Level 1: Same Wire, Two Turns Instead of One: Field ×2?
A circle or a square?
The same length of wire, carrying the same current, is bent once into a circle and once into a square. Which gives the stronger magnetic field at its centre?
For a wire of length L, the circle has radius L/2π. Each side of the square is L/4 long, at a distance L/8 from the centre, and each side gives the field of a finite straight wire whose ends lie at 45° on either side.
The ratio is 8√2/π² ≈ 1.15, so the square gives about 15% more field than the circle. Its sides come closer to the centre than the circle does, which more than makes up for its far corners.
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