(a)It is clear from symmetry that the field vanishes at the center.

(b)\(E=\frac{qz}{4 \pi \epsilon_0 (z^2+R^2)^{3/2}}\)

\(z=\infty\)

\(E=0\)

(c)\(\frac{d}{dz}\frac{qz}{4 \pi \epsilon_0 (z^2+R^2)^{3/2}}=\frac{q}{4 \pi \epsilon_0}\frac{R^2-2z^2}{(z^2+R^2)^{5/2}}=0\)

\(z=+\frac{R}{\sqrt{2}}=0.707R\)

(d)\(E=\frac{qz}{4 \pi \epsilon_0 (z^2+R^2)^{3/2}}\)

\(z=+\frac{R}{\sqrt{2}}\)

\(E=4.33*10^7 N/C\)