Electric field of a part of a hemisphere

[ermia]

Homework Statement

There is a problem in krotov which I need help solving it.(3.4)
the solution of krotov is not obvious for me.
The problem's statement is uploaded as an image.
It says we want to find the electric feild of a part of a hemisphere with angle alpha at its center.

Relevant Equations

Maybe nothing but vector summation.

I tried gauss law.
And the fact that if alpha is less than pi/2 we can say that we have two parts with angle alpha and one other part which has a normal field at the center.
But non of them helped me answer.

The problem's solution says that we can use the fact that our section has longitudinal and transverse symmetry. But how it has such a symmetry?

 

[TSny]

Imagine a plane that is perpendicular to the diameter AB and also contains the line OC. By symmetry, the net electric field at O must lie in this plane. I think this might be the "transverse symmetry". The plane transversely cuts the region in half.

Imagine a plane that contains the diameter AB and bisects the angle $\alpha$. This plane contains the colored region ADB. It also cuts the charged region in half. By symmetry, the net electric field at O must lie in this plane. I think this might be the "longitudinal" symmetry.

Anyway, whatever we call these planes, the net field at O must lie in both planes. That is, the field must lie along the intersection of these two planes.

Even though we have these two symmetry planes, there is not enough symmetry to use Gauss' law. But Krotov avoids integration by using a very clever argument.