Lec4 - Winter 2008 Lecture 4 Gauss' Law for electric flux...

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Winter 2008 Lecture 4 Page 1 of 4 Lecture 4 Gauss’ Law for electric flux density Up to this point in the course, we have used Coulomb’s law to determine the effect of electric charge on an environment. The advantage of this approach is that the linearity of electric fields insure that using Coulomb’s law will always work (for any charge distribution). However, the complexity of the geometry of the problem, can make some integrals very difficult to solve and the use of superposition may be cumbersome for some problems. It is possible to attack the problem using a different relationship. Gauss’ Law of conservation of electric flux : An experimental law that determined that the net migration of charge (electric flux ( 29 e Ψ ) out of a closed surface ( S ) must be equal to the total electric charge contained within the volume ( Q encl ) of the enclosed surface. This is written as ∫∫ = Ψ = S e encl S d D Q . The electric flux density ( D ) is proportional to the electric field ( E ) and is given by E D e = . As an exercise, we can check that the units for flux density (C/m 2 ), the permittivity of the environment ( e ) (F/m) and the electric field (V/m) are in agreement. We know that capacitance is V C F = , so that the permittivity is m F Vm C m V m C E D = = = = 2 e . So the units agree. This type of exercise of often useful when you are trying to determine if your approach is correct, it is a good sign if the units agree. Also we note that the electric flux density is parallel to the electric fields when the medium is isotropic (i.e. the permittivity is not a tensor). The advantage to using Gauss’ law is apparent when the charge distribution is highly symmetric about a regular form (sphere, cylinder, rectangular box, …). With high symmetry the integrals may be solved quickly. So when you are attempting to use Gauss law, it is best to define the surface of integration so that the flux density is either perpendicular and constant over the surface or tangential to the surface. Since for isotropic media the electric field and flux are parallel, choosing the surface so that the line of action is either perpendicular or parallel will make the problem less complicated.
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Lec4 - Winter 2008 Lecture 4 Gauss' Law for electric flux...

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