Classical Electrodynamics · Problem 5.15

Maxwell Equations, Macroscopic Electromagnetism, Conservation Laws — John David Jackson

Physics — E&M · two-parallel-wires-magnetic-scalar-potential

Problem

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

\mathbf{H}

in terms of a magnetic scalar potential

\Phi_M

, with

\mathbf{H} = -\nabla\Phi_M

. (a) If the wires are parallel to the

z

axis with positions,

x = \pm d/2

y = 0

, show that in the limit of small spacing, the potential is approximately that of a two-dimensional dipole,

\Phi_M \approx -\frac{Id\sin\phi}{2\pi\rho} + O(d^2/\rho^2)

where

\rho

and

\phi

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

\mu = \mu_r \mu_0

. Determine the magnetic scalar potential in the three regions,

0 < \rho < a

a < \rho < b

, and

\rho > 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

F = \frac{4\mu_r b^2}{(\mu_r + 1)^2 b^2 - (\mu_r - 1)^2 a^2}

Relate your result to Problem 5.14. (c) Assuming that

\mu_r \gg 1

, and

b = a + t

, where the thickness

t \ll b

, write down an approximate expression for

F

and determine its numerical value for

\mu_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.