In order for you to get a little used to the differential form of Gauss's law. solve the following problems. Problem 1: Infinite wire and Gauss's law The electric field outside an infinite line that runs along the z-axis and carries a line charge density A is equal to E = 21 4 Teo in cylindrical coordinates. (a) Find the divergence of the E-field for s >0 b) Calculate the electric flux out of an imaginary "Gaussian" cylinder of length L. and radius a, centered around the z-axis. Do this in two different ways to check yourself: by direct integration, and using Gauss's law. c) Given parts (a) and (b). what is the divergence of this E-field? (Hint: your first step should be to derive the divergence in cylindrical coordinates. Also. be aware your answer cannot be zero everywhere! Why not?' Problem 2 The electric field intensity in free space is F(r) = 1A22 + int + Cx2 z where A = 3V /m3, B = 2V /m2 ,C = 1V/0 . Using the differential form relating divergence of the electric field and density. what is the charge r = i2 - #2 m? Problem 3 Given the form of the form of the field E(x.y,z) = 4 x yi + 2(x2 + z2)/^/ + 4yz$ find the charge enclosed in the region delimited by 0<x<2; 0<y<3: 0<z<5.
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