I solved the following question(Answer is correct):
Find the force with which two hemisperical parts of a uniformly charged hollow sphere repel each other?(charge density: +$\sigma$)
Answer:
Radial force on strip is: $$dF=\left(\frac{\sigma^2}{2\epsilon_0}\right)(2\pi R^2\sin\theta d\theta)$$ Total force on hemisphere is: $$F=\int dF\cos\theta=\int_0^{\frac{\pi}2}\frac{\pi R^2\sigma^2}{\epsilon_0}\sin\theta\cos\theta d\theta$$ $$F=\frac{\pi R^2\sigma^2}{2\epsilon_0}$$
My question is: If the hollow sphere is uniformly charged on one half with a uniform charge density $+\sigma_1$ and its other half is also charged at charge density +$\sigma_2$.Now find the force with which the two halves repel?
Similiar to previous question force can be given as: $$F=\int \frac1{4\pi\epsilon_0}\sigma_1\sigma_2dS_1dS_2$$ But it doesn't give the answer, or I must say it can not be manipulated to an integrable form.Note that this is not a problem of integration.It is simple manipulation, as my teacher says, he suggests using previous problem.
Answer
$$F=\frac{\pi R^2\sigma_1\sigma_2}{2\epsilon_0}$$