Use direct integration to find electric field inside a uniformly charged non-conducting solid sphere. The radius of the sphere is R, observing point is at a way from center of the sphere while a<R.
Use Coulomb's law only. No Gauss law is allowed. You may use Gauss law to verify your result.
The Attempt at a Solution
Performing the triple integral is quite sweating. I managed to carry it out. My answer agrees with Gauss's law. I'm not completely satisfied with the way I do it.
My "brutal force" solution can be found at: http://gradovec.com/scratches/solid-sph.pdf
In the final integration with respect to r, I found that there is a jump discontinuity(also a pole) in the integrand at r=a. hence need to handle it carefully.
My questions are:
1) Did I handle the improper integral correctly?
2) Is there any textbook demonstrate how to carry out the direct integration? I can't find any in popular undergrad and graduate level textbook.. Why this is not popular?
3) I'm thinking of using dirac delta function, but don't see how to setup the integral. Can the integral be simplified by using elementary distribution theory? If yes, how?
4) Is there other simpler way to handle the singularity in the integrand?
Thank you for your time.
P.S. This is not a homework question. I've done my degree 17 years ago.