{ "cells": [ { "cell_type": "markdown", "id": "markdown-000", "metadata": {}, "source": [ "# Coulomb’s law as field generator\n", "\n", "[Read the lesson](https://tirta.co/a-byte-of-physics/coulombs-law-field-generator). 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": [ "Coulomb's law gives the electrostatic force between ideal point charges. The same law\n", "generates an electric field when one charge is treated as the source and the force is\n", "reported per unit positive test charge. Multiple sources combine by vector addition.\n", "\n", "The geometry needs two positions: the source and the observation point. An unsigned\n", "distance gives the magnitude but loses the direction that determines attraction,\n", "repulsion, and cancellation.\n", "\n", "## Force between two point charges\n", "\n", "Let charge $q_1$ be at position\n", "$\\mathbf r_1$ and charge\n", "$q_2$ be at position\n", "$\\mathbf r_2$. Define the\n", "displacement from charge one to charge two as\n", "\n", "\n", "$$\n", "\\mathbf r_{21}=\\mathbf r_2-\\mathbf r_1\n", "$$\n", "\n", "\n", "The force exerted by charge one on charge two in vacuum is\n", "\n", "\n", "$$\n", "\\mathbf F_{1\\to2}\n", " =\\frac{1}{4\\pi\\epsilon_0}\n", " \\frac{q_1q_2}{\\lVert\\mathbf r_{21}\\rVert^3}\n", " \\mathbf r_{21}\n", "$$\n", "\n", "\n", "The factor containing the displacement has magnitude equal to the inverse square of\n", "distance and points from the source toward the target. The product\n", "$q_1q_2$ supplies\n", "the sign. Like signs produce a force along the displacement, which is repulsive.\n", "Opposite signs reverse the vector, producing attraction.\n", "\n", "The force on charge one from charge two is equal and opposite:\n", "\n", "\n", "$$\n", "\\mathbf F_{2\\to1}=-\\mathbf F_{1\\to2}\n", "$$\n", "\n", "\n", "This follows because reversing the displacement reverses the unit direction while the\n", "charge product remains the same.\n", "\n", "## Coulomb constant and units\n", "\n", "It is common to define\n", "\n", "\n", "$$\n", "k_e=\\frac{1}{4\\pi\\epsilon_0}\n", " \\approx8.988\\times10^9\\,\n", " \\mathrm{N\\,m^2/C^2}\n", "$$\n", "\n", "\n", "Here $\\epsilon_0$ is the vacuum\n", "permittivity. With charges in coulombs and distance in meters, the force is in newtons.\n", "\n", "In a homogeneous linear material, a simplified macroscopic model may replace\n", "$\\epsilon_0$ with the material\n", "permittivity $\\epsilon$. Real\n", "dielectrics can be nonuniform, anisotropic, nonlinear, or frequency-dependent, so a\n", "single constant is not universally adequate.\n", "\n", "## From force to electric field\n", "\n", "The electric field is the force per unit positive test charge in the limit that the test\n", "charge does not disturb the sources:\n", "\n", "\n", "$$\n", "\\mathbf E(\\mathbf r)\n", " =\\lim_{q_{\\mathrm{test}}\\to0}\n", " \\frac{\\mathbf F(\\mathbf r)}{q_{\\mathrm{test}}}\n", "$$\n", "\n", "\n", "For one source charge $q$ at\n", "$\\mathbf r_s$, define\n", "\n", "\n", "$$\n", "\\mathbf R=\\mathbf r-\\mathbf r_s\n", "$$\n", "\n", "\n", "Then\n", "\n", "\n", "$$\n", "\\mathbf E(\\mathbf r)\n", " =k_eq\\frac{\\mathbf R}{\\lVert\\mathbf R\\rVert^3}\n", "$$\n", "\n", "\n", "The field has SI units of newtons per coulomb, equivalent to volts per meter. A positive\n", "source produces an outward field; a negative source produces an inward field. The test\n", "charge sign does not belong in this field expression. It enters later when calculating\n", "the force $\\mathbf F=q_{\\mathrm{test}}\\mathbf E$.\n", "\n", "## A numerical example\n", "\n", "Place a positive source of\n", "$q=2.00\\ \\mathrm{nC}$ at the origin\n", "and observe the field at\n", "$x=0.300\\ \\mathrm{m}$.\n", "The magnitude is\n", "\n", "\n", "$$\n", "E=k_e\\frac{|q|}{r^2}\n", " =\\frac{(8.988\\times10^9)(2.00\\times10^{-9})}\n", " {(0.300)^2}\n", " \\approx2.00\\times10^2\\ \\mathrm{N/C}\n", "$$\n", "\n", "\n", "Because the source is positive and the point lies on the positive horizontal axis, the\n", "field points in the positive horizontal direction. Moving the observation point to twice\n", "the distance would divide the magnitude by four. Changing only the source sign would\n", "reverse the vector without changing its magnitude.\n", "\n", "## Superposition for many sources\n", "\n", "Electric fields add as vectors. For point sources\n", "$q_i$ at positions\n", "$\\mathbf r_i$,\n", "\n", "\n", "$$\n", "\\mathbf E(\\mathbf r)\n", " =\\sum_i k_eq_i\n", " \\frac{\\mathbf r-\\mathbf r_i}\n", " {\\lVert\\mathbf r-\\mathbf r_i\\rVert^3}\n", "$$\n", "\n", "\n", "Calculate each contribution with its own signed charge and displacement, then add\n", "components. Adding magnitudes is correct only when all vectors point in the same\n", "direction.\n", "\n", "Two equal positive charges have zero field at their midpoint because their equal field\n", "vectors point oppositely. Two equal and opposite charges do not cancel there: both field\n", "vectors point from the positive charge toward the negative charge and therefore add.\n", "This pair of symmetry cases is an effective sign test.\n", "\n", "Superposition is linear in source charge. Doubling every source charge doubles the field\n", "everywhere away from the sources. Moving one source changes only that source's\n", "contribution before the sum is recomputed.\n", "\n", "## Field and potential cross-check\n", "\n", "The electrostatic potential of point charges, with zero chosen at infinity, is\n", "\n", "\n", "$$\n", "V(\\mathbf r)\n", " =\\sum_i\\frac{k_eq_i}\n", " {\\lVert\\mathbf r-\\mathbf r_i\\rVert}\n", "$$\n", "\n", "\n", "The field is related to potential by\n", "\n", "\n", "$$\n", "\\mathbf E=-\\boldsymbol{\\nabla}V\n", "$$\n", "\n", "\n", "Potential is a scalar, so equal and opposite charges have zero potential at their\n", "midpoint. The field there is generally nonzero because the spatial slopes of the two\n", "potential contributions add. Checking a numerical field against a finite-difference\n", "gradient away from sources can reveal displacement and sign errors.\n", "\n", "## The point-source singularity\n", "\n", "At the exact source position,\n", "$\\lVert\\mathbf R\\rVert=0$,\n", "and the ideal point-charge field diverges. Returning zero hides the singularity and\n", "creates a false physical statement.\n", "\n", "A strict calculation should reject that observation point. A display may omit a small\n", "region or cap arrow lengths, but it should label the choice. If a finite-radius charge\n", "distribution replaces the point source, use the field derived for that distribution\n", "rather than silently adding an arbitrary denominator.\n", "\n", "The singularity is a limitation of the ideal point model. Real sources have structure,\n", "and classical electrostatics is not a complete description at arbitrarily small scales.\n", "\n", "## A narrow field algorithm\n", "\n", "The vector form translates directly into a sequence of short operations." ] }, { "cell_type": "code", "id": "code-003", "metadata": {}, "source": [ "import numpy as np\n", "\n", "k_e = 8.9875517923e9\n", "sources = [(np.array([-1.0, 0.0]), 1e-9), (np.array([1.0, 0.0]), -1e-9)]\n", "probe = np.array([0.0, 1.0])\n", "field = np.zeros(2)\n", "for position, charge in sources:\n", " offset = probe - position\n", " radius = np.linalg.norm(offset)\n", " if radius == 0:\n", " raise ValueError(\"The probe cannot coincide with a point source.\")\n", " field += k_e * charge * offset / radius**3\n", "print(\"Electric field (N/C):\", field)\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-004", "metadata": {}, "source": [ "The signed magnitude deliberately retains the charge sign. Alternatively, use the vector\n", "divided by the cube of its magnitude exactly as in the displayed equation. Do not apply\n", "the sign twice.\n", "\n", "## Verification and model limits\n", "\n", "Test one positive source on each coordinate axis. The field should point outward with the\n", "expected component sign. Repeat with a negative source. Check inverse-square scaling at\n", "two distances and linear scaling with charge.\n", "\n", "Use equal-source symmetry to test cancellation and opposite-source symmetry to test\n", "addition. Verify that source order does not change the sum. Test Newton's third law for a\n", "two-charge force calculation, and compare field-derived force with direct Coulomb force.\n", "\n", "Finally, compare the field with the negative numerical gradient of potential at points\n", "that remain safely away from sources. Repeat the gradient calculation with smaller grid\n", "spacing to distinguish a coding error from finite-difference error.\n", "\n", "Coulomb's instantaneous point-source law is an electrostatic model. Moving charges,\n", "radiation, relativistic delay, conductor response, polarization, and quantum structure\n", "require additional physics. A two-dimensional arrow diagram is only a slice through the\n", "three-dimensional field, and visual scaling must remain separate from calculated values.\n", "\n", "## A worked Python notebook\n", "\n", "Calculate the outward field magnitude of a positive one nanocoulomb point charge. The question for this calculation is precise: how does **electric field magnitude**\n", "change when we change **distance from source**, 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 undefined at the source, and direction must be restored when using the magnitude in a vector calculation. 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", "0.5 to 3 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 distance from source, expressed in m. The output is\n", "electric field magnitude, expressed in N/C. 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", "$$\n", "f(q) = \\frac{8.9875517923}{q^{2}}\n", "$$\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 1 m, the expression gives 8.98755 N/C. 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(\"8.9875517923/q**2\", locals={\"q\": q})\n", "reference_input = 1.0\n", "reference_output = float(expression.subs(q, reference_input))\n", "print(\"Input:\", reference_input, \"m\")\n", "print(\"Predicted electric field magnitude:\", reference_output, \"N/C\")\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", "35.9502 N/C. The midpoint, at 1.75 m, gives\n", "2.93471 N/C. The upper endpoint gives 0.998617 N/C.\n", "Read these together with the assumptions: they describe different choices of distance from source\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([0.5, 1.75, 3.0])\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} N/C\")\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 distance from source; the vertical axis shows electric field magnitude. 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(0.5, 3.0, 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(\"Distance from source (m)\")\n", "ax.set_ylabel(\"Electric field magnitude (N/C)\")\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 magnitude change near the reference input? Its units are the output unit divided by\n", "the input unit. For this calculation, the derivative at 1 m is\n", "-17.9751 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", "$$\n", "f'(q) = - \\frac{17.9751035846}{q^{3}}\n", "$$\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 = 1.0125\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, \"N/C\")\n", "print(\"Full expression:\", full_prediction, \"N/C\")\n", "print(\"Approximation difference:\", full_prediction - linear_prediction, \"N/C\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-012", "metadata": {}, "source": [ "Here the changed input is 1.0125 m. The tangent estimate is\n", "8.76286 N/C, while the full expression gives\n", "8.76701 N/C. 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 = 2.5\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", "distance from source inside the stated interval, predict whether electric field magnitude 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 undefined at the source, and direction must be restored when using the magnitude in a vector calculation. 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 N/C:\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(3.0) - evaluate(0.5))\n", "for points in [101, 201, 401]:\n", " grid = np.linspace(0.5, 3.0, 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 5.3, “Coulomb’s Law”](https://openstax.org/books/university-physics-volume-2/pages/5-3-coulombs-law).\n", "- [OpenStax, *University Physics Volume 2*, Section 5.4, “Electric Field”](https://openstax.org/books/university-physics-volume-2/pages/5-4-electric-field).\n", "- [Benjamin Crowell, *Light and Matter*, Chapter 21, “Electricity and Circuits,” and Chapter 22, “Fields”](https://open.umn.edu/opentextbooks/textbooks/light-and-matter).\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": "coulombs-law-field-generator.ipynb", "provenance": [] }, "tirta": { "lesson": "https://tirta.co/a-byte-of-physics/coulombs-law-field-generator", "source_sha256": "3c71bc6cf78714a1949e73ef2fdda0532611b3e6b572ee4f12b7ccc5968b787d" } }, "nbformat": 4, "nbformat_minor": 5 }