Comment on “Magnetic field calculation for arbitrarily shaped planar wires” [Am. J. Phys. 68(3), 254–258 (2000)]

In the paper in the title of this comment, J. A. Miranda¹ (JAM) derives a simple and compact expression to compute the magnetic field caused by a current in a plane loop of wire at a point lying in the plane.

Starting from the Biot–Savart law

where μ₀ is the permeability of free space, $ds$ is an element of length (pointing in the direction of current flow) of a wire which carries a current I, $r̂$ is the unit vector pointing from the element of length to the observation point $O$⁠, and r is the distance from the element of length to the observation point, he obtains the following simple expression for the magnitude of the total magnetic flux density at point $O$⁠:

The expression is conveniently written in terms of the wire's shape $r=r(θ)$ in polar coordinates, where r is the distance of the point on the wire from the origin at $O$⁠, and θ is the counterclockwise angle made by the line joining the point and the origin, and the reference x-line (usually horizontal). JAM illustrates the usefulness of formula (2), Eq. (4) in his paper, calculating the magnetic field at specific points in the wire's plane due to currents flowing in conic curves, spirals, and harmonically deformed circular circuits.

While formula (2) is certainly correct and generally valid (see also Ref. 2), the derivation has a weak point, as we discuss later. The following procedure is at once simpler and more general.

The element of length of a wire $ds$ in polar coordinates can be decomposed into

where $−r̂$ is the unit vector pointing in the direction of increasing r [compare with the definition of $r̂$ in Eq. (1)], $θ̂$ is the unit vector tangent to the circle of radius r, pointing in the direction of increasing θ. The sign ± is connected with the direction of current flow. From this, we get

where $ẑ=−r̂×θ̂$ is the unit vector normal to both $−r̂$ and $θ̂$⁠. By combining Eqs. (1) and (4), we immediately obtain the simple and compact expression (2).

JAM states that Eq. (4) can be obtained with the help of Fig. 1 in his paper. The figure is reproduced here as Fig. 1(a). JAM writes: “Denoting the angle between the vectors ds and $r̂$ by φ… we readily see that θ = φ − π/2.” (page 255). In point of fact, this statement is in disagreement with Fig. 1(a). It is obvious that if the relation is true, the figure should be modified. The modified version is shown in Fig. 1(b).

The statement given in a caption of Fig. 1 in the JAM paper

means that the length of the arc $rdθ$ is equal to the length of the segment $ds cos θ$ [see Fig. 1(b)]. Let us check when it is possible. The differential of arc length ds of a curve $r=r(θ)$ is given by

Our task is to find a function $r=r(θ)$ for an arbitrarily shaped wire which fulfills $rdθ=ds cos θ$⁠. By combining this with Eq. (5) we obtain the following ordinary differential equation:

where $r′=dr/dθ$⁠. After simple manipulations, we get

Equation (7) has two solutions depending on the sign before the $tan θ$

which are equations of a straight line, the horizontal line in Fig. 1(a), and a circle whose diameter lies on the polar axis $θ=0$ with one end at the origin $O$⁠, respectively. The point $(r0,θ0)$ represents the position of the element $ds$⁠. This can be easily seen if we express Eqs. (9a) and (9b) in Cartesian coordinates x–y. Substituting

we get

for Eq. (9a), and

for Eq. (9b). The functions in Eq. (9) satisfy $rdθ=ds cos θ$⁠, while, for example, the function $r(θ)=2f/(1−cos θ)$⁠, considered in Ref. 1 for a parabolic wire, does not (f is the distance from the parabola's vertex to its focus located at the origin $O$⁠).

The derivation given by JAM is valid only for an infinitely long straight wire or a circular wire passing through the observation point $O$ (or for fragments of such wires), not for arbitrarily shaped wires. The statement cited above that $θ=φ−π/2$ is generally not true. The major limitation of JAM's derivation is simply the assumption that the angle between the dotted line and the element of length of a wire $ds$ in Fig. 1(a) is equal to the polar angle θ. If JAM had used any other notation for that angle, his derivation would be general (Griffiths² and Zangwill³). Regardless of the fact that the main formula [Eq. (4) in Ref. 1] and all subsequent results presented in the JAM paper are correct, and that the author should be commended for a good paper, the simple use of Eq. (3) proposed here offers a more general derivation.

Warmest thanks to my friend Marek Ziolkowski and younger colleague Marcin Ziolkowski for stimulating discussions.