Two coaxial loops of radii 0.5 m and 0.05 m are separated by a distance 0.5 m and carry currents 2A and 1A 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),vecB_(1)=mu_(0)/(4pi)(2piR_(1)^(2))/((R_(1)^(2)+x^(2))^(3//2))(veci)`

and as the magnetic moment of coil `C_(2)`
`vecM_(2)=I_(2)vecS_(2)=I_(2)piR_(2)^(2)(veci)`
and as its radius `R_(2)lltx` so treating it as a dipole in the field of coil `C_(1)`, its potential energy
`U=-vecM_(2)*vecB_(1)=mu_(0)/(4pi)(2pi^(2)R_(1)^(2)R_(2)^(2)I_(1)I_(2))/((R_(1)^(2)+x^(2))^(3//2))" "["as"veci*veci=1]`
and hence the force between the coil `C_(1)andC_(2)`.
`F=-(dU)/(dx)=mu_(0)/(4pi)(2pi^(2)R_(1)^(2)R_(2)^(2)I_(1)I_(2))d/(dx)[(R_(1)^(2)+x^(2))^(-3//2)]`
i.e., `F=-u_(0)/(4pi)(6pi^(2)R_(1)^(2)R_(2)^(2)I_(1)I_(2)x)/((R_(1)^(2)+x^(2))^(3//2))` ----(101)
So, `F=-10^(-7)(6pi^(2)xx(0.5)^(2)xx(0.5)^()xx1xx2xx(0.5))/([(0.5)^(2)+(0.5)^(2)]^(5//2))=-2.09xx10^(-8)N`
The negative sign her implies that hte 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 attract while opposite repel.