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Unformatted text preview: Electricity & Magnetism February 3, 2011 Robert Haussman Notes Symmetries and Gauss’ Law Margin Notes H In this first course in Electricity and Magnetism, many students are exposed to symmetry arguments for the first time and often have a bit of trouble with the concepts. Since the book for the course does not discuss symmetries in any real detail, this document (hopefully) should serve as a clarification or supplement to the material. † 1 Motivation To begin, recall Gauss’ Law: The total electric flux Φ E through any closed surface S defined by its out ward normal ˆ n is proportional to the total charge contained within the volume bounded by S. In particular, Φ E = I S ( ~ E · ˆ n )d A = Q enc ε . (1) I Gauss’ Law: An important point to remember is that Gauss’ Law is true for any closed surface S. To make Gauss’ Law useful, we can take advantage of this fact and choose a convenient ! Key Point: Gauss’ Law is true for any closed surface S, so we can choose whatever is most convenient. surface so as to make the flux integral as simple or trivial as possible. The chosen surface is often referred to as a Gaussian surface . But how do we know in advance what surface to use? To see how this works, we will look at the ~ E · ˆ n term in the integrand. Ideally, we would like a surface such that ~ E · ˆ n is a number – that is, the electric field is constant along the surface. Then, that term can be brought out of the integral, with the remaining integral simply being the surface area of S. In general, the electric field will depend on some parameters λ i , so if we can find a surface defined by assigning values to parameters as well as a the unit normal ˆ n parallel to ~ E, then I S ( ~ E ( λ i ) · ˆ n )  {z } # d A = E ( λ i ) I S d A  {z } surface area = E ( λ i ) A. To reiterate, by looking at the form of ~ E we determined a surface S such that the electric field was constant along it. Then with our knowledge of the charge distribution, the electric field at the surface is just the enclosed charge divided by the surface area. This method is great, as long as we already know the form of ~ E. However, in many appli cations of Gauss’ Law, we are just given the charge distribution and want to determine the electric field. The question then becomes, given a charge distribution, what form must the electric field take? That is, what parameters does ~ E depend on and in what direction does it point? The answer is to examine the symmetries of the distribution! † A rather surprising (and unfortunate) discovery is that most textbooks really only mention symmetry arguments in passing or as an obvious fact (e.g. “... by symmetry, we know the field must have this form.. .”). In fact, I have yet to find a satisfying exposition that clearly highlights the different types of symmetries and how to (correctly) take advantage of them when employing Gauss’ Law – hence why I decided to write out these notes. Symmetries and Gauss’ Law...
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 Rotational symmetry, Gauss’ Law

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