Two infinite straight wires run parallel to the -axis

PHYSICS · Multi-part · Fermi University Physics · Electricity, Magnetism & Electromagnetism · Magnetostatics · Magnetostatics

The Boundary-Value Problem, Ch The vector potential and the magnetostatic boundary-value problem

Problem

Two infinite straight wires run parallel to the

z

-axis, separated by

d=4.0\,\mathrm{cm}

: one at

x=+d/2

y=0

carrying current

I=15\,\mathrm{A}

in the

+z

direction, the other at

x=-d/2

y=0

carrying the same current

I

in the

-z

direction (antiparallel wires). This is the magnetic analogue of two infinite line charges

+\lambda_\ell

and

-\lambda_\ell

met earlier. ((a)) Write the vector potential of a single infinite wire carrying current

I_0

in the

+z

direction, at perpendicular distance

r

, and use it, exactly as the two line charges' potentials were superposed, to write

A_z(x,y)

for the pair of wires above in terms of the distances

r_1

(to the

+z

wire) and

r_2

(to the

-z

wire). ((b)) Evaluate

A_z

at the point

(x,y)=(3.0,\,2.0)\,\mathrm{cm}

(taking the midpoint between the wires as the origin). ((c)) Explain why the answer does not depend on the reference radius

s_0

, and identify the corresponding feature of the two-line-charge potential.