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 Homework Statement

The electric field outside and an infinitesimal distance away from a
uniformly charged spherical shell, with radius R and surface charge
density σ, is given by Eq. (1.42) as σ/0. Derive this in the following
way.
(a) Slice the shell into rings (symmetrically located with respect to
the point in question), and then integrate the field contributions
from all the rings. You should obtain the incorrect result of
##\frac{\sigma}{2\epsilon_0}##.
(b) Why isn’t the result correct? Explain how to modify it to obtain
the correct result of ##\frac{\sigma}{2\epsilon_0}##. Hint: You could very well have performed
the above integral in an effort to obtain the electric
field an infinitesimal distance inside the shell, where we know
the field is zero. Does the above integration provide a good
description of what’s going on for points on the shell that are
very close to the point in question?
 Homework Equations
 Coulomb's Law
Hi! I need help with this problem. I tried to do it the way you can see in the picture. I then has this:
##dE_z=dE\cdot \cos\theta## thus ##dE_z=\frac{\sigma dA}{4\pi\epsilon_0}\cos\theta=\frac{\sigma 2\pi L^2\sin\theta d\theta}{4\pi\epsilon_0 L^2}\cos\theta##.
Then I integrated and ended up with ##E=\frac{\sigma}{2\epsilon_0}\int \sin\theta\cos\theta d\theta##. The problem is that I don't know what are the limits of integrations, I first tried with ##\pi##, but I got 0. What am I doing wrong?