A proton accelerator produces a narrow beam of protons, each having an initial speed of v_0. The beam is directed towards an initially uncharged distant metal sphere of radius R. The sphere is fixed and centered at point O. The initial path of the beam is at a distance of (R/2) from the centre, as indicated in the diagram. The protons in the beam that collide with the sphere get absorbed and cause it to become uniformly charged. The subsequent potential field at the accelerator due to the sphere can be neglected. Assume the mass of the proton as m_P and the charge on it as e. (a) After a long time, when the potential of the sphere reaches a constant value, sketch the trajectory of proton in the beam. (b) Once the potential of the sphere has reached its final, constant value, find the minimum speed v of a proton along its trajectory path. (c) Find the limiting electric potential of the sphere.