{
  "cells": [
    {
      "cell_type": "markdown",
      "id": "markdown-000",
      "metadata": {},
      "source": [
        "# Energy in fields\n",
        "\n",
        "[Read the lesson](https://tirta.co/a-byte-of-physics/energy-in-fields). 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": [
        "An electric configuration can store potential energy. That energy can be calculated from\n",
        "charges and electric potential, from the work required to assemble the charges, or from\n",
        "the electric field throughout space. These descriptions must agree when they use the\n",
        "same sources, material model, boundaries, and reference state.\n",
        "\n",
        "Field energy is especially useful because it assigns energy to regions of space. A\n",
        "numerical value still depends on the integration domain, grid cells, permittivity, and\n",
        "treatment of ideal singular sources.\n",
        "\n",
        "## Building a configuration of point charges\n",
        "\n",
        "For two point charges separated by distance\n",
        "$r_{12}$, the\n",
        "electrostatic interaction energy, with zero chosen at infinite separation, is\n",
        "\n",
        "\n",
        "$$\n",
        "U_{12}\n",
        "  =\\frac{1}{4\\pi\\epsilon_0}\n",
        "  \\frac{q_1q_2}{r_{12}}\n",
        "$$\n",
        "\n",
        "\n",
        "Like charges have positive interaction energy. Work must be supplied to bring them closer\n",
        "against their repulsion. Opposite charges have negative interaction energy because the\n",
        "electric force releases energy as they approach from infinity.\n",
        "\n",
        "For several point charges, count each pair once:\n",
        "\n",
        "\n",
        "$$\n",
        "U\n",
        "  =\\frac{1}{4\\pi\\epsilon_0}\n",
        "  \\sum_{i<j}\\frac{q_iq_j}{r_{ij}}\n",
        "$$\n",
        "\n",
        "\n",
        "An equivalent form uses the potential created by all other charges at each charge:\n",
        "\n",
        "\n",
        "$$\n",
        "U=\\frac12\\sum_i q_iV_{\\mathrm{others}}(\\mathbf r_i)\n",
        "$$\n",
        "\n",
        "\n",
        "The factor one half prevents double counting, because the potential at charge\n",
        "$i$ contains the contribution from charge\n",
        "$j$ and the reverse term appears again\n",
        "when the indices swap. A charge's own point-source potential is excluded from this\n",
        "interaction formula.\n",
        "\n",
        "## Energy density in a vacuum electric field\n",
        "\n",
        "In vacuum, the electrostatic energy density is\n",
        "\n",
        "\n",
        "$$\n",
        "u_E=\\frac12\\epsilon_0\\lVert\\mathbf E\\rVert^2\n",
        "$$\n",
        "\n",
        "\n",
        "Here $u_E$ is measured in\n",
        "joules per cubic meter, $\\epsilon_0$\n",
        "is the vacuum permittivity, and\n",
        "$\\mathbf E$ is measured in volts\n",
        "per meter or newtons per coulomb.\n",
        "\n",
        "The energy in a specified volume is\n",
        "\n",
        "\n",
        "$$\n",
        "U_E=\\int_Vu_E\\,dV\n",
        "  =\\frac12\\epsilon_0\n",
        "  \\int_V\\lVert\\mathbf E\\rVert^2\\,dV\n",
        "$$\n",
        "\n",
        "\n",
        "The square makes the local density nonnegative and independent of field direction.\n",
        "Doubling the field magnitude at a point multiplies its energy density by four.\n",
        "\n",
        "The integral over all space agrees with electrostatic assembly energy only when\n",
        "self-energy and boundary terms are handled consistently. Ideal point charges make that\n",
        "qualification essential.\n",
        "\n",
        "## Capacitor energy connects the descriptions\n",
        "\n",
        "For a capacitor with charge magnitude $Q$,\n",
        "potential difference $V$,\n",
        "and capacitance $C$, the stored energy is\n",
        "\n",
        "\n",
        "$$\n",
        "U=\\frac12QV\n",
        "  =\\frac{Q^2}{2C}\n",
        "  =\\frac12CV^2\n",
        "$$\n",
        "\n",
        "\n",
        "These forms follow from $Q=CV$.\n",
        "Use the form matching the quantity held fixed. Inserting a dielectric can change\n",
        "capacitance, and the energy response differs depending on whether the capacitor remains\n",
        "connected to a voltage source or is isolated at fixed charge.\n",
        "\n",
        "For a large parallel-plate capacitor with plate area\n",
        "$A$, separation\n",
        "$d$, and uniform linear dielectric\n",
        "permittivity $\\epsilon$,\n",
        "\n",
        "\n",
        "$$\n",
        "C=\\frac{\\epsilon A}{d},\n",
        "  \\qquad E=\\frac{V}{d}\n",
        "$$\n",
        "\n",
        "\n",
        "Substituting into the capacitor energy gives\n",
        "\n",
        "\n",
        "$$\n",
        "U\n",
        "  =\\frac12\\frac{\\epsilon A}{d}V^2\n",
        "  =\\frac12\\epsilon E^2(Ad)\n",
        "$$\n",
        "\n",
        "\n",
        "The quantity $Ad$ is the\n",
        "volume between the plates. The result identifies the linear-dielectric energy density as\n",
        "one half the permittivity times the squared field, neglecting fringing near the plate\n",
        "edges.\n",
        "\n",
        "## Linear dielectric form\n",
        "\n",
        "In a linear, nondispersive dielectric,\n",
        "\n",
        "\n",
        "$$\n",
        "u_E=\\frac12\\mathbf E\\cdot\\mathbf D\n",
        "$$\n",
        "\n",
        "\n",
        "The electric displacement field\n",
        "$\\mathbf D$ satisfies\n",
        "$\\mathbf D=\\epsilon\\mathbf E$\n",
        "for an isotropic material, reducing the expression to\n",
        "$u_E=\\epsilon E^2/2$.\n",
        "\n",
        "This simple form is not universal. For a nonlinear material, the energy density depends\n",
        "on the path used to build the field and is written as an integral of field against the\n",
        "increment of displacement. Dispersive and lossy media require still more care because\n",
        "energy can be stored in and transferred to microscopic material degrees of freedom.\n",
        "\n",
        "## Why negative interaction energy is not a contradiction\n",
        "\n",
        "The field-energy density is nonnegative, but the interaction energy of opposite point\n",
        "charges is negative relative to the convention of zero energy at infinite separation.\n",
        "The statements refer to different bookkeeping.\n",
        "\n",
        "The total squared field contains individual self-field terms and cross terms:\n",
        "\n",
        "\n",
        "$$\n",
        "\\lVert\\mathbf E_1+\\mathbf E_2\\rVert^2\n",
        "  =E_1^2+E_2^2+2\\mathbf E_1\\cdot\\mathbf E_2\n",
        "$$\n",
        "\n",
        "\n",
        "The ideal self-energy terms are positive and divergent for point charges. The finite\n",
        "cross term produces the interaction energy and can be negative for opposite charges.\n",
        "Subtracting the unchanged infinite self-energy reference leaves a negative interaction\n",
        "energy without requiring negative local vacuum field density.\n",
        "\n",
        "For extended charges with finite size, self-energy can be finite. A charged conducting\n",
        "spherical shell of radius $R$ and\n",
        "total charge $Q$ has\n",
        "\n",
        "\n",
        "$$\n",
        "U_{\\mathrm{self}}\n",
        "  =\\frac{1}{4\\pi\\epsilon_0}\\frac{Q^2}{2R}\n",
        "$$\n",
        "\n",
        "\n",
        "As the radius shrinks toward zero, this classical energy diverges, revealing the\n",
        "breakdown of the point-source self-energy model.\n",
        "\n",
        "## Numerical field-energy integration\n",
        "\n",
        "On a grid, the volume integral becomes a weighted sum:\n",
        "\n",
        "\n",
        "$$\n",
        "U_{\\mathrm{domain}}\n",
        "  \\approx\\sum_i\n",
        "  \\frac12\\epsilon_i\n",
        "  \\lVert\\mathbf E_i\\rVert^2\n",
        "  \\Delta V_i\n",
        "$$\n",
        "\n",
        "\n",
        "Each cell needs a positive volume, a field sample, and the correct material law. Sampling\n",
        "only a two-dimensional slice and multiplying by an unstated thickness does not produce a\n",
        "three-dimensional energy."
      ]
    },
    {
      "cell_type": "code",
      "id": "code-003",
      "metadata": {},
      "source": [
        "import numpy as np\n",
        "\n",
        "epsilon_0, area, gap, voltage = 8.8541878128e-12, 0.02, 0.001, 100.0\n",
        "field = voltage / gap\n",
        "energy_density = 0.5 * epsilon_0 * field**2\n",
        "volume = area * gap\n",
        "energy = energy_density * volume\n",
        "capacitance = epsilon_0 * area / gap\n",
        "assert np.isclose(energy, 0.5 * capacitance * voltage**2)\n",
        "print(f\"Uniform capacitor field energy: {energy:.6g} J\")\n"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "id": "markdown-004",
      "metadata": {},
      "source": [
        "For an anisotropic linear material, replace scalar permittivity multiplication with the\n",
        "appropriate dot product involving the displacement field.\n",
        "\n",
        "## Domain and cutoff convergence are different\n",
        "\n",
        "A finite computational domain omits field energy outside its boundary. Enlarge the domain\n",
        "while holding source geometry, inner resolution, and boundary conditions fixed. If the\n",
        "total changes materially, the smaller domain was not a good approximation to all space.\n",
        "\n",
        "An inner cutoff around a point charge addresses a different issue. At radius\n",
        "$r$, the field energy\n",
        "density scales as the inverse fourth power of radius, while a spherical shell volume\n",
        "scales as radius squared times thickness. The radial energy integral therefore behaves\n",
        "like\n",
        "\n",
        "\n",
        "$$\n",
        "\\int_{r_{\\min}}\\frac{dr}{r^2}\n",
        "  \\propto\\frac{1}{r_{\\min}}\n",
        "$$\n",
        "\n",
        "\n",
        "Shrinking the cutoff makes the result grow without bound. That is not failed grid\n",
        "convergence; it is the expected divergence of the ideal point-charge model. Compare mesh\n",
        "refinement at a fixed physical cutoff, and compare cutoff choices separately as changes\n",
        "to the source model.\n",
        "\n",
        "## Cross-checks and energy flow\n",
        "\n",
        "For an ideal parallel-plate capacitor, compare the grid integral with\n",
        "$CV^2/2$.\n",
        "Refine the grid, enlarge the domain, and include edge fringing if the geometry is finite.\n",
        "The two methods should approach agreement under the same boundary conditions.\n",
        "\n",
        "Check local scaling by multiplying the field by two and confirming that energy density\n",
        "multiplies by four. Reverse the field direction and confirm that the energy density is\n",
        "unchanged. Reject negative cell volumes and invalid material parameters.\n",
        "\n",
        "When fields change with time, energy can cross the domain boundary. Full electromagnetic\n",
        "accounting uses the Poynting vector and includes both electric and magnetic energy. A\n",
        "changing domain total without a boundary-flux and source-work ledger is incomplete.\n",
        "\n",
        "Field energy is not merely a colored map. It is a quantitative accounting method whose\n",
        "answer is inseparable from source structure, material response, domain, boundaries, and\n",
        "the resolution at which those ingredients are represented.\n",
        "\n",
        "## A worked Python notebook\n",
        "\n",
        "Evaluate the electric energy density in vacuum before integrating over volume. The question for this calculation is precise: how does **electric energy density**\n",
        "change when we change **uniform field magnitude**, 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",
        "Material polarisation needs an appropriate constitutive energy relation. 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 200000 V/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 uniform field magnitude, expressed in V/m. The output is\n",
        "electric energy density, expressed in J/m^3. 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",
        "$$\n",
        "f(q) = 4.4270939064 \\cdot 10^{-12} q^{2}\n",
        "$$\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 100000 V/m, the expression gives 0.0442709 J/m^3. 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(\"8.8541878128e-12*q**2/2\", locals={\"q\": q})\n",
        "reference_input = 100000.0\n",
        "reference_output = float(expression.subs(q, reference_input))\n",
        "print(\"Input:\", reference_input, \"V/m\")\n",
        "print(\"Predicted electric energy density:\", reference_output, \"J/m^3\")\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",
        "0 J/m^3. The midpoint, at 100000 V/m, gives\n",
        "0.0442709 J/m^3. The upper endpoint gives 0.177084 J/m^3.\n",
        "Read these together with the assumptions: they describe different choices of uniform field magnitude\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, 100000.0, 200000.0])\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} V/m -> {output_value:.6g} J/m^3\")\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 uniform field magnitude; the vertical axis shows electric energy density. 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, 200000.0, 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(\"Uniform field magnitude (V/m)\")\n",
        "ax.set_ylabel(\"Electric energy density (J/m^3)\")\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 energy density change near the reference input? Its units are the output unit divided by\n",
        "the input unit. For this calculation, the derivative at 100000 V/m is\n",
        "8.85419e-07 in those ratio units. Locally, the output increases 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",
        "$$\n",
        "f'(q) = 8.8541878128 \\cdot 10^{-12} q\n",
        "$$\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 = 101000.0\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, \"J/m^3\")\n",
        "print(\"Full expression:\", full_prediction, \"J/m^3\")\n",
        "print(\"Approximation difference:\", full_prediction - linear_prediction, \"J/m^3\")\n"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "id": "markdown-012",
      "metadata": {},
      "source": [
        "Here the changed input is 101000 V/m. The tangent estimate is\n",
        "0.0451564 J/m^3, while the full expression gives\n",
        "0.0451608 J/m^3. 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 = 200000.0\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",
        "uniform field magnitude inside the stated interval, predict whether electric energy density 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: Material polarisation needs an appropriate constitutive energy relation. 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 J/m^3:\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(200000.0) - evaluate(0.0))\n",
        "for points in [101, 201, 401]:\n",
        "    grid = np.linspace(0.0, 200000.0, 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 7.1, “Electric Potential Energy”](https://openstax.org/books/university-physics-volume-2/pages/7-1-electric-potential-energy).\n",
        "- [OpenStax, *University Physics Volume 2*, Section 8.3, “Energy Stored in a Capacitor”](https://openstax.org/books/university-physics-volume-2/pages/8-3-energy-stored-in-a-capacitor).\n",
        "- [OpenStax, *University Physics Volume 2*, Sections 8.4–8.5, dielectric materials](https://openstax.org/books/university-physics-volume-2/pages/8-4-capacitor-with-a-dielectric).\n",
        "- [Benjamin Crowell, *Light and Matter*, 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)."
      ]
    }
  ],
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