A thin rod of length \(2 \ell\) has a linear charge density that is \(\lambda_{0}\) at the left end but decreases linearly with distance going from left to right in such a way that the charge on the entire rod is zero.

(a) What is the magnitude of the electric field along the rod's axis at a position \(\mathrm{P}\) that is a distance \(d>\ell\) to the right of the center of the rod?

(b) Does your expression simplify to a familiar result for the limiting case \(d \gg \ell\) ? (Hint: The dipole moment of the rod has magnitude \(p=2 \lambda_{0} \ell^{2} / 3\).)