Past exam of the physics course of the year 1 of the natural sciences course of the University of Cambridge / 2022 / Section B / 10 / d / Solution

To calculate magnetic-field cancellation of an arc and tangent leads away from the plane, put $a^2=R^2+h^2$. Each point on the arc is distance $a$ from $(0,0,h)$, and the axial component of $d\mathbf{l}\times \mathbf{s}$ still has magnitude $R^{2}d\varphi$. Thus its axial magnetic field isFor either tangent lead the axial numerator remains $Rds$, while the cubed distance is $(R^2+h^2+s^2{)}^{3/2}$. HenceAdding both leads givesFor $ϵ=h/R\ll 1$, a binomial series yieldsThe exact condition for cancellation of the axial component isFor example, keeping $\theta =2$ would leave a small positive axial magnetic field proportional to $h^2/R^2$. Both contributions weaken with height, but the arc factor is $(1+{ϵ}^2{)}^{-3/2}$ whereas the lead factor is $(1+{ϵ}^2{)}^{-1}$. The arc therefore loses a greater fraction of its axial magnetic field. Increasing its angle compensates for this loss. Physically, the arc is all at the same finite radius, whereas distant elements of the semi-infinite leads undergo a smaller fractional change in distance on moving a little out of the plane.

The cancellation here concerns $B_z$ only; off the plane it does not assert that every component of the magnetic field vanishes. The small-height result also stays within the tangent-lead geometry shown in the original paper.