Potentials of thin and thick wires (26 marks) (a) (3 marks) using gauss's law, show that in cylindrical coordinates the electrie field around an infinitely long, thin wire coincident with the z-axis and with carrying charge a per unit length is given by 2reo s (b) (4 marks) somewhat arbitrarily choosing the zero reference for potential to occur at a radial distance of so from the wire, derive an expression for the electric potential at s. further, show that choosing either so - 0 or so-oo as the zero reference for potential is problematic. (e) (6 marks) consider now a pair of infinitely-long, thin wires which are parallel and separated by a distance d. suppose one carries a charge λ per unit length while the other carries a charge-a per unit length. analysing the problem by using cylindrical coordinates with the z axis parallel to both wires and situated midway between them, find the potential everywhere. show that it is now possible to take s → oo as the zero reference for potential. show further that the potential is also zero over the midplane between the wires. (d) (8 marks) show that the vo equipotential surfaces are cylinders with radius c2) and centre at coth (2(hit: it suffices why? to show that the equipotentials in a 2d plane perpen- dicular to the z-axis are circles. despite having posed this problem in cylindrical coordinates in part (e), i find it much easier to recognise the equation for a cirele if at this point the problem is recast into cartesian coordinates.) consider now a pair of infinitely-long, thick, conducting wires, each or radius r, which are parallel with centres separated by a distance d. suppose one is set at potential vo and the other at potential -vo (e) (2 marks) explain how the result obtained in part (d) provides a "short-cut" to solving this problem (hint: think in the spirit of the "method of images". note: you need only provide a qualitative explanation here; the detailed analysis is part ( (f) (3 marks) derive expression(s) for the potential everywhere, in eylindrical coordinates with origin at the midplane between the wires and the z-axis running parallel to the wires.