Gravitational potential due to a "cap" (a portion of a hollow sphere) at a given point 
In this problem, I understand that the modulus of the gravitational fields must be equal, however I'm not quite able to understand the calculation of potential due to each portion.
I tried calculating the potential by considering differential rings and integrating over the angle subtended at the given point... but it gets very messy and is clearly out of my scope. Is there an analytical way to approach this problem?
 A: I will try to give you a hint to this problem first, and if it's not clear after that how to solve the problem, I will provide a more detailed answer. 
Let's introduce a cartesian system of coordinates with axis $OZ$ pointing along the direction $AB$. Then you can write the expression for the potential at point A:
$$V(A) = - \int_{left \ part \ of \ sphere } \frac{G }{|\mathbf r_A - \mathbf r|} dm (\mathbf r) -  \int_{right \ part \ of \ sphere } \frac{G }{|\mathbf r_A - (\mathbf r + \mathbf r_{AB})|} dm (\mathbf r)$$
Where the variable of integration $dm(\mathbf r)$ is taken to belong to the sphere as if it was not divided into two pieces and the fact that the pieces were separated is taken into account in the addition $\mathbf r _{AB} = (0,0,z_0)$.
Now you can approach the problem analytically by expanding the second integral with assumption that $z_0\gg R$, where $R$ is the radius of the sphere. You just need to expand the denumerator in the second integral into series and then make the estimations.
You don't need to calculate the integral $V(A)$ to obtain the answer. After you expand it into the series, you shoul write a similar expression for $V(B)$ and then compare them.
