In the figure four long straight wires are perpendicular to

In the figure, four long straight wires are perpendicular to the page, and their cross sections form a square of edge length a = 5.78 cm. Each wire carries 16.4 A, and all the currents are out of the page. What is the magnitude of the net magnetic force per unit length on wire 1?

Solution

set 1 - 2, The wires carry the same current I in the same direction.

magnitude = mu₀ I² / (2 pi a) per unit length.

Force is in the positive direction. F₂ = [ mu₀ I² / (2 pi a) ] i

set 1-3, Force is in the negative y direction. F₃ = -[ mu₀ I² / (2 pi a) ] j

set 1-4, The two are now a distance a sqrt 2 apart, and the attractive force is in the direction i - j

A unit vector in this direction is 1 / sqrt 2 (i - j)

F₄ = mu₀ I² / (2 pi a sqrt 2) * (1 / sqrt 2)(i - j) = mu₀ I² /(4 pi a) (i - j)

The total force per unit length is F₂ + F₃ + F₄

= [ 3 mu₀ I² / (4 pi a) ] (i - j)

= [ 3 * 4 pi * 10^-7 * 16.4² ] / [4 pi * 5.78 * 10^-2 ]

= 0.00139 N/m = 1.39 * 10^-3 N/m