Question 10/01

AREA MAXIMIZATION

A flexible electric wire-loop (of fixed length) (black line in the picture) is placed on a horizontal plane and is free to slide on it. At certain points rings (shown in red) are anchoring the wire to the plane. The wire can slide through the rings. A battery ensures that a constant current is flowing through the ring. Strong magnetic field B is applied perpendicular to the plane.

(a) Show that the wire will assume shape maximizing the area surrounded by the wire (consistent with the constraints imposed by the rings).

(b) What happens if the rings are placed in such a way that they force the current loop to cross itself?

This problem appears (in a more "commercial" form: maximization of area surrounded by a fence of given length, which must pass through given points) in the book Laws of Electromagnetism by A.A. Borovoi, E.B. Finkel'shtein and A.N. Kheruvimov ("Nauka", Moscow, 1970).

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