An infinitely long positively charged wire carries a uniform linear charge density \(\lambda\). A charged particle, carrying a charge \(q\), is placed at a distance \(2 d\) from the wire.

(a) What is the relation between \(q\) and \(\lambda\) so that the magnitude of the electric field at the midpoint between the particle and the wire is pointing toward the charge and has magnitude \(2 E\), where \(E\) is the magnitude of the electric field produced by the wire?

(b) At what positions is the vector sum of the electric fields generated by the wire and by the particle equal to zero?

(c) What is the magnitude of the electric field produced by the wire at the locations identified in part \((b)\) ? (Express your results in terms of \(E\).)