Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 2 / 16D (Electromagnetism)

The Maxwell equation $\nabla \times \mathbf{E}=-\partial \mathbf{B}/\partial t$, integrated over a fixed spanning surface $S$, gives by Stokes theoremThus the electromotive force and magnetic flux obey Faraday's law

By cylindrical symmetry the field is azimuthal and constant on a circle of radius $r$. The Ampère's law for a steady current givessoIn the plane $x=0$ with $y>0$, it points in the $-x$ direction.

At time $t$, let the centre of the loop have coordinate $Y=d+vt$. With normal $+{\mathbf{e}}_x$, the magnetic flux isFaraday's law and Ohm's law therefore give a current of magnitudeThe original flux points into the plane and decreases in magnitude as the loop recedes. By Lenz's law, the induced current reinforces the into-plane field: it is clockwise when viewed from the $+x$ side.

Translation parallel to the wire leaves every point of the loop at the same distance from the wire, so its magnetic flux is constant. Equivalently, the line integrals of the motional field $\mathbf{u}\times \mathbf{B}$ cancel on opposite sides. Hence