Question

Two coaxial circular loops of raadii `r_(1)` and `r_(2)` are separated by a distance `x` and carry currents `i_(1)` and `i_(2)` respectively. Calculate the mutual inductance. What is the force between the loops ?

Answer 1

Magnetic field due to loop `(1)` at `(2)`
`B_(1)=(mu_(0)i_(1)r_(1)^(2))/(2(r_(1)^(2)+x^(2))^(3//2))` , along the axis
Flux passing through `(2)`
`phi=B_(1)A_(2)=(mu_(0)i_(1)r_(1)^(2))/(2(r_(1)^(2)+x^(2))^(3//2))pir_(2)^(2)`
Mutual inductance
`M=(phi_(2))/(i_(1))=(mu_(0)pir_(1)^(2)r_(2)^(2))/(2(r_(1)^(2)+x^(2))^(3//2))`
Magnetic moment of loop `(2)`
`M_(2)=i_(2).pir_(2)^(2)` , along the axis
`P.E.` of loop `(2)` ltbr. `U=-vec(M)_(2).vec(M)_(1)=-M_(2)B_(1)=-(i_(2)pir_(2)^(2)mu_(0)i_(1)r_(1)^(2))/(2(r^(2)+x^(2))^(3//2))`
`F=-(dU)/(dx)=(mu_(0)i_(1)i_(2)pir_(1)^(2)r_(2)^(2))/(2)(-(3)/(2))(r^(2)+x^(2))^(-5//2)(2x)`
`=-(3mu_(0)i_(1)i_(2)pir_(1)^(2)r_(2)^(2)x)/(2(r^(2)+x^(2))^(5//2))`
`-ve` sign shows that foorce is attractive.