{ "cells": [ { "cell_type": "markdown", "id": "markdown-000", "metadata": {}, "source": [ "# Wave properties of light\n", "\n", "[Read the lesson](https://tirta.co/a-byte-of-physics/wave-properties-of-light). This notebook includes the lesson text and its Python examples in reading order. Run the setup cell, then run the remaining cells from top to bottom. In Colab, use File → Save a copy in Drive to keep your edits. You can run and edit this view without changing the course source. Examples use illustrative data unless stated otherwise." ] }, { "cell_type": "code", "id": "code-001", "metadata": {}, "source": [ "import importlib.metadata\n", "import subprocess\n", "import sys\n", "\n", "requirements = {\"numpy\": (2, 0), \"sympy\": (1, 13), \"matplotlib\": (3, 8)}\n", "to_install = []\n", "for package, minimum in requirements.items():\n", " try:\n", " installed = importlib.metadata.version(package)\n", " version = tuple(int(part) for part in installed.split(\".\")[:2])\n", " except importlib.metadata.PackageNotFoundError:\n", " version = (0, 0)\n", " if version < minimum:\n", " to_install.append(f\"{package}>={minimum[0]}.{minimum[1]}\")\n", "if to_install:\n", " subprocess.check_call([sys.executable, \"-m\", \"pip\", \"install\", *to_install])\n", "\n", "import numpy as np\n", "import sympy as sp\n", "import matplotlib\n", "import matplotlib.pyplot as plt\n", "\n", "print(\"Python\", sys.version.split()[0])\n", "print(\"NumPy\", np.__version__)\n", "print(\"SymPy\", sp.__version__)\n", "print(\"Matplotlib\", matplotlib.__version__)\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-002", "metadata": {}, "source": [ "Light is an electromagnetic wave with frequency, wavelength, phase, amplitude, direction,\n", "and polarization. These properties are related but not interchangeable. Frequency sets\n", "the rate of field oscillation. Wavelength sets spatial repetition. Amplitude controls\n", "field strength and, through its square, intensity. Phase locates a wave within its cycle.\n", "\n", "The values needed in a model depend on the question. Refraction needs frequency and a\n", "material index. Interference needs relative phase and coherence. Diffraction needs\n", "wavelength and aperture geometry. Polarization needs two transverse components.\n", "\n", "## Frequency, period, wavelength, and speed\n", "\n", "For a periodic wave,\n", "\n", "\n", "$$\n", "f=\\frac{1}{T}\n", "$$\n", "\n", "\n", "where frequency $f$ is measured in hertz\n", "and period $T$ in seconds. Wave speed,\n", "wavelength, and frequency obey\n", "\n", "\n", "$$\n", "v=\\lambda f\n", "$$\n", "\n", "\n", "In vacuum,\n", "\n", "\n", "$$\n", "c=299\\,792\\,458\\ \\mathrm{m/s}\n", "$$\n", "\n", "\n", "and\n", "\n", "\n", "$$\n", "c=\\lambda_0f\n", "$$\n", "\n", "\n", "The vacuum light speed is exact in SI. The vacuum wavelength\n", "$\\lambda_0$ follows from the frequency.\n", "\n", "## A sinusoidal field and phase\n", "\n", "A linearly polarized plane-wave component can be written\n", "\n", "\n", "$$\n", "E(x,t)=E_0\\cos(kx-\\omega t+\\phi_0)\n", "$$\n", "\n", "\n", "The wave number and angular frequency are\n", "\n", "\n", "$$\n", "k=\\frac{2\\pi}{\\lambda},\\qquad\n", " \\omega=2\\pi f\n", "$$\n", "\n", "\n", "The phase is\n", "\n", "\n", "$$\n", "\\phi(x,t)=kx-\\omega t+\\phi_0\n", "$$\n", "\n", "\n", "Increasing position by one wavelength or time by one period changes phase by a whole\n", "cycle. A common phase shift changes where the cycle starts but not frequency, wavelength,\n", "amplitude, or polarization shape." ] }, { "cell_type": "code", "id": "code-003", "metadata": {}, "source": [ "import numpy as np\n", "\n", "c, frequency = 299792458.0, 5e14\n", "wavelength = c / frequency\n", "k, omega = 2 * np.pi / wavelength, 2 * np.pi * frequency\n", "position, time = 1e-6, 1e-15\n", "phase = k * position - omega * time\n", "print(f\"Vacuum wavelength: {wavelength*1e9:.2f} nm\")\n", "print(f\"Phase: {phase:.3f} rad\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-004", "metadata": {}, "source": [ "## Propagation through a medium\n", "\n", "For an ideal medium with refractive index\n", "$n$,\n", "\n", "\n", "$$\n", "n=\\frac{c}{v_p}\n", "$$\n", "\n", "\n", "so the phase velocity and medium wavelength are\n", "\n", "\n", "$$\n", "v_p=\\frac{c}{n},\\qquad\n", " \\lambda=\\frac{\\lambda_0}{n}\n", "$$\n", "\n", "\n", "At a stationary interface, frequency stays the same because the fields on both sides\n", "must share the same temporal oscillation. Speed and wavelength change together. A\n", "photon's energy also depends on frequency, so it does not drop merely because wavelength\n", "is shorter in the medium.\n", "\n", "## Worked wavelength conversion\n", "\n", "Take frequency\n", "$f=6.00\\times10^{14}\\ \\mathrm{Hz}$.\n", "Its vacuum wavelength is\n", "\n", "\n", "$$\n", "\\lambda_0=\\frac{c}{f}\n", " =\\frac{2.998\\times10^8\\ \\mathrm{m/s}}\n", " {6.00\\times10^{14}\\ \\mathrm{Hz}}\n", " \\approx5.00\\times10^{-7}\\ \\mathrm m\n", "$$\n", "\n", "\n", "This is five hundred nanometres. In an ideal medium with\n", "$n=1.50$, the phase speed\n", "is about $2.00\\times10^8\\ \\mathrm{m/s}$\n", "and the wavelength is about\n", "$333\\ \\mathrm{nm}$.\n", "The frequency remains the original value.\n", "\n", "## Transverse fields\n", "\n", "In a source-free vacuum plane wave, the electric field, magnetic field, and propagation\n", "direction are mutually perpendicular:\n", "\n", "\n", "$$\n", "\\mathbf E\\perp\\mathbf B,\n", " \\qquad\n", " \\mathbf E\\perp\\mathbf k,\n", " \\qquad\n", " \\mathbf B\\perp\\mathbf k\n", "$$\n", "\n", "\n", "Their amplitudes satisfy\n", "\n", "\n", "$$\n", "E_0=cB_0\n", "$$\n", "\n", "\n", "and $\\mathbf E\\times\\mathbf B$\n", "points with energy propagation. Light has two independent transverse polarization\n", "directions.\n", "\n", "## Amplitude and intensity\n", "\n", "For a sinusoidal vacuum plane wave, average intensity is\n", "\n", "\n", "$$\n", "I=\\frac{1}{2}\\varepsilon_0cE_0^2\n", " =\\varepsilon_0cE_{\\mathrm{rms}}^2\n", "$$\n", "\n", "\n", "Doubling electric-field amplitude quadruples intensity. Intensity is an energy-flow rate\n", "per area, not field amplitude. The peak amplitude and root-mean-square amplitude differ\n", "by a factor of the square root of two for a sinusoid, so formulas must identify which one\n", "they use.\n", "\n", "## Electromagnetic spectrum\n", "\n", "Radio, microwave, infrared, visible, ultraviolet, X-ray, and gamma radiation are ranges\n", "of the same electromagnetic phenomenon. Boundaries between names are conventional and\n", "overlap across applications. Frequency or vacuum wavelength is a clearer physical label\n", "than a band name.\n", "\n", "Visible color perception depends on the spectrum reaching the eye and on biological\n", "response. Monochromatic wavelength is not the same thing as a screen RGB value. Many\n", "different spectra can produce similar perceived color, and wavelengths outside the\n", "visible range have no visible color.\n", "\n", "The quantum energy of one photon is\n", "\n", "\n", "$$\n", "E_{\\mathrm{photon}}=hf=\\frac{hc}{\\lambda_0}\n", "$$\n", "\n", "\n", "Greater frequency means greater energy per photon. Classical wave intensity instead\n", "describes energy per area per time and can vary independently through photon arrival rate.\n", "\n", "## Dispersion and group velocity\n", "\n", "Real refractive index often depends on angular frequency:\n", "\n", "\n", "$$\n", "n=n(\\omega)\n", "$$\n", "\n", "\n", "The phase velocity is\n", "\n", "\n", "$$\n", "v_p=\\frac{\\omega}{k}\n", "$$\n", "\n", "\n", "while the group velocity of a narrowband wave packet is\n", "\n", "\n", "$$\n", "v_g=\\frac{d\\omega}{dk}\n", "$$\n", "\n", "\n", "In a nondispersive vacuum they are equal. In a dispersive medium they can differ, causing\n", "frequency components to accumulate different phases and pulses to broaden or reshape.\n", "Neither quantity should be described casually as information speed in every strongly\n", "absorbing or anomalously dispersive situation.\n", "\n", "## Coherence\n", "\n", "Interference requires stable relative phase. Temporal coherence describes how long a\n", "wave remains phase-correlated with a delayed copy of itself. Spatial coherence describes\n", "phase correlation across different positions on a wavefront. A perfectly monochromatic\n", "infinite plane wave is completely coherent but unphysical; real sources have finite\n", "bandwidth, duration, and extent.\n", "\n", "A rough narrowband coherence length scale is\n", "\n", "\n", "$$\n", "L_c\\sim\\frac{c}{\\Delta f}\n", "$$\n", "\n", "\n", "The numerical factor and detailed shape depend on how bandwidth and coherence are defined.\n", "\n", "## Polarization\n", "\n", "Polarization records the relative amplitudes and phase of two transverse electric-field\n", "components. Linear polarization traces a line, circular polarization traces a circle,\n", "and the general state traces an ellipse. Unpolarized or partially polarized light requires\n", "statistical information rather than one deterministic field pair.\n", "\n", "Polarization is independent of frequency in the sense that waves of many frequencies can\n", "share a polarization state, but real optical elements transform polarization differently\n", "across wavelength.\n", "\n", "## Wave packets and finite signals\n", "\n", "A sinusoidal wave extends forever and carries no localized message. A finite pulse is a\n", "superposition of frequencies. Narrowing a pulse in time generally requires broader\n", "frequency bandwidth. The phase relationships among those frequencies determine the pulse\n", "shape.\n", "\n", "A pulse can also have a carrier frequency and a slowly varying envelope. Sampling must\n", "resolve the carrier if the instantaneous field matters, or use a justified envelope model\n", "when only slower behavior matters.\n", "\n", "## Verification and limits\n", "\n", "Check units in $v=\\lambda f$. At fixed\n", "frequency, lowering phase speed must shorten wavelength. Advancing one wavelength or one\n", "period must change phase by a full cycle. For a vacuum plane wave, verify transverse fields,\n", "the amplitude ratio, and energy-flow direction.\n", "\n", "The simple relations assume linear propagation and clearly defined frequency. Real media\n", "can absorb, scatter, disperse, be anisotropic, move, or respond nonlinearly. Very short\n", "pulses need broadband models. Classical waves predict field propagation and interference;\n", "quantum models are needed for discrete photon emission, absorption, and detection. The\n", "two descriptions agree on many measurable averages but answer different questions.\n", "\n", "## A worked Python notebook\n", "\n", "Convert wavelengths across a visible-light example range into frequencies. The question for this calculation is precise: how does **frequency**\n", "change when we change **vacuum wavelength**, while keeping the other choices fixed?\n", "The earlier sections explain the physical relationship. Here we turn one part of that\n", "relationship into a calculation that you can run, inspect, and change in Google Colab.\n", "\n", "Wavelength changes on entering a medium, while frequency is set by the source and interface conditions. Keep that sentence beside the results. A formula can return a number outside\n", "its useful range, and a plotting library will usually draw that number without asking\n", "whether the physical assumptions still hold. The selected interval is from\n", "4e-07 to 7e-07 m. It gives us a specific domain to discuss\n", "instead of leaving the graph's range to a software default.\n", "\n", "Use the notebook link above the lesson, run the setup cell, and then run the cells in\n", "order. The code below is also included there. NumPy handles the arrays of numerical\n", "values, SymPy keeps the equation symbolic, and Matplotlib draws the results. If these\n", "libraries or notebook cells are unfamiliar, start with\n", "[Your first physics notebook](https://tirta.co/a-byte-of-physics/your-first-physics-notebook).\n", "\n", "### State the relationship before choosing a graph\n", "\n", "In this small example, `q` stands for vacuum wavelength, expressed in m. The output is\n", "frequency, expressed in Hz. The equation below uses the numerical values in those\n", "units. Any fixed coefficients therefore carry the units needed to make the relation\n", "consistent. A value supplied in another unit must be converted before it enters this\n", "expression.\n", "\n", "\n", "$$\n", "f(q) = \\frac{299792458}{q}\n", "$$\n", "\n", "\n", "This is a function of one changing input. It is not a statement that the physical\n", "system has only one variable. We deliberately hold the other inputs fixed so that one\n", "cause of variation is visible. Changing several inputs at once would make it harder\n", "to tell which change produced the output difference. Write down the held-fixed choices\n", "before trying a different experiment.\n", "\n", "At 5e-07 m, the expression gives 5.99585e+14 Hz. This is our\n", "reference calculation. It is a prediction of the stated example, not a new measurement.\n", "Its displayed digits help us compare implementations. They should not be read as a\n", "claim that physical inputs are known to six significant figures." ] }, { "cell_type": "code", "id": "code-005", "metadata": {}, "source": [ "import numpy as np\n", "import sympy as sp\n", "import matplotlib.pyplot as plt\n", "\n", "q = sp.Symbol(\"q\", real=True)\n", "expression = sp.sympify(\"299792458/q\", locals={\"q\": q})\n", "reference_input = 5e-07\n", "reference_output = float(expression.subs(q, reference_input))\n", "print(\"Input:\", reference_input, \"m\")\n", "print(\"Predicted frequency:\", reference_output, \"Hz\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-006", "metadata": {}, "source": [ "`sp.Symbol` declares a mathematical symbol. It does not assign a numerical measurement\n", "to that symbol. `subs` then replaces the symbol with our chosen input, and `float`\n", "converts the resulting symbolic number into a standard Python numerical value.\n", "Keeping these steps separate lets us inspect the equation before numerical substitution.\n", "It also prevents a later plot from becoming the only surviving record of the model.\n", "\n", "### Compare a few cases you can explain\n", "\n", "Before drawing hundreds of points, calculate three. The lower endpoint gives\n", "7.49481e+14 Hz. The midpoint, at 5.5e-07 m, gives\n", "5.45077e+14 Hz. The upper endpoint gives 4.28275e+14 Hz.\n", "Read these together with the assumptions: they describe different choices of vacuum wavelength\n", "within the same example, with the other inputs held fixed.\n", "\n", "These three values are a useful first check, but they do not establish the shape between\n", "them. Two endpoints can be equal even when the curve bends or oscillates in between.\n", "Likewise, a large change between endpoints does not tell us whether the relationship is\n", "linear. We need the equation and intermediate samples to answer those questions." ] }, { "cell_type": "code", "id": "code-007", "metadata": {}, "source": [ "evaluate = sp.lambdify(q, expression, modules=\"numpy\")\n", "inputs = np.array([4e-07, 5.5e-07, 7e-07])\n", "outputs = np.asarray(evaluate(inputs), dtype=float)\n", "outputs = np.broadcast_to(outputs, inputs.shape)\n", "for input_value, output_value in zip(inputs, outputs):\n", " print(f\"{input_value:.6g} m -> {output_value:.6g} Hz\")\n", "assert np.all(np.isfinite(outputs))\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-008", "metadata": {}, "source": [ "`lambdify` turns the symbolic expression into a numerical function. We explicitly select\n", "NumPy as its numerical backend so it can accept an array. The same expression now has\n", "two useful forms: one for algebra and one for calculating many cases. The assertion\n", "checks that the chosen inputs produce finite outputs. It would catch an infinity or an\n", "undefined numerical result, but a finite value can still be physically inappropriate.\n", "\n", "### Make a graph that answers the question\n", "\n", "The horizontal axis shows vacuum wavelength; the vertical axis shows frequency. Every plotted\n", "point is a prediction of the same expression. We mark the reference case separately\n", "so that a reader can connect the earlier printed value to its place on the curve.\n", "The axis labels include units because a shape without a scale cannot tell us the size\n", "of a physical effect." ] }, { "cell_type": "code", "id": "code-009", "metadata": {}, "source": [ "input_grid = np.linspace(4e-07, 7e-07, 301)\n", "output_grid = np.asarray(evaluate(input_grid), dtype=float)\n", "output_grid = np.broadcast_to(output_grid, input_grid.shape)\n", "fig, ax = plt.subplots(figsize=(7, 4))\n", "ax.plot(input_grid, output_grid, label=\"Stated model\")\n", "ax.scatter([reference_input], [reference_output], label=\"Reference case\", zorder=3)\n", "ax.set_xlabel(\"Vacuum wavelength (m)\")\n", "ax.set_ylabel(\"Frequency (Hz)\")\n", "ax.legend()\n", "ax.grid(alpha=0.25)\n", "fig.tight_layout()\n", "plt.show()\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-010", "metadata": {}, "source": [ "Look for a sign change, a turning point, or a region where the curve becomes steep.\n", "Connect each feature to the expression rather than to the colour of the line. If the\n", "curve has a minimum or maximum, changing the input in opposite directions may have\n", "similar effects on the output near that point. If it is nearly straight over a short\n", "interval, a local linear approximation may be useful even when the complete expression\n", "is nonlinear.\n", "\n", "Increasing the number of samples makes the line smoother and may reveal features that\n", "a coarse grid missed. It does not improve the physical inputs or enlarge the model's\n", "validity range. For quantities that can take only discrete values, use the line as a\n", "guide to the relationship and interpret only the allowed input values. A plotting\n", "function does not enforce such restrictions for you.\n", "\n", "### Measure the local sensitivity\n", "\n", "A derivative asks a more specific question than the whole graph: how quickly does\n", "frequency change near the reference input? Its units are the output unit divided by\n", "the input unit. For this calculation, the derivative at 5e-07 m is\n", "-1.19917e+21 in those ratio units. Locally, the output decreases as the\n", "input increases. This statement is local; the sign or size of the slope can be different\n", "elsewhere on the curve.\n", "\n", "\n", "$$\n", "f'(q) = - \\frac{299792458}{q^{2}}\n", "$$\n", "\n", "\n", "For a small input change, multiply the derivative by that change to estimate the output\n", "change. Then compare that estimate with a fresh evaluation of the full expression.\n", "The distinction between these two calculations matters: the derivative gives a tangent\n", "approximation, while substitution evaluates the chosen expression at the new point." ] }, { "cell_type": "code", "id": "code-011", "metadata": {}, "source": [ "slope_expression = sp.diff(expression, q)\n", "slope_at_reference = float(slope_expression.subs(q, reference_input))\n", "changed_input = 5.015e-07\n", "linear_prediction = reference_output + slope_at_reference * (\n", " changed_input - reference_input\n", ")\n", "full_prediction = float(evaluate(changed_input))\n", "print(\"Symbolic derivative:\", slope_expression)\n", "print(\"Tangent prediction:\", linear_prediction, \"Hz\")\n", "print(\"Full expression:\", full_prediction, \"Hz\")\n", "print(\"Approximation difference:\", full_prediction - linear_prediction, \"Hz\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-012", "metadata": {}, "source": [ "Here the changed input is 5.015e-07 m. The tangent estimate is\n", "5.97786e+14 Hz, while the full expression gives\n", "5.97792e+14 Hz. If the two values are close, it means the chosen\n", "input change is small enough for this local approximation at this point. It does not\n", "mean the entire model is linear. If the slope is zero, a quadratic or higher-order term\n", "may supply the leading change instead.\n", "\n", "This is also the beginning of uncertainty propagation. If the input has a small stated\n", "standard uncertainty, the magnitude of the slope times that uncertainty estimates one\n", "contribution to output uncertainty. Do not automatically use the same rule for a wide\n", "interval, a discontinuity, a turning point with zero first derivative, or several\n", "correlated uncertain inputs. Those situations need more of the model than one slope.\n", "\n", "### Check the derivative with numerical differences\n", "\n", "We can estimate the same slope from values on either side of the reference point.\n", "Subtract the lower output from the upper output and divide by twice the input step.\n", "Parentheses are essential: the entire output difference must be divided by the entire\n", "input interval. Without them, Python can execute a different mathematical expression\n", "while still producing a plausible number.\n", "\n", "The calculation below uses three decreasing step sizes. Each estimate is compared with\n", "the symbolic derivative, so the reference is independent of the numerical difference\n", "formula. Near an endpoint, it is acceptable here to evaluate a small extension only\n", "because the displayed mathematical expression is smooth there; such a check does not\n", "assert that extrapolated values describe additional physical cases." ] }, { "cell_type": "code", "id": "code-013", "metadata": {}, "source": [ "input_span = 3e-07\n", "for fraction in [1e-2, 1e-3, 1e-4]:\n", " step = input_span * fraction\n", " upper = float(evaluate(reference_input + step))\n", " lower = float(evaluate(reference_input - step))\n", " numerical_slope = (upper - lower) / (2 * step)\n", " difference = numerical_slope - slope_at_reference\n", " print(f\"step={step:.6g}: slope={numerical_slope:.6g}, difference={difference:.6g}\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-014", "metadata": {}, "source": [ "For a smooth function, reducing the step often reduces the approximation error at first.\n", "Very small steps can eventually make subtraction of nearly equal floating-point values\n", "less reliable. Therefore “smaller” is not an unlimited guarantee of “better.” Compare\n", "the observed behaviour over a useful range of steps instead of choosing the smallest\n", "number the computer can represent.\n", "\n", "An absolute slope difference is useful when the expected slope is near zero. A relative\n", "difference would divide by a very small reference and could look enormous even when the\n", "absolute discrepancy is tiny. Away from zero, a relative difference can help compare\n", "quantities with very different scales. The tolerance should reflect the question and\n", "the numerical method, rather than a universal decimal threshold copied into every test.\n", "\n", "### Change the example and preserve the reasoning\n", "\n", "Make one copy of the reference calculation before editing it. Choose a second value of\n", "vacuum wavelength inside the stated interval, predict whether frequency will rise or fall,\n", "and then evaluate the expression. Place the prediction and the result in a text cell.\n", "If they disagree, identify whether the mistake was in the sign, the unit conversion,\n", "the algebra, or the assumption about how the curve behaves.\n", "\n", "Next choose a shorter plotting interval around the reference point. Compare the local\n", "curve with its tangent approximation. Then widen the interval again. The difference\n", "shows why a useful local estimate need not describe the whole range. Keep the axis\n", "labels and numerical scale visible when comparing the plots; automatic axis limits can\n", "make two different effects appear equally large.\n", "\n", "Finally, return to the physical limit stated at the start: Wavelength changes on entering a medium, while frequency is set by the source and interface conditions. Describe one change\n", "to the setup that would violate that limit. State what new input, equation, or observation\n", "would be needed before trusting an answer in the changed situation. This last step\n", "connects the Python experiment to the physics instead of treating successful execution\n", "as the only criterion for a useful result.\n", "\n", "Before saving, restart the runtime and run all cells in order. A result that depends on\n", "an earlier, now-deleted input cell is not reproducible from the saved notebook. Keep\n", "unrounded values for later calculations, round only the reported output, and label any\n", "new data as measured, assumed, or generated. The graph and printed values should remain\n", "traceable to the same expression and the same choices of inputs.\n", "\n", "### Recover the output change by integrating the slope\n", "\n", "There is another way to check that the expression and its derivative describe the same\n", "relationship. Add up the small output changes predicted by the slope across the chosen\n", "input interval. In the continuous limit, this integral equals the output at the upper\n", "endpoint minus the output at the lower endpoint. The result has units of Hz:\n", "the input unit in the integration cancels the input unit in the denominator of the slope.\n", "\n", "This statement is about the mathematics of this particular relationship. It does not\n", "assign a new physical meaning to an arbitrary area under a graph. For example, an area\n", "under force versus distance represents work, while an area under a different pair of\n", "quantities may represent something else or have no standard name. Always carry the\n", "axis units through the calculation before naming an integral." ] }, { "cell_type": "code", "id": "code-015", "metadata": {}, "source": [ "slope_function = sp.lambdify(q, slope_expression, modules=\"numpy\")\n", "endpoint_change = float(evaluate(7e-07) - evaluate(4e-07))\n", "for points in [101, 201, 401]:\n", " grid = np.linspace(4e-07, 7e-07, points)\n", " slopes = np.asarray(slope_function(grid), dtype=float)\n", " slopes = np.broadcast_to(slopes, grid.shape)\n", " integrated_change = np.trapezoid(slopes, grid)\n", " print(\n", " points,\n", " \"integrated change:\",\n", " integrated_change,\n", " \"endpoint change:\",\n", " endpoint_change,\n", " )\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-016", "metadata": {}, "source": [ "The trapezoidal rule joins neighbouring slope samples with straight segments and adds\n", "their signed areas. A negative contribution remains negative; replacing every slope\n", "by its magnitude would calculate a different quantity. Increasing the sample count\n", "provides a practical refinement check. If the values do not settle toward the endpoint\n", "difference, first inspect the interval, singularities, and array ordering.\n", "\n", "Do not conclude that two agreeing calculations prove the model is physically complete.\n", "Both use the same selected relationship, so an omitted effect would be omitted from\n", "both. Their agreement is evidence that the numerical tools implement that relationship\n", "consistently. Comparing with an independent experiment or a more appropriate physical\n", "model is a separate test.\n", "\n", "Keep this comparison in the notebook when you change the expression. It is particularly\n", "useful after adding a coefficient, changing a sign convention, or converting an input\n", "unit. A code cell that produces a smooth graph can conceal these mistakes; a second\n", "calculation of the same output change gives you another place to notice them.\n", "\n", "## Sources\n", "\n", "- [OpenStax, *University Physics Volume 2*, Chapter 16, “Electromagnetic Waves”](https://openstax.org/books/university-physics-volume-2/pages/16-introduction).\n", "- [OpenStax, *University Physics Volume 3*, Chapter 1, “The Nature of Light”](https://openstax.org/books/university-physics-volume-3/pages/1-introduction).\n", "- [Benjamin Crowell, *Light and Matter*](https://open.umn.edu/opentextbooks/textbooks/light-and-matter).\n", "- [Ralph A. Llewellyn and Paul A. Tipler, *Modern Physics*](https://web.pdx.edu/~pmoeck/books/Tipler_Llewellyn.pdf).\n", "- [Physics LibreTexts, “Optics”](https://phys.libretexts.org/Bookshelves/Optics)." ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "name": "python", "version": "3.12" }, "colab": { "name": "wave-properties-of-light.ipynb", "provenance": [] }, "tirta": { "lesson": "https://tirta.co/a-byte-of-physics/wave-properties-of-light", "source_sha256": "19b7c7ee4d8567bf7cf1dc79e04126dbcc5f500e69be689761e98561f9fd048a" } }, "nbformat": 4, "nbformat_minor": 5 }