Consider two long, straight wires, parallel to the z axis, spaced a distance d apart and carrying currents I in opposite directions. Describe the magnetic field H in terms of a magnetic scalar potential ?_M, with H = ? ??_?.

(a) If the wires are parallel to the z axis with positions, x = ?d/2, ? = 0, show that in the limit of small spacing, the potential is approximately that of a two- dimensional dipole,

Where ? and ? are the usual polar coordinates

(b) The closely spaced wires are now centered in a hollow right circular cylinder of steel, of inner (outer) radius a (b) and magnetic permeability ? = ?_??₀. Determine the magnetic scalar potential in the three regions, 0 b. Show that the field outside the steel cylinder is a two-dimensional dipole field, as in part a, but with a strength reduced by the factor

Relate your result to Problem 5.14.

(c) Assuming that ?_r >> 1, and b = a + t, where the thickness t r?= 200 (typical of steel at 20 G), b = 1.25 cm, t = 3 mm. The shielding effect is relevant for reduction of stray fields in residential and commercial 60 Hz, 110 or 220 V wiring. The figure illustrates the shielding effect for a/b = 0.9, ?_r = 100.

Id sin o O(dip) 2 'M