \times \mathbf{E}}{c} }[/math]. Now if you notice the first two terms in the above E-field equation can be dropped at large [math]\displaystyle{ r }[/math] because they drop off with the distance squared. Calculating, we have that our electric field in a wave is [math]\displaystyle{ \mathbf{E} = -\frac{q}{4\pi\epsilon_0 c^2} \frac{d^2 \mathbf{e}_{r'}}{dt^2} }[/math]. The second derivative term is just the derivative of our velocity, so we get that [math]\displaystyle{ \mathbf{E} = \frac{q}{4\pi\|\vec{r}|epsilon_0 c^2}\sin(t) \vec{k} }[/math]. Using the B formula we get that since [math]\displaystyle{ r }[/math] is very large, the [math]\displaystyle{ e_r }[/math] vector and the [math]\displaystyle{ \mathbf{E} }[/math] vector are perpendicular, leading us to calculate B to be [math]\displaystyle{ \mathbf{B} = -\frac{q}{4\pi\epsilon_0|\vec{r}|c^3}\sin(t) \vec{i} }[/math]. So, from our investigations we conclude (by way of the Lorentz formula) that an oscillating point charge will cause a similar oscillation in the same direction over large distances. This movement is how our eyes detect light!
A Computational Model
https://www.glowscript.org/#/user/sboebel3012/folder/MyPrograms/program/EMWaveVisualization
Examples
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