Asolid metallic sphere of radius a carries total charge q. no other charges are nearby. the electric field just outside its surface is keq/a2 radially outward. at this close point, the uniformly charged surface of the sphere looks exactly like a uniform flat sheet of charge. is the electric field here given by σ/ε0 or σ/2ε0? . σ. ε0. σ. 2ε0.

The best and most correct answer among the choices you provided is σ/2ε0. The solution is as follows:

If you use a right circular cylinder as the gaussian surface with the"flat end" oriented parallel to the surface fo the sphere (assuming your small radial distance) then 

Integral(E*dA) = q/e0 = 1/e0 Integral( s dA) 

E = s/e0

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