Two long thin parallel conductors C and D of the shape as shown in figure. Carry currents `I_1` and `I_2`. The separation between the conductors is a, the width of the right hand conductor is equal to b. Both the conductors are lying in one plane. Find the magnetic interaction force between them reduced to a unit length.

Explanation

Consider an elementary strip `dx` of conductor D at a distance x from conductor C. Mangnetic field induction at a point on this strip due to current through the conductor C is
`B=(mu_0)/(4pi)(2I_1)/(x)` acting perpendicular to the plane, downwards
Current through the elementary strip, `dI=(I_2dx)/(b)`
Force on unit length of this elementary strip.
`dF=dIxxdlBsin90^@`
`:. dF=(I_2dx)/(b)xx1xx(mu_0)/(4pi)(2I_1)/(x)=(mu_0)/(4pi)(2I_1I_2)/(b)(dx)/(x)`
Total force per unit length of interaction is
`F=int dF=(mu_0)/(4pi)(2I_1I_2)/(b)int_(a)^((a+b))`
`=(mu_0)/(4pi)(2I_1I_2)/(b)log_e(((a+b))/(a))`