What is the magnetic force acting on the finite wire due to the fixed infinite wire shown here?

Strategy: We cannot use formula `vec(F) = i vec(l) xx vec(B)` to find the force as the magnetic field created by the fixed infinite wire is non-uniform. We have to consider an element on the finite wire, find force on it and integrate to obtain the net force.

Explanation

Let us consider an element of length dx on the finite wire at a distance x from the infinite wire as shown.
Force on the element,
`vec(dF) = ivec(dl) xx vec(B) = i_(2) (dxhat(i) xx (mu_(0) i_(1) (-hat(k)))/(2pi x)) = (mu_(0) i_(1) i_(2) dx)/(2pi x) hat(j)`
Net force on the wire,
`vec(F) = int dvec(F) = int_(r)^(r + a) (mu_(0) i_(1) i_(2))/(2pi x) dx hat(j)`
`= (mu_(0)i_(1)i_(2))/(2pi x) int_(r)^(r +a) (dx)/(x) hat(j)`
`= (mu_(0) i_(1)i_(2))/(2pi x) [ln x]_(r)^(r +a)`
`= (mu_(0) i_(1)^(2))/(2pi x) ln ((r + a)/(r))`