ece331 homework four solution, Assignments of Electromagnetic Engineering

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Homework on Gauss’s Law 1. There are electric charges in a certain region of space. In cylindrical coordinates (r, ¢, z) the distribution of charges is characterized as py =5re* ~— (C/m*) Find the electric field E induced by the charges. Solution The charge density is a function of r only. On a coaxial cylinder the charge will produce an electrostatic field in a way like a infinite line charge. So the field E will have the component E, and the other components Ey, and E, are zero. E=E, Thus we construct a Gaussian surface S$ as a coaxial cylinder, with a vertical surface of radius be r and length / plus a top and a bottom discs of the same radius. We want to use Gauss’s law to figure out the field E. f5-a-2 The differential surface d& in the top dise is Zrd¢édr and that in the bottom disc —Zrdgdr. The dot product of the field and ds is zero with both the discs. So, the surface integral on the left hand side is to be on the vertical wall of the cylinder. Further, we argue that the field at a distance r from the z axis is constant. Therefore, the left side of the Gauss’s law is [ #E,- (?rdbdz) = 7 By | / ddide'= Sire «Bi, wall ‘We compute the right hand side to count the total charge inside S$. a= [ nv =| f [tore yraraeas (90) = ant f bre“"dr 0 = 10nl[-r?e~" + 2(1— "(1+ 7))] 95