30:T512,md`Lambert's cosine law says that the amount of light energy arriving at a surface is proportional to the cosine of the angle between the light direction and the surface normal [1]. This law is typically used in radiometry and computer graphics, and this notebook is to help build a geometric intuition behind the rule. To demonstrate Lambert's cosine law, assume we have a light source with length ${tex`A_0 = ${A0.toFixed( 3 )}`} ${tex`m`} (meters) that emits a constant ${tex`216000J`} (joules) over the course of an hour. We can calculate the light sources flux via: ${tex`\Phi = ${J}J / ${s}s \approx ${W}W`} (watts) [2] We want to determine the amount of flux ${tex`\Phi`} illuminating the surface ${tex`A`}. This is given by ${tex`E = \dfrac{\Phi}{A}`}, which is typically called irradiance. In the construction below we can see that the light direction from ${tex`A_0`} makes an angle ${tex`\theta = ${theta.toFixed( 2 )}`} radians with the surface normal ${tex`n`}:`31:Tabb,board = { const board = JXG.JSXGraph.initBoard('graph', { axis: false, grid: true, keepaspectratio: true, showCopyright: false, showNavigation: false }); const a = board.create('point', [0, -4], { visible: false }); const b = board.create('point', [15, -4], { visible: false }); const c = board.create('point', [2, 2], { visible: false }); const d = board.create('point', [15, 2], { label: { fontSize: 16 }, name: 'n', size: 0, visible: true }); const e = board.create('point', [16, -4], { visible: false }); const l0 = board.create('line', [a, b], { straightFirst: false, straightLast: false }); l0.isDraggable = false; const l1 = board.create('line', [c, b], { firstArrow: false, lastArrow: true, straightFirst: true, straightLast: false, strokeColor: 'black' }); const l2 = board.create('parallel', [l1, a], { firstArrow: true, lastArrow: false, straightFirst: false, straightLast: false, strokeColor: 'black' }); l2.isDraggable = false; const l3 = board.create('perpendicularsegment', [l1, a], { strokeColor: '#e0e000' }); l3.isDraggable = false; const mp0 = board.create('midpoint', [l3], { visible: false }); const mpt = board.create('text', [-0.5, 0.5, 'A0'], { anchor: mp0, color: '#e0e000', fontSize: 16 }); const f = board.create('perpendicularpoint', [c, l3], { visible: false }); const l4 = board.create('line', [b, e], { firstArrow: false, lastArrow: true, straightFirst: false, straightLast: false, strokeColor: 'black' }); l4.isDraggable = false; const l5 = board.create('arrow', [b, d], { strokeColor: 'black' }); l5.isDraggable = false; board.create('angle', [a, f, b], { label: { visible: false } }); board.create('angle', [e, b, d], { label: { visible: false } }); const a0 = board.create('angle', [b, a, f], { label: { color: '#f7a22d', visible: true }, radius: 1.0 }); a0.setLabelText('θ'); const a1 = board.create('angle', [d, b, c], { label: { color: '#f7a22d', visible: true }, radius: 1.0 }); a1.setLabelText('θ'); const a2 = board.create('angle', [f, b, a], { label: { color: '#f7a22d', visible: true }, radius: 1.5 }); a2.setLabelText('π - θ'); const slider = board.create( 'slider', [[17, 2], [17, -4], [0, 0.25 * Math.PI, 0.5 * Math.PI]], { name: 'θ' } ); const t0 = board.create('text', [7.1, -4.5, 'A'], { color: 'blue', fontSize: 16 }); t0.isDraggable = false; a1.setAngle(() => { const theta = slider.Value(); mutable theta = theta; mutable A0 = l3.L() + 100; return theta; }); board.update(); return board; }32:[] 2e:{"title":"Lambert's Cosine Law","cells":[{"id":0,"value":"md`# Lambert's Cosine Law`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1129,"value":"$30","pinned":false,"hidden":false,"mode":"ojs"},{"id":551,"value":"viewof graph = html`

`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2031,"value":"md`Using the fact that all angles in a triangle add up to π [3] we can see that \n${tex`\\cos(\\theta) = ${Math.cos(theta).toFixed(\n 3\n)}`}, and using the definition of cosine [4] we get:`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1934,"value":"tex`\\cos(\\theta) = \\dfrac{A_0}{A} \\Rightarrow A = \\dfrac{A_0}{\\cos(\\theta)}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1940,"value":"md`We can now solve for the irradiance ${tex`E`}:`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1946,"value":"tex`E = \\dfrac{\\Phi}{A} = \\dfrac{\\Phi}{\\dfrac{A_0}{\\cos(\\theta)}} = \\dfrac{\\Phi \\cos(\\theta)}{A_0} = \\dfrac{(${W})(${Math.cos(\n theta\n).toFixed(3)})}{${A0.toFixed(3)}} = ${E.toFixed(3)} \\frac{W}{m}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2017,"value":"md`Try moving the slider in the construction to change ${tex`\\theta`}. Notice how ${tex`E`} changes in proportion with ${tex`\\cos(\\theta)`} just as Lambert's cosine law states.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2038,"value":"md`# Appendix`","pinned":false,"hidden":false,"mode":"ojs"},{"id":836,"value":"$31","pinned":false,"hidden":false,"mode":"ojs"},{"id":1253,"value":"mutable theta = Math.PI / 4","pinned":false,"hidden":false,"mode":"ojs"},{"id":1858,"value":"mutable A0 = 110.60660171779821","pinned":false,"hidden":false,"mode":"ojs"},{"id":1905,"value":"J = 216000","pinned":false,"hidden":false,"mode":"ojs"},{"id":1908,"value":"s = 3600","pinned":false,"hidden":false,"mode":"ojs"},{"id":1900,"value":"W = J / s","pinned":false,"hidden":false,"mode":"ojs"},{"id":1958,"value":"E = ((W * Math.cos(theta)) / A0) * 10","pinned":false,"hidden":false,"mode":"ojs"},{"id":824,"value":"JXG = {\n const JXG = await require('https://cdnjs.cloudflare.com/ajax/libs/jsxgraph/0.99.7/jsxgraphcore.js');\n window.JXG = JXG;\n return JXG;\n}","pinned":false,"hidden":false,"mode":"ojs"}],"files":"$W32","stdlib":"1","titling":"auto"}