# Magnetic Field of a Current Loop

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A electrical current moving around a circular loop of radius , shown in yellow from a lateral point of view, produces a magnetic field, with lines of force shown as blue loops. For clarity, only lines of force in the vertical plane bisecting the ring () are shown.

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Contributed by: S. M. Blinder (March 2011)

Open content licensed under CC BY-NC-SA

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## Details

The vector potential for a ring of current can be solved exactly in terms of complete elliptic integrals. Transforming to Cartesian coordinates in the vertical plane bisecting the ring, the following approximation is found to be indistinguishable graphically from the exact solution: . The magnetic induction is given by , so that and . Since this is a magnetostatic problem, the magnetic induction can also be represented in terms of a scalar potential , so , . The magnetic lines of force are represented by contours orthogonal to those of the potential , thus . This equation can be satisfied by

Reference: J. D. Jackson, *Classical Electrodynamics, *3rd ed., New York: John Wiley & Sons, 1999 pp. 181–186.

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