An infinitely long current carrying wire and a small current carrying wire and a small current carrying loop are in the plane of thye paper as shown. The ardius of the loop is a and distance of its centre from the wire is d (dgtgta). If the loop applies a force F on the wire then :

Options

A. `F prop (a/d)`
B. `F prop (a/d)^(2)`
C. `F=0`
D. `F prop ((a^(2))/(d^(3)))`

Answer

B

Explanation

Considering the loop as magnetic dipole `m=I(pia^(2))` Field due to the dipole at the element dx
`vecB=(mu_(0))/(4pi) m/(r^(3))`
Force on the element
`dF=I_(0)(dx)B=I_(0)dx(mu_(0))/(4pi)(Ipia^(2))/((d^(2)+x^(2))^(3//2))`
`F=intdF=(mu_(0)I_(0)Ia^(2))/rint(dx)/((d^(2)+x^(2))^(3//2))`
`x/d=tan thetaimplies dx = d sec^(2) theta d theta :. int (dx)/((d^(2)+x^(2))^(3//2))=int(d sec^(2)theta d theta)/(d^(3) sec^(3) theta)=1/(d^(2)) int cos theta d theta`
`:. F=(mu_(0)I_(o)Ia^(2))/r I/(d^(2)) int_(-pi//2)^(pi//2) cos theta d theta = (mu_(0)I_(0)Ia^(2))/(2d^(2))`