A single, thin, square loop of side length carries a steady current in the…
PHYSICS · Multi-part · AP · Physics C: Electricity and Magnetism · Magnetic Fields and Electromagnetism · Magnetic Fields and Electromagnetism
Problem
A single, thin, square loop of side length L carries a steady current I in the counter-clockwise direction when viewed from above the plane of the loop. The loop lies in the xy-plane with its centre at the origin and its sides parallel to the coordinate axes.
Throughout this problem treat each side as an infinitely thin straight segment; ignore the curvature at the corners.
(a) Derive an expression for the magnitude of the magnetic field at the centre of the square due to one side only, in terms of I, L, and fundamental constants.
(b) Using symmetry and your result from part (a), obtain the total magnetic field (magnitude and direction) at the centre of the square produced by the entire loop.
(c) A circular loop of radius r carries the same current I and has the same total length of wire as the square loop. Determine the ratio of the magnetic-field magnitude at the centre of the square to that at the centre of the circle.