Talk:PlanetPhysics/Biot Savart Law

Original TeX Content from PlanetPhysics Archive

%%% This file is part of PlanetPhysics snapshot of 2011-09-01 %%% Primary Title: Biot-Savart law %%% Primary Category Code: 41.20.Gz %%% Filename: BiotSavartLaw.tex %%% Version: 3 %%% Owner: bloftin %%% Author(s): bloftin %%% PlanetPhysics is released under the GNU Free Documentation License. %%% You should have received a file called fdl.txt along with this file. %%% If not, please write to gnu@gnu.org. \documentclass[12pt]{article} \pagestyle{empty} \setlength{\paperwidth}{8.5in} \setlength{\paperheight}{11in}

\setlength{\topmargin}{0.00in} \setlength{\headsep}{0.00in} \setlength{\headheight}{0.00in} \setlength{\evensidemargin}{0.00in} \setlength{\oddsidemargin}{0.00in} \setlength{\textwidth}{6.5in} \setlength{\textheight}{9.00in} \setlength{\voffset}{0.00in} \setlength{\hoffset}{0.00in} \setlength{\marginparwidth}{0.00in} \setlength{\marginparsep}{0.00in} \setlength{\parindent}{0.00in} \setlength{\parskip}{0.15in}

\usepackage{html}

% this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners.

% almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts}

% used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic}

% there are many more packages, add them here as you need them

% define commands here

\begin{document}

The Biot-Savart law is a \htmladdnormallink{physical law}{http://planetphysics.us/encyclopedia/PrincipleOfCorrespondingStates.html} with applications in both \htmladdnormallink{Electromagnetism}{http://planetphysics.us/encyclopedia/Electromagnetism.html} and aerodynamics. As originally formulated, the law describes the \htmladdnormallink{magnetic field}{http://planetphysics.us/encyclopedia/NeutrinoRestMass.html} set up by a steady current density. More recently, by a simple analogy between \htmladdnormallink{magnetostatics}{http://planetphysics.us/encyclopedia/Magnetostatics.html} and \htmladdnormallink{fluid dynamics}{http://planetphysics.us/encyclopedia/LQG2.html}, the same law has been used to calculate the \htmladdnormallink{velocity}{http://planetphysics.us/encyclopedia/Velocity.html} of air induced by vortex lines in aerodynamic \htmladdnormallink{systems}{http://planetphysics.us/encyclopedia/SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence.html}.

The Biot-Savart law is fundamental to magnetostatics just as \htmladdnormallink{Coulomb's law}{http://planetphysics.us/encyclopedia/CoulombsLaw.html} is to electrostatics. The Biot-Savart law follows from and is fully consistent with Amp\`ere's law, much as Coulomb's law follows from \htmladdnormallink{Gauss' Law}{http://planetphysics.us/encyclopedia/GausssLaw.html}.

In particular, if we define a differential element of current

$$I d{\bf l} $$

then the corresponding differential element of magnetic field is

$$ d{\bf B} = \frac{\mu_0}{4 \pi}\frac{I {\bf dl} \times {\bf\hat{r}}}{r^2} $$

where

I is the current, measured in amperes

$\hat{r}$ is the unit displacement \htmladdnormallink{vector}{http://planetphysics.us/encyclopedia/Vectors.html} from the element to the \htmladdnormallink{field}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html} point and the integral is over the current distribution

{\bf Examples} \begin{itemize} \item \htmladdnormallink{quarter loop example of Biot-Savart law}{http://planetphysics.us/encyclopedia/QuarterLoopExampleOfBiotSavartLaw.html} \item \htmladdnormallink{loop example of Biot-Savart law}{http://planetphysics.us/encyclopedia/LoopExampleOfBiotSavartLaw.html} \end{itemize} {\bf References}

[2] Jackson, D. "Classical Electrodynamics", John Wiley and Sons, Inc., 1975.

This entry is a derivative of the Biot-Savart law article \htmladdnormallink{from Wikipedia, the Free Encyclopedia}{http://en.wikipedia.org/wiki/Biot-Savart\%27s_Law}. Authors of the orginial article include: Salsb, Tim Starling, Metacomet, JabberWok and Toby Bartels. History page of the original is \htmladdnormallink{here}{http://en.wikipedia.org/w/index.php?title=Biot-Savart_law\&action=history}

\end{document}