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Electrostatics - Gauss's Law, Field due to spherical charge

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This problem is from a book I have, and has three parts. I have attached both the problem (p23.jpg) and the solution (a23.jpg). I've tried different approaches, but I think I'm either not using the correct formula, or not setting up the integrals correctly. I'd appreciate someone showing me how to arrive at the solutions. Thanks for your help.

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Solution Summary

The charge distributed on the volume of a sphere is given and the electric field strength is calculated.

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