{ "cells": [ { "cell_type": "markdown", "id": "markdown-000", "metadata": {}, "source": [ "# Electromagnetic wave solutions\n", "\n", "[Read the lesson](https://tirta.co/a-byte-of-physics/em-wave-solutions). 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": [ "Maxwell's equations allow electric and magnetic fields to propagate through empty space\n", "without nearby charge or current. A changing electric field produces magnetic curl, and\n", "a changing magnetic field produces electric curl. The coupled disturbance travels at\n", "the speed of light.\n", "\n", "A plane wave is the simplest exact solution. It extends infinitely across planes\n", "perpendicular to its direction of travel, so it is an idealization, but it exposes the\n", "essential geometry, phase, amplitude relation, and wave speed used to understand more\n", "realistic beams and pulses.\n", "\n", "## Wave equations from Maxwell's equations\n", "\n", "In vacuum without charge or current, Maxwell's equations include\n", "\n", "\n", "$$\n", "\\boldsymbol\\nabla\\times\\mathbf E\n", " =-\\frac{\\partial\\mathbf B}{\\partial t}\n", "$$\n", "\n", "\n", "and\n", "\n", "\n", "$$\n", "\\boldsymbol\\nabla\\times\\mathbf B\n", " =\\mu_0\\varepsilon_0\n", " \\frac{\\partial\\mathbf E}{\\partial t}\n", "$$\n", "\n", "\n", "Taking another curl, substituting the companion equation, and using the zero-divergence\n", "constraints gives\n", "\n", "\n", "$$\n", "\\nabla^2\\mathbf E\n", " =\\mu_0\\varepsilon_0\n", " \\frac{\\partial^2\\mathbf E}{\\partial t^2}\n", "$$\n", "\n", "\n", "and\n", "\n", "\n", "$$\n", "\\nabla^2\\mathbf B\n", " =\\mu_0\\varepsilon_0\n", " \\frac{\\partial^2\\mathbf B}{\\partial t^2}\n", "$$\n", "\n", "\n", "Comparing these with the standard wave equation identifies the propagation speed\n", "\n", "\n", "$$\n", "c=\\frac{1}{\\sqrt{\\mu_0\\varepsilon_0}}\n", "$$\n", "\n", "\n", "The fact that this value matches the measured speed of light is the central theoretical\n", "connection: light is an electromagnetic wave.\n", "\n", "## A sinusoidal plane wave\n", "\n", "Consider a wave traveling in positive x. One valid choice of fields is\n", "\n", "\n", "$$\n", "\\mathbf E(x,t)\n", " =E_0\\cos(kx-\\omega t+\\phi_0)\\,\n", " \\hat{\\mathbf y}\n", "$$\n", "\n", "\n", "and\n", "\n", "\n", "$$\n", "\\mathbf B(x,t)\n", " =B_0\\cos(kx-\\omega t+\\phi_0)\\,\n", " \\hat{\\mathbf z}\n", "$$\n", "\n", "\n", "The wave number, angular frequency, wavelength, frequency, and period satisfy\n", "\n", "\n", "$$\n", "k=\\frac{2\\pi}{\\lambda},\\qquad\n", " \\omega=2\\pi f,\\qquad\n", " c=\\frac{\\omega}{k}=\\lambda f\n", "$$\n", "\n", "\n", "Changing the phase constant $\\phi_0$\n", "shifts the pattern in space or time without changing its wavelength, frequency, or\n", "amplitude.\n", "\n", "## Field geometry\n", "\n", "For a source-free plane wave,\n", "\n", "\n", "$$\n", "\\mathbf k\\cdot\\mathbf E=0,\n", " \\qquad\n", " \\mathbf k\\cdot\\mathbf B=0,\n", " \\qquad\n", " \\mathbf E\\cdot\\mathbf B=0\n", "$$\n", "\n", "\n", "The electric field, magnetic field, and propagation direction are mutually perpendicular.\n", "Their orientation follows\n", "\n", "\n", "$$\n", "\\hat{\\mathbf k}\n", " =\\frac{\\mathbf E\\times\\mathbf B}\n", " {|\\mathbf E\\times\\mathbf B|}\n", "$$\n", "\n", "\n", "For the fields above, positive y crossed with positive z gives positive x. Reversing one\n", "field while leaving the other fixed reverses the direction of energy propagation and no\n", "longer describes the same positive-x solution.\n", "\n", "Maxwell's curl equations also fix the amplitude ratio:\n", "\n", "\n", "$$\n", "E_0=cB_0\n", "$$\n", "\n", "\n", "Electric field and magnetic field reach maxima, minima, and zero at the same phase in a\n", "traveling vacuum plane wave.\n", "\n", "## Reading the phase\n", "\n", "A surface of constant phase satisfies\n", "\n", "\n", "$$\n", "kx-\\omega t+\\phi_0=\\mathrm{constant}\n", "$$\n", "\n", "\n", "Differentiating this condition gives\n", "\n", "\n", "$$\n", "\\frac{dx}{dt}=\\frac{\\omega}{k}=c\n", "$$\n", "\n", "\n", "The sign in the phase tells the travel direction. A pattern\n", "$\\cos(kx-\\omega t)$ moves toward\n", "positive x, while $\\cos(kx+\\omega t)$\n", "moves toward negative x. This is easier to verify by tracking a constant phase than by\n", "memorizing the signs.\n", "\n", "## Worked frequency and field example\n", "\n", "Take a vacuum wave with wavelength\n", "$600\\ \\mathrm{nm}$.\n", "Its frequency is\n", "\n", "\n", "$$\n", "f=\\frac{c}{\\lambda}\n", " =\\frac{2.998\\times10^8\\ \\mathrm{m/s}}\n", " {600\\times10^{-9}\\ \\mathrm m}\n", " \\approx5.00\\times10^{14}\\ \\mathrm{Hz}\n", "$$\n", "\n", "\n", "If the electric amplitude is\n", "$E_0=300\\ \\mathrm{V/m}$,\n", "the magnetic amplitude is\n", "\n", "\n", "$$\n", "B_0=\\frac{E_0}{c}\n", " \\approx1.00\\times10^{-6}\\ \\mathrm T\n", "$$\n", "\n", "\n", "The ratio is fixed by vacuum propagation, not by independently choosing two amplitudes.\n", "\n", "## Evaluating a plane wave in Python" ] }, { "cell_type": "code", "id": "code-003", "metadata": {}, "source": [ "import numpy as np\n", "\n", "c, frequency, amplitude = 299792458.0, 1e6, 10.0\n", "position, time = 100.0, 1e-7\n", "k, omega = 2 * np.pi * frequency / c, 2 * np.pi * frequency\n", "phase = k * position - omega * time\n", "electric = np.array([0.0, amplitude * np.cos(phase), 0.0])\n", "magnetic = np.cross([1.0, 0.0, 0.0], electric) / c\n", "print(\"E (V/m):\", electric, \"B (T):\", magnetic)\n", "assert np.isclose(np.linalg.norm(electric), c * np.linalg.norm(magnetic))\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-004", "metadata": {}, "source": [ "Constructing the magnetic field from the cross product enforces its direction relative\n", "to propagation and the electric field. Still check that the electric amplitude is\n", "perpendicular to the propagation vector; otherwise the input is not a source-free plane\n", "wave.\n", "\n", "## General vector form\n", "\n", "For propagation wave vector $\\mathbf k$,\n", "a complex notation is often convenient:\n", "\n", "\n", "$$\n", "\\mathbf E(\\mathbf r,t)\n", " =\\operatorname{Re}\\left\\{\n", " \\mathbf E_0e^{i(\\mathbf k\\cdot\\mathbf r-\\omega t)}\n", " \\right\\}\n", "$$\n", "\n", "\n", "The physical field is the real part. Complex amplitudes simplify phase relations and\n", "polarization; they do not mean the measured electric field has an imaginary component.\n", "In vacuum,\n", "\n", "\n", "$$\n", "\\mathbf B_0\n", " =\\frac{1}{\\omega}\\mathbf k\\times\\mathbf E_0\n", " =\\frac{1}{c}\\hat{\\mathbf k}\\times\\mathbf E_0\n", "$$\n", "\n", "\n", "## Waves in a simple material\n", "\n", "In a linear, homogeneous, isotropic, nonconducting medium with permittivity\n", "$\\varepsilon$ and permeability\n", "$\\mu$, the ideal wave speed is\n", "\n", "\n", "$$\n", "v=\\frac{1}{\\sqrt{\\mu\\varepsilon}}\n", "$$\n", "\n", "\n", "The field ratio becomes the wave impedance\n", "\n", "\n", "$$\n", "\\frac{E_0}{H_0}\n", " =\\sqrt{\\frac{\\mu}{\\varepsilon}}\n", "$$\n", "\n", "\n", "Real materials can be dispersive, so permittivity and permeability depend on frequency.\n", "Then phase velocity, group velocity, absorption, and pulse shape require more than one\n", "constant refractive index.\n", "\n", "## Plane waves, beams, and pulses\n", "\n", "An infinite plane wave has infinite transverse extent and infinite total energy, so no\n", "finite source produces it exactly. A wide beam can resemble a plane wave near its center\n", "over a limited region. A pulse is a superposition of frequencies and occupies a finite\n", "range of positions.\n", "\n", "In a nondispersive vacuum, all frequency components have the same speed. In a dispersive\n", "medium, different frequencies accumulate different phase, so a pulse can spread or\n", "reshape. The simple sinusoidal solution remains a useful basis, but it is not a complete\n", "description of every signal.\n", "\n", "## Initial fields must form one consistent wave\n", "\n", "Specifying an electric-field shape at one instant is not enough to choose whether a wave\n", "travels left, travels right, or forms a standing pattern. The accompanying magnetic field\n", "and time derivative carry that information. For the positive-x wave above, the magnetic\n", "field must have the stated magnitude, phase, and positive-z direction. Reversing its sign\n", "while retaining the electric field produces a negative-x traveling solution instead.\n", "\n", "This matters when constructing a pulse. An arbitrary localized electric profile paired\n", "with zero magnetic field is generally not a single traveling pulse. It decomposes into\n", "left- and right-traveling parts so that Maxwell's equations and the initial conditions can\n", "both hold. A calculation that seeds only the shape visible on screen can therefore split\n", "even when the numerical method is correct.\n", "\n", "Boundary data must be compatible as well. Fixing every field component independently at\n", "an entrance can overconstrain Maxwell's equations. A meaningful incoming-wave boundary\n", "prescribes independent wave information and lets the field relations set the rest. At an\n", "open edge, an absorbing condition attempts to let outgoing energy leave without\n", "reflection; it is an approximation and should be tested across the frequencies and\n", "angles present in the signal.\n", "\n", "Standing waves arise when equal-frequency waves travel in opposite directions. Their\n", "electric and magnetic patterns need not peak at the same positions or times as one\n", "another, even though each traveling component has fields in phase. Thus the statement\n", "that electric and magnetic fields are in phase applies to a single traveling plane wave,\n", "not automatically to every electromagnetic field configuration.\n", "\n", "When checking a computed wave, compare more than its crest location. Check wavelength,\n", "frequency, propagation direction, field ratio, transverse geometry, phase, and energy\n", "flux. A boundary-reflected wave can preserve local frequency while corrupting direction\n", "and amplitude. These independent checks make the analytic plane wave a useful reference\n", "rather than merely a familiar picture.\n", "\n", "## Verification and limits\n", "\n", "Check that substituting the fields into both curl equations gives the same dispersion\n", "relation $\\omega=ck$. Verify\n", "zero divergence, mutual perpendicularity, correct cross-product direction, equal phase,\n", "and $E_0/B_0=c$. Track a\n", "constant phase and confirm that it moves the expected direction at light speed.\n", "\n", "The plane-wave solution assumes source-free, linear, homogeneous space. It omits finite\n", "antenna geometry, near fields, wavefront curvature, material interfaces, absorption,\n", "dispersion, nonlinearity, and quantum photon detection. Numerical grids add phase error,\n", "boundary reflection, and limited resolution. A visually moving stripe can pass none of\n", "the analytic checks above.\n", "\n", "## A worked Python notebook\n", "\n", "A one megahertz plane wave has peak electric field ten volts per metre and propagates in vacuum. The question for this calculation is precise: how does **electric field at time zero**\n", "change when we change **position**, 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", "The field is transverse and its magnetic magnitude is the electric magnitude divided by light speed. 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", "0 to 300 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 position, expressed in m. The output is\n", "electric field at time zero, expressed in V/m. 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", "$$\n", "f(q) = 10 \\cos{\\left(0.00667128190396304 \\pi q \\right)}\n", "$$\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 100 m, the expression gives -5.01255 V/m. 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(\"10*cos(2*pi*1e6*q/299792458)\", locals={\"q\": q})\n", "reference_input = 100.0\n", "reference_output = float(expression.subs(q, reference_input))\n", "print(\"Input:\", reference_input, \"m\")\n", "print(\"Predicted electric field at time zero:\", reference_output, \"V/m\")\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", "10 V/m. The midpoint, at 150 m, gives\n", "-9.99998 V/m. The upper endpoint gives 9.99991 V/m.\n", "Read these together with the assumptions: they describe different choices of position\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([0.0, 150.0, 300.0])\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} V/m\")\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 position; the vertical axis shows electric field at time zero. 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(0.0, 300.0, 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(\"Position (m)\")\n", "ax.set_ylabel(\"Electric field at time zero (V/m)\")\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", "electric field at time zero change near the reference input? Its units are the output unit divided by\n", "the input unit. For this calculation, the derivative at 100 m is\n", "-0.181353 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", "$$\n", "f'(q) = - 0.0667128190396304 \\pi \\sin{\\left(0.00667128190396304 \\pi q \\right)}\n", "$$\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 = 101.5\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, \"V/m\")\n", "print(\"Full expression:\", full_prediction, \"V/m\")\n", "print(\"Approximation difference:\", full_prediction - linear_prediction, \"V/m\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-012", "metadata": {}, "source": [ "Here the changed input is 101.5 m. The tangent estimate is\n", "-5.28458 V/m, while the full expression gives\n", "-5.28206 V/m. 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 = 300.0\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", "position inside the stated interval, predict whether electric field at time zero 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: The field is transverse and its magnetic magnitude is the electric magnitude divided by light speed. 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 V/m:\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(300.0) - evaluate(0.0))\n", "for points in [101, 201, 401]:\n", " grid = np.linspace(0.0, 300.0, 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*, Section 16.1, “Maxwell’s Equations and Electromagnetic Waves”](https://openstax.org/books/university-physics-volume-2/pages/16-1-maxwells-equations-and-electromagnetic-waves).\n", "- [OpenStax, *University Physics Volume 2*, Section 16.2, “Plane Electromagnetic Waves”](https://openstax.org/books/university-physics-volume-2/pages/16-2-plane-electromagnetic-waves).\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, “Electricity and Magnetism”](https://phys.libretexts.org/Bookshelves/Electricity_and_Magnetism)." ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "name": "python", "version": "3.12" }, "colab": { "name": "em-wave-solutions.ipynb", "provenance": [] }, "tirta": { "lesson": "https://tirta.co/a-byte-of-physics/em-wave-solutions", "source_sha256": "c1d07470818c605b3d672ecb42f3adb4f581e46c70747e92a01c360bcc467329" } }, "nbformat": 4, "nbformat_minor": 5 }