Past exam of the physics B course of the University of Cambridge / 2018 / Paper 1 / Section B / B7 / b / Solution

At $(x,0,0)$ the field is perpendicular to the loop plane. Its Biot-Savart numerator isThe leading denominator variation must be kept:The integral of $-xd{y}^{\prime }$ is zero, but its product with $3x^{\prime }/x$ is not. Keeping all terms of the leading nonzero order givesFor the counterclockwise rectangle, $\oint x^{\prime }d{y}^{\prime }=\Delta x\Delta y$ and $\oint y^{\prime }d{x}^{\prime }=-\Delta x\Delta y$. HenceThis transverse field is opposite to the magnetic dipole moment. Replacing every denominator by a constant before examining the term proportional to $x$ would give the wrong answer.