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 a wire of thickness `dx` at distance `x` from long wire as shown.
Current in this wire
`i' = (i_(2))/(b)dx`
Force per unit length between wires
`dF = (mu_(0)i_(1)i')/(2pi x)`
`F = (mu_(0)i_(1)i_(2))/(2pi b) int_(a)^(a + b) (dx)/(x)`
`= (mu_(0)i_(1)i_(2))/(2pi b)|log_(e) x|_(a)^(a + b)`
`= (mu_(0)i_(1)i_(2))/(2pi b){log_(e)(a + b) - log_(e)(a)}`
`= (mu_(0)i_(1)i_(2))/(2pi b)log_(e)((a + b)/(a))`
`= (mu_(0)i_(1)i_(2))/(2pi b)log_(e)(1 + (b)/(a))`