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Unformatted text preview: Chapter 33 Gauss’s Law 296 33 Gauss’s Law When asked to find the electric flux through a closed surface due to a specified nontrivial charge distribution, folks all too often try the immensely complicated approach of finding the electric field everywhere on the surface and doing the integral of E h dot A d h over the surface instead of just dividing the total charge that the surface encloses by e o . Conceptually speaking, Gauss’s Law states that the number of electric field lines poking outward through an imaginary closed surface is proportional to the charge enclosed by the surface . A closed surface is one that divides the universe up into two parts: inside the surface, and, outside the surface. An example would be a soap bubble for which the soap film itself is of negligible thickness. I’m talking about a spheroidal soap bubble floating in air. Imagine one in the shape of a tin can, a closed jar with its lid on, or a closed box. These would also be closed surfaces. To be closed, a surface has to encompass a volume of empty space. A surface in the shape of a flat sheet of paper would not be a closed surface. In the context of Gauss’s law, an imaginary closed surface is often referred to as a Gaussian surface . In conceptual terms, if you use Gauss’s Law to determine how much charge is in some imaginary closed surface by counting the number of electric field lines poking outward through the surface, you have to consider inwardpoking electric field lines as negative outwardpoking field lines. Also, if a given electric field line pokes through the surface at more than one location, you have to count each and every penetration of the surface as another field line poking through the surface, adding +1 to the tally if it pokes outward through the surface, and − 1 to the tally if it pokes inward through the surface. So for instance, in a situation like: we have 4 electric field lines poking inward through the surface which, together, count as − 4 outward field lines, plus, we have 4 electric field lines poking outward through the surface which together count as +4 outward field lines for a total of 0 outwardpoking electric field lines through the closed surface. By Gauss’s Law, that means that the net charge inside the Gaussian surface is zero . Closed Surface E Chapter 33 Gauss’s Law 297 The following diagram might make our conceptual statement of Gauss’s Law seem like plain old common sense to you: The closed surface has the shape of an egg shell. There are 32 electric field lines poking outward through the Gaussian surface (and zero poking inward through it) meaning there must (according to Gauss’s Law) be a net positive charge inside the closed surface. Indeed, from your understanding that electric field lines begin, either at positive charges or infinity, and end, either at negative charges or infinity, you could probably deduce our conceptual form of Gauss’s Law....
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 Fall '08
 RABE
 Physics, Charge, Gauss' Law, Electric charge, Surface, Gauss’s Law

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