Charge distribution conductor in electric field

Surface charge density of a conductor at a particular region on its surface is inversely proportional to the radius of curvature at that region. But this is valid only when there is no external electric field.

How will we calculate if external electric field is also present? Please explain with any example.
I think I have to integrate taking segments of different radius of curvatures on that conductor, but for each dx the force due to external electric filed will be at different direction so we have to take components in x and y direction.