Past exam of the physics course of the year 1 of the natural sciences course of the University of Cambridge / 2014 / Paper 1 / E15 / c / Solution

For steady currents in vacuum, the Biot-Savart law sums the contributions of current elements:It is useful for known wire geometries; a circular loop of radius $a$ gives $B={\mu }_{0}I/(2a)$ at its centre.

Ampère's circuital law in magnetostatics isWith sufficient symmetry it finds fields directly: a long straight wire gives $B={\mu }_{0}I/(2\pi r)$, while a long solenoid has interior $B\simeq {\mu }_{0}nI$ and a small exterior field away from its ends. Without symmetry, a circulation value alone does not determine the local field. For time-dependent electric fields, use the Ampère-Maxwell equation with the displacement-current term ${\mu }_0{ϵ}_{0}d{\Phi }_E/dt$ as well as conduction current.