Past exam of the mathematics course of the University of Cambridge / 2023 / ib / Paper 3 / 15D / d / Solution

Measure $z$ from the centre of the loop of radius $R$, toward the second loop. Between the loops the two axial fields have opposite directions. Their magnitudes areEquating them and cancelling common factors givesTaking the power $2/3$ yields $(D-z{)}^2=4z^2$. Since $0<z<D$, we have $D-z=2z$, and thereforeThus the magnetic-field cancellation between oppositely driven coaxial loops occurs one third of the separation from the smaller loop.