{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "Chapter 9 of [A guided tour of mathematical methods for the physical sciences](http://www.cambridge.org/nz/academic/subjects/physics/mathematical-methods/guided-tour-mathematical-methods-physical-sciences-3rd-edition#KUoGXYx5FTwytcUg.97) introduces the theorem of Stokes. In Chapter 7, you saw a special vector field, for which the divergence and curl are zero. This gave rise to a model for Hurricane winds and the magnetic field around a wire, captured in one of Maxwell's equations representing the Biot-Savart Law." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Biot-Savart Law\n", "The current in an long wire creates a magnetic field around it. If we align the wire with the z-axis, the field cannot vary with $z$, or $\\phi$, so that ${\\bf B}(r,\\phi,z) = {\\bf B}(r)$. The divergence of magnetic fields, and of incompressible flows, is zero, we found in Chapter 7 that\n", "\n", "$$ {\\bf B} = \\frac{A}{r} \\hat{\\mathbf \\phi}$$\n", "\n", "But how do we find the value of $A$? This is where we apply the theorem of Stokes to the Biot-Savart Law: " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$ \\nabla\\times{\\bf B} = \\mu_0{\\bf J}.$$\n", "Integrating this\n", "over a disk $d{\\bf S}$ centered on the wire of radius $r$ in the plane perpendicular to the wire, we have:\n", "$$ \\iint_S \\nabla\\times{\\bf B} \\cdot d{\\bf S} = \\iint_S\\mu_0{\\bf J}\\cdot d{\\bf S}.$$\n", "\n", "![figure 1](https://ndownloader.figshare.com/files/10244490)\n", "\n", "For the left hand side, we apply Stokes Theorem:\n", "$$ \\iint_S \\nabla\\times{\\bf B} \\cdot d{\\bf S} = \\oint_C {\\bf B}\\cdot d{\\bf r} = 2\\pi r{\\bf B},$$\n", "because $\\bf B$ is constant for each circular path $C$ around the wire.\n", "\n", "For the right hand side, we recognize that as long as the radius of the disk $r$ is larger than the thickness of our wire:\n", "$$\\iint_S\\mu_0{\\bf J}\\cdot d{\\bf S}= \\mu_0 I,$$\n", "where $I$ is the current. Putting these two results together, we find that the magnetic field is \n", "$$ {\\bf B}(r) = \\frac{\\mu_0 I}{2\\pi r} \\hat{\\mathbf \\phi}.$$ Breaking the radius $r$ and the unit vector $\\hat{\\phi}$ down in an x- and y-component (see Problem c of Section 7.2 of the book), we define a python function for the three Cartesian components of the magnetic field: " ] }, { "cell_type": "code", "execution_count": 22, "metadata": {}, "outputs": [], "source": [ "def B(I,x,y): \n", " mu0 = 1.26 * 10**(-6) \n", " r = np.sqrt((x)**2+(y)**2)\n", " c = mu0*I/(2*np.pi) \n", " Bx = -y*c/r**2\n", " By = x*c/r**2 \n", " Bz = z*0 \n", " return Bx,By,Bz" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Then, we set up a grid of points:" ] }, { "cell_type": "code", "execution_count": 62, "metadata": {}, "outputs": [], "source": [ "from mpl_toolkits.mplot3d import axes3d\n", "import matplotlib.pyplot as plt\n", "import numpy as np\n", "%matplotlib notebook\n", "\n", "X = np.linspace(-1,1,12)\n", "Y = np.linspace(-1,1,12)\n", "Z = np.linspace(-1,1,12)\n", "x,y,z = np.meshgrid(X,Y,Z)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The magnetic field on the grid points around the wire with current $I$ is:" ] }, { "cell_type": "code", "execution_count": 63, "metadata": {}, "outputs": [], "source": [ "I = 200000\n", "Bx,By,Bz = B(I,x,y) " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "And finally, we plot of the magnetic field (our first 3D plot with matplotlib!):" ] }, { "cell_type": "code", "execution_count": 64, "metadata": {}, "outputs": [ { "data": { "application/javascript": [ "/* Put everything inside the global mpl namespace */\n", "window.mpl = {};\n", "\n", "\n", "mpl.get_websocket_type = function() {\n", " if (typeof(WebSocket) !== 'undefined') {\n", " return WebSocket;\n", " } else if (typeof(MozWebSocket) !== 'undefined') {\n", " return MozWebSocket;\n", " } else {\n", " alert('Your browser does not have WebSocket support.' +\n", " 'Please try Chrome, Safari or Firefox ≥ 6. 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