A finite conductor `AB` carrying current `i` is placed near a fixed very long wire current carrying `i_(0)` as shown in the figure.Find the point of application and magnitude of the net ampere force on the conductor `AB`.What happens to the conductor `AB` if it is free to move. (Neglect gravitational field)

Answer

`F=mu_(0)/(2pi)(i_(0)i)ln(1+l/a)` in the direction of `i_(0)`
`x=l/(ln(1+l/a))`,where `x` is the perpendicular distance from the wire `i_(0)`.It will move upward and will roted in the clockwise direction.

Explanation

Force on differential length `dF=i.(mu_(0)i_(0))/(2pi) dx`
Net force `F=(mu_(0)i_(0))/(2pi) underset(a)overset(l+a)int (dx)/x`
`=(mu_(0)i_(0))/(2pi) ln ((l+a)/a)`
Toque about point of application `=0`
Let point of application is at a distance `x_(0)` from wire `i_(0)` .
`d tau=(mu_(0)i_(0)i(x-x_(0)))/(2pi x)dx`
`tau=(mu_(0)i_(0)i)/(2pi)underset(a)overset(l+a)intdx`
`rArr [x-x_(0)lnx]_(a)^(l+a)=0 rArr l+a-x_(0)ln(l+a)-a+x_(0) ln a -0`
`x_(0)ln ((l+a)/a)=l`
`x_(0)=l/(ln(1+l/a))`