A particle carrying a charge of \(6 \mathrm{nC}\) is at the center of a thick conducting spherical shell of inner radius \(9 \mathrm{~cm}\) and outer radius \(10 \mathrm{~cm}\).

(a) What is the surface charge density on the outer surface of the shell? 

(b) What is the electrostatic potential (relative to zero at infinity) on the outer surface of the shell?

(c) Compute the potential difference \(\Delta V\) between the outer surface of the shell and a point A located at a distance \(d=1 \mathrm{~mm}\) outside it.

(d) Use the approximate expression \(E=\Delta V / d\) to calculate the electric field at the surface of the shell. Compare your expression with what you would have obtained by considering the shell as an infinitely wide conducting plane carrying the same charge density.

(e) Evaluate the error you made compared to the exact result for the electric field at A. \(\cdot\)