{ "cells": [ { "cell_type": "markdown", "id": "markdown-000", "metadata": {}, "source": [ "# Lorentz force and particle motion\n", "\n", "[Read the lesson](https://tirta.co/a-byte-of-physics/lorentz-force-simulations). 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": [ "Electric and magnetic fields act together through the Lorentz force. An electric field\n", "can speed up or slow down a charged particle. A magnetic field deflects a moving charge\n", "sideways, perpendicular to both the velocity and the field. This difference explains\n", "why magnetic fields bend ideal particle paths without changing their speed.\n", "\n", "This page develops the physical model and its computational checks without an interactive\n", "simulation. The goal is to understand what a calculated trajectory means, which\n", "quantities remain constant, and how charge sign and vector orientation determine the\n", "motion.\n", "\n", "## The Lorentz force\n", "\n", "A particle with charge $q$ and velocity\n", "$\\mathbf v$ in electric and magnetic\n", "fields experiences\n", "\n", "\n", "$$\n", "\\mathbf F=q\\left(\\mathbf E+\\mathbf v\\times\\mathbf B\\right)\n", "$$\n", "\n", "\n", "The electric part is\n", "$q\\mathbf E$ and acts even when the\n", "particle is momentarily at rest. The magnetic part is\n", "$q\\mathbf v\\times\\mathbf B$ and vanishes\n", "for a stationary particle or for motion parallel to the magnetic field.\n", "\n", "Newton's second law gives the nonrelativistic acceleration:\n", "\n", "\n", "$$\n", "\\mathbf a=\\frac{q}{m}\n", " \\left(\\mathbf E+\\mathbf v\\times\\mathbf B\\right)\n", "$$\n", "\n", "\n", "The mass must be positive. The charge may be positive, negative, or zero. Reversing the\n", "charge reverses both electric and magnetic force while leaving the fields unchanged." ] }, { "cell_type": "code", "id": "code-003", "metadata": {}, "source": [ "import numpy as np\n", "\n", "charge, mass = 1.0, 1.0 # Scaled units.\n", "velocity = np.array([1.0, 0.0, 0.0])\n", "electric, magnetic = np.zeros(3), np.array([0.0, 0.0, 1.0])\n", "force = charge * (electric + np.cross(velocity, magnetic))\n", "assert np.isclose(np.dot(force, velocity), 0.0)\n", "print(\"Magnetic force:\", force)\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-004", "metadata": {}, "source": [ "## Magnetic-force magnitude and direction\n", "\n", "The magnitude of the magnetic part is\n", "\n", "\n", "$$\n", "F_B=|q|vB\\sin\\theta\n", "$$\n", "\n", "\n", "The angle $\\theta$\n", "lies between the velocity and magnetic field. The force is largest when they are\n", "perpendicular and zero when they are parallel or antiparallel.\n", "\n", "For a positive charge, the direction is that of\n", "$\\mathbf v\\times\\mathbf B$.\n", "For a negative charge, reverse it. In a right-handed basis,\n", "\n", "\n", "$$\n", "\\hat{\\mathbf x}\\times\\hat{\\mathbf z}\n", " =-\\hat{\\mathbf y}\n", "$$\n", "\n", "\n", "Thus a positive charge moving in positive x through a positive z magnetic field initially\n", "accelerates in negative y. Reversing either charge or field reverses the curvature;\n", "reversing both preserves it.\n", "\n", "## Why a magnetic field does no work\n", "\n", "The instantaneous rate of work by a force is\n", "\n", "\n", "$$\n", "P=\\mathbf F\\cdot\\mathbf v\n", "$$\n", "\n", "\n", "For the magnetic force,\n", "\n", "\n", "$$\n", "P_B=q\\left(\\mathbf v\\times\\mathbf B\\right)\\cdot\\mathbf v=0\n", "$$\n", "\n", "\n", "A cross product is perpendicular to the velocity, so the dot product is zero. Therefore\n", "an ideal magnetic field alone changes the direction of momentum but not the particle's\n", "speed or kinetic energy. The electric field supplies the total work rate:\n", "\n", "\n", "$$\n", "\\frac{dK}{dt}=q\\mathbf E\\cdot\\mathbf v\n", "$$\n", "\n", "\n", "This identity is a powerful diagnostic. If a calculation with\n", "$\\mathbf E=0$ shows steadily growing\n", "speed, the error comes from the numerical update or implementation, not magnetic work.\n", "\n", "## Circular motion in a uniform magnetic field\n", "\n", "Suppose the magnetic field is uniform and the velocity is perpendicular to it. The\n", "magnetic force magnitude is $|q|vB$\n", "and supplies the centripetal force:\n", "\n", "\n", "$$\n", "\\frac{mv^2}{r}=|q|vB\n", "$$\n", "\n", "\n", "Solving for the radius gives\n", "\n", "\n", "$$\n", "r=\\frac{mv}{|q|B}\n", "$$\n", "\n", "\n", "The angular frequency and period are\n", "\n", "\n", "$$\n", "\\omega_c=\\frac{|q|B}{m},\\qquad\n", " T=\\frac{2\\pi m}{|q|B}\n", "$$\n", "\n", "\n", "In this ideal nonrelativistic model, the frequency does not depend on speed, while the\n", "radius is proportional to speed. Charge sign determines the direction of rotation, not\n", "the radius or period magnitude.\n", "\n", "For example, doubling the magnetic field halves both the radius and period for the same\n", "particle and speed. Doubling the mass doubles both. Doubling the perpendicular speed\n", "doubles the radius but leaves the period unchanged.\n", "\n", "## Helical motion\n", "\n", "Split velocity into components parallel and perpendicular to the field:\n", "\n", "\n", "$$\n", "\\mathbf v=\\mathbf v_{\\parallel}+\\mathbf v_{\\perp}\n", "$$\n", "\n", "\n", "The parallel component produces no magnetic force and remains constant in a uniform\n", "field. The perpendicular component rotates in a circle. Together they produce a helix\n", "whose axis follows the magnetic field.\n", "\n", "The distance advanced parallel to the field during one period, called the pitch, is\n", "\n", "\n", "$$\n", "p=v_{\\parallel}T\n", " =\\frac{2\\pi m v_{\\parallel}}{|q|B}\n", "$$\n", "\n", "\n", "If the parallel component is zero, the helix becomes a circle. If the perpendicular\n", "component is zero, the radius is zero and the particle moves in a straight line along\n", "the field.\n", "\n", "## Combined electric and magnetic fields\n", "\n", "An electric field can change kinetic energy, so combined fields produce richer motion.\n", "One important case has perpendicular electric and magnetic fields. If a particle velocity\n", "satisfies\n", "\n", "\n", "$$\n", "\\mathbf E+\\mathbf v\\times\\mathbf B=\\mathbf 0\n", "$$\n", "\n", "\n", "then the electric and magnetic forces cancel. For mutually perpendicular velocity and\n", "fields, the speed selected by the magnitudes is\n", "\n", "\n", "$$\n", "v=\\frac{E}{B}\n", "$$\n", "\n", "\n", "The direction must still satisfy the vector equation. A speed alone cannot determine\n", "force cancellation.\n", "\n", "Crossed-field selectors and magnetic curvature can be combined to distinguish charged\n", "particles by charge-to-mass ratio. After selection, a measured circular radius obeys\n", "\n", "\n", "$$\n", "\\frac{|q|}{m}=\\frac{v}{Br}\n", "$$\n", "\n", "\n", "## Work and electric potential\n", "\n", "If the electric field is electrostatic, its work can be expressed through electric\n", "potential. For motion between two points,\n", "\n", "\n", "$$\n", "\\Delta K=-\\Delta U=-q\\Delta V\n", "$$\n", "\n", "\n", "A particle accelerated from rest through a potential difference of magnitude\n", "$|\\Delta V|$ gains\n", "kinetic energy\n", "\n", "\n", "$$\n", "K=|q|\\,|\\Delta V|\n", "$$\n", "\n", "\n", "Combining this with $K=mv^2/2$\n", "gives a speed only when the nonrelativistic approximation remains valid.\n", "\n", "## Computing a trajectory carefully\n", "\n", "A trajectory calculation advances position and velocity from the force. Because magnetic\n", "acceleration depends on velocity, a generic integrator that works well for position-only\n", "forces may not preserve magnetic energy. The algorithm and step size are part of the\n", "reported model." ] }, { "cell_type": "code", "id": "code-005", "metadata": {}, "source": [ "# Boris rotation in a constant magnetic field, with E=0.\n", "position, velocity, dt = np.zeros(3), np.array([1.0, 0.0, 0.0]), 0.01\n", "for _ in range(1000):\n", " t = charge * magnetic * dt / (2 * mass)\n", " s = 2 * t / (1 + np.dot(t, t))\n", " intermediate = velocity + np.cross(velocity, t)\n", " velocity = velocity + np.cross(intermediate, s)\n", " position += velocity * dt\n", "assert np.isclose(np.linalg.norm(velocity), 1.0)\n", "print(\"Final position and speed:\", position, np.linalg.norm(velocity))\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-006", "metadata": {}, "source": [ "For uniform magnetic fields, structure-preserving methods such as the Boris particle\n", "pusher are often used because they control artificial energy drift better than a naive\n", "explicit Euler update. Whatever method is chosen, compare it with the analytic radius,\n", "frequency, and constant-speed result.\n", "\n", "## Numerical and physical checks\n", "\n", "For zero electric field, monitor\n", "\n", "\n", "$$\n", "\\epsilon_{\\perp}\n", " =\\mathbf v\\cdot\\mathbf F_B,\n", " \\qquad\n", " \\epsilon_K(t)=K(t)-K(0)\n", "$$\n", "\n", "\n", "Both should remain close to zero at the accuracy scale of the calculation. Halve the\n", "time step and compare position and velocity at identical physical times. Verify that\n", "reversing charge or magnetic field reverses rotation. Verify that velocity parallel to\n", "the field produces no magnetic force. Do not judge accuracy from the smoothness of a\n", "plotted curve.\n", "\n", "## Relativistic and other limits\n", "\n", "At speeds approaching the speed of light, use relativistic momentum\n", "$\\mathbf p=\\gamma m\\mathbf v$ and\n", "\n", "\n", "$$\n", "\\frac{d\\mathbf p}{dt}\n", " =q\\left(\\mathbf E+\\mathbf v\\times\\mathbf B\\right)\n", "$$\n", "\n", "\n", "The relativistic cyclotron frequency decreases with increasing energy through the\n", "Lorentz factor, so the nonrelativistic speed-independent result no longer holds.\n", "\n", "The ideal single-particle model also omits radiation by accelerating charges,\n", "collisions, collective plasma fields, material interactions, quantum effects, and field\n", "back-reaction. Nonuniform fields can produce drifts and magnetic-mirror behavior that a\n", "uniform-field circle does not describe. State these limits before interpreting a\n", "calculated path as a laboratory prediction.\n", "\n", "## A worked Python notebook\n", "\n", "An electron travels perpendicular to a uniform magnetic field at one million metres per second. The question for this calculation is precise: how does **cyclotron radius**\n", "change when we change **magnetic 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", "The radius uses charge magnitude; the charge sign determines the direction of gyration. 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", "0.001 to 0.02 T. 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 magnetic field magnitude, expressed in T. The output is\n", "cyclotron radius, expressed in 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", "$$\n", "f(q) = \\frac{5.6856301026295 \\cdot 10^{-6}}{q}\n", "$$\n", "\n", "\n", "This is a function of one changing input. It is not a statement that the physical\n", "system has only one variable. We deliberately hold the other inputs fixed so that one\n", "cause of variation is visible. Changing several inputs at once would make it harder\n", "to tell which change produced the output difference. Write down the held-fixed choices\n", "before trying a different experiment.\n", "\n", "At 0.01 T, the expression gives 0.000568563 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-007", "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(\"9.1093837e-31*1e6/(1.602176634e-19*q)\", locals={\"q\": q})\n", "reference_input = 0.01\n", "reference_output = float(expression.subs(q, reference_input))\n", "print(\"Input:\", reference_input, \"T\")\n", "print(\"Predicted cyclotron radius:\", reference_output, \"m\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-008", "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", "0.00568563 m. The midpoint, at 0.0105 T, gives\n", "0.000541489 m. The upper endpoint gives 0.000284282 m.\n", "Read these together with the assumptions: they describe different choices of magnetic 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-009", "metadata": {}, "source": [ "evaluate = sp.lambdify(q, expression, modules=\"numpy\")\n", "inputs = np.array([0.001, 0.0105, 0.02])\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} T -> {output_value:.6g} m\")\n", "assert np.all(np.isfinite(outputs))\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-010", "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 magnetic field magnitude; the vertical axis shows cyclotron radius. 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-011", "metadata": {}, "source": [ "input_grid = np.linspace(0.001, 0.02, 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(\"Magnetic field magnitude (T)\")\n", "ax.set_ylabel(\"Cyclotron radius (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-012", "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", "cyclotron radius change near the reference input? Its units are the output unit divided by\n", "the input unit. For this calculation, the derivative at 0.01 T is\n", "-0.0568563 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", "$$\n", "f'(q) = - \\frac{5.6856301026295 \\cdot 10^{-6}}{q^{2}}\n", "$$\n", "\n", "\n", "For a small input change, multiply the derivative by that change to estimate the output\n", "change. Then compare that estimate with a fresh evaluation of the full expression.\n", "The distinction between these two calculations matters: the derivative gives a tangent\n", "approximation, while substitution evaluates the chosen expression at the new point." ] }, { "cell_type": "code", "id": "code-013", "metadata": {}, "source": [ "slope_expression = sp.diff(expression, q)\n", "slope_at_reference = float(slope_expression.subs(q, reference_input))\n", "changed_input = 0.010095\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, \"m\")\n", "print(\"Full expression:\", full_prediction, \"m\")\n", "print(\"Approximation difference:\", full_prediction - linear_prediction, \"m\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-014", "metadata": {}, "source": [ "Here the changed input is 0.010095 T. The tangent estimate is\n", "0.000563162 m, while the full expression gives\n", "0.000563212 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-015", "metadata": {}, "source": [ "input_span = 0.019\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-016", "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", "magnetic field magnitude inside the stated interval, predict whether cyclotron radius 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 radius uses charge magnitude; the charge sign determines the direction of gyration. 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 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-017", "metadata": {}, "source": [ "slope_function = sp.lambdify(q, slope_expression, modules=\"numpy\")\n", "endpoint_change = float(evaluate(0.02) - evaluate(0.001))\n", "for points in [101, 201, 401]:\n", " grid = np.linspace(0.001, 0.02, 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-018", "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 11.1, “Magnetism and Its Historical Discoveries”](https://openstax.org/books/university-physics-volume-2/pages/11-1-magnetism-and-its-historical-discoveries).\n", "- [OpenStax, *University Physics Volume 2*, Section 11.3, “Motion of a Charged Particle in a Magnetic Field”](https://openstax.org/books/university-physics-volume-2/pages/11-3-motion-of-a-charged-particle-in-a-magnetic-field).\n", "- [OpenStax, *University Physics Volume 2*, Section 11.4, “Magnetic Force on a Current-Carrying Conductor”](https://openstax.org/books/university-physics-volume-2/pages/11-4-magnetic-force-on-a-current-carrying-conductor).\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": "lorentz-force-simulations.ipynb", "provenance": [] }, "tirta": { "lesson": "https://tirta.co/a-byte-of-physics/lorentz-force-simulations", "source_sha256": "ed287d8be74614bbf80170c2926d2b1b4fa1152ff921d48caeb0ddcb9c7ff18b" } }, "nbformat": 4, "nbformat_minor": 5 }