An infinitely long current carrying wire and a small current carrying loop are in the plane of the paper as shown. The radius of the loop is a and distance of its centre from the wire is d (d>>a). 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((d^(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)(I pi a^(2))/((d^(2)+x^(2))^(3//2))`
`F=int dF=(mu_(0)I_(0)Ia^(2))/4 int(dx)/((d^(2)+x^(2))^(3//2))`
`x/d = tan theta implies 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_(0)Ia^(2))/4 I/(d^(2)) int_(-pi//2)^(pi//2) cos theta d theta=(mu_(0)I_(0) Ia^(2))/(2d^(2))`