Three point charges are fixed collinearly in air. Charge q is at the origin, charge 4q at x=a, and another charge q at x=2a.
a. (i) For a point P located on the x–axis between the first two charges at a coordinate x (where 0<x<a), write the expression for the net electric field E(x) in terms of q, a, and x.
b. (ii) Set E(x)=0 and solve for x to find the position(s) between the first two charges where the electric field vanishes.
c. (iii) Verify algebraically whether any additional point of zero field exists on the line segment between x=a and x=2a or outside the outer charges. Provide logical reasoning for your conclusion.