A conductor carries a current I parallel to a current strip of current per unit width j and width w, as shown in figure . Find an expression for the force per unit length on the conductor. Discuss the result when the width w approaches infinity.

Explanation

We consider a differential segment of thickness `dx`, carrying current `j dx`. The attractive force per unit length is
`(dF)/(L)=(mu_(0)I(j dx))/(2pir)`
A symmetrical segment at `-x` exerts same magnitude of force, the `x-` components cancel and the resultant force is
`(dF)/(L)=(mu_(0)I(j dx))/(2pir)cos theta`
`=(mu_(0)Ijdx)/(2pi sqrt(h^(2)+x^(2)))xx(h)/(sqrt(h^(2)+x^(2)))`
Integrating over half width of strip.
`(F)/(L)=(mu_(0)Ijh)/(pi)int_(0)^(w//2)(dx)/(h^(2)+x^(2))=(mu_(0)Ij)/(pi)tan^(-1)((w)/(2h))`
When `w rarr oo , (F)/(L)rarr(mu_(0)Ij)/(2),` the force is attractive as expected for parallel currents.