Two coaxial loops of radii 0.5 m and 0.05 m are separated by distance 0.5 m and carry currents 2 A and 1 A respectively. What is the force between the loops

Explanation

As shown in figure, the field due to coil `C_1` at the centre of `C_2,barB_1=mu_0/(4pi)(2piR_1^2)/((R_1^2+x^2)^(3//2))(bari)`

and as the magnetic moment of coil `C_2`
`barM_2=I_2barS_2=I_2piR_2^2(bari)`
and as its radius `R_2ltltx` so treating it as a dipole in the field of coil `C_1`, its potential energy
`U=-barM_2.barB_1=(mu_0)/(4pi)(2pi^2R_1^2R_2^2I_1I_2)/((R_1^2+x^2)^(3//2))" "["as"bari.bari=1`]
and hence the force between the coils `C_1 and C_2`
`F=-(DU)/(dx)=mu_0/(4pi)(2pi^2R_1^2R_2^2I_1I_2)d/(dx)[(R_1^2+x^2)^(-3//2)]`
i.e., `F=-mu_0/(4pi)(6pi^2R_1^2R_2^2I_1I_2x)/((R_1^2+x^2)^(5//2)).....(82)`
So, `F=-10^(-7)(6pi^2xx(0.5)^2xx(0.05)^2xx1xx2xx(0.5))/([(0.5)^2+(0.5)^2]^(5//2))=-2.09xx10^(-8)N`
The negative sign here implies that the force is attractive and if the current in the two coils had been in opposite directions the force will be positive i.e., repulsive, which is in accordance with the fact that similar currents attracts while oppsite repel