A hole of radius R 1 = 0.4 m is drilled alongthe symmetry axis of an infinite, n

Question

A hole of radius R1 = 0.4 m is drilled alongthe symmetry axis of an infinite, non-conducting solid cylinder ofradius R2 = 2.7 m. The resulting annularcylinder is given a uniform volume charge density r = +1.1 µC/m3.

(a) Calculate the magnitude of the electric fieldE at the following values of r, wherer is the radial distance measured perpendicularly from theaxis of symmetry (z-axis).

At r = 0.2 m: E = N/C   

At r = 0.6 m: E = N/C   

At r = 5.4 m: E = N/C    

HELP:  Rather than approaching thisproblem as a "standard" Gauss's Law problem with a "thick"cylindrical shell of charge, it is helpful to view it as asuperposition of two cylindrical charge distributions. Do you seehow? (The benefits will become clear in the second part of theproblem.)
HELP:  Good, you see now that theoriginal charge distribution is equivalent to the superposition ofa solid infinite cylinder of radius R2 andcharge density +r with a solidinfinite cylinder of radius R1 and chargedensity -r, with the axes ofsymmetry of the two cylinders coincident.

Suppose instead the hole was parallel to the z-axisalong the line (x, y) = (0, 0.6) m.

(b) Calculate the resulting x- andy-components of the electric field at the point (x, y, z)= (0, 5.4, 0) m.

Ex = N/C   

Ey = N/C   

(c) Calculate also the x- and y-components ofthe electric field at the point (x, y, z) = (0, 0.6, 0) m.

Ex = N/C   

Ey = N/C    

HELP:  Does the given chargedistribution have the prerequisite symmetry to be solvable directlyas a "standard" Gauss's Law problem? If not, what superposition of"symmetrical" charge distributions is equivalent to the givencharge distribution?
HELP:  Proceed by superimposing (adding)the electric fields of each of two symmetrical chargedistributions. Be sure to draw a diagram to assist you incalculating the relevant distances properly.

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