The magnetic dipole moment of a compact conserved current and of the declared ideal thin loop

Let the compact smooth current class and the declared ideal models be those of The steady-current (magnetostatic) regime, the compact smooth current class and the declared ideal noncompact models, with the units of Framework-local SI units, the vacuum constants and the exact/measured status of their values.

(a) [Compact smooth currents]. Let $J\in C_c^{\mathrm{\infty }}(\mathrm{\Omega };{\mathbb{R}}^3)$ with $\mathrm{div}J=0$, extended by zero to ${\mathbb{R}}^3$. The magnetic dipole moment of $J$ is the vector $m:=\frac{1}{2}{\int }_{{\mathbb{R}}^3}y\times J(y)dy\in {\mathbb{R}}^3,$ of unit A m${}^2$. The integral is computed componentwise over any nondegenerate closed bounding rectangle $Q$ containing $\mathrm{supp}J$, using the continuous functions $y\mapsto (y\times J(y){)}_i$ on $Q$ and their zero extension outside $Q$. Its value is independent of the chosen larger bounding rectangle by finite rectangular subdivision and vanishing of the integrand off its support; the support itself need not be Jordan measurable. Thus this is a Riemann integral of continuous compactly supported functions (The Riemann integral of a bounded function over a bounded Jordan measurable set, A continuous real function on a compact Jordan measurable set is Riemann integrable over that set, The cross product in ${\mathbb{R}}^3$); it is absolutely finite because $J$ is bounded with compact support.

(b) [Declared ideal thin loop]. For a closed embedded $C^2$ curve $\gamma :\mathbb{R}\mathrm{/}L\mathbb{Z}\to {\mathbb{R}}^3$ with $\mathrm{\mid }{\gamma }^{\mathrm{\prime }}\mathrm{\mid }\equiv 1$ and $I\in \mathbb{R}$, the magnetic moment of the ideal loop current of The steady-current (magnetostatic) regime, the compact smooth current class and the declared ideal noncompact models(c1) is $m:=\frac{I}{2}{\int }_0^L\gamma (s)\times {\gamma }^{\mathrm{\prime }}(s)ds,$ again a componentwise Riemann integral over the compact interval $[0,L]$ of continuous functions; for the circle $\gamma (s)=a(\cos (s\mathrm{/}a),\sin (s\mathrm{/}a),0)$ traversed in the positive orientation this is $m=I\pi a^2{e}_3$.

(c) [Status and scope]. In both clauses $m$ is a source moment of the declared current model: it is not a rest mass, not an intrinsic spin and not an observed quantity. The definition asserts no far-field property of $m$ (that is the content of Far-field dipole expansion of a compact conserved current with a uniform explicit remainder under its hypotheses), no agreement between the two clauses in a common limit, and no moment for currents outside the two classes; no higher multipole is defined here. The finite Riemann moment definitions and circular-loop evaluation themselves are choice-free; no distribution-embedding or surface-measure theorem is used to compute these moments.

Well-definedness and units. In (a), $y\mapsto y\times J(y)$ is continuous on the compact rectangle $Q$ and vanishes off $\mathrm{supp}J$, so each component is Riemann integrable on $Q$ by A continuous real function on a compact Jordan measurable set is Riemann integrable over that set and bounded there, hence the integral of The Riemann integral of a bounded function over a bounded Jordan measurable set is finite. Two bounding rectangles are compared through a common larger rectangle subdivided at their face coordinates: every added cell lies outside $\mathrm{supp}J$, where the integrand vanishes, so finite additivity of the integral (Linearity, monotonicity, the absolute-value estimate and coordinate-slice additivity for the Riemann integral in ${\mathbb{R}}^m$) identifies both values without integrating over the support itself. The cross product is that of The cross product in ${\mathbb{R}}^3$. Since $y$ carries the metre and $J$ the ampere per square metre, the product $y\times J$ carries A m${}^{-1}$ and the volume integral A m${}^2$ (Framework-local SI units, the vacuum constants and the exact/measured status of their values). In (b), $\gamma$ and ${\gamma }^{\mathrm{\prime }}$ are continuous on the compact circle and periodic, so each component of $\gamma \times {\gamma }^{\mathrm{\prime }}$ is a continuous periodic function on $[0,L]$ and the componentwise Riemann integral exists; $\mathrm{\mid }{\gamma }^{\mathrm{\prime }}\mathrm{\mid }\equiv 1$ makes the parametrization unit-speed so that the value is independent of an orientation-preserving change of unit-speed parameter; reversing orientation reverses the tangent current unless the signed current parameter $I$ is also reversed. The circular evaluation is the componentwise computation $\gamma \times {\gamma }^{\mathrm{\prime }}=a{e}_3$ and ${\int }_0^{2\pi a}a{e}_{3}ds=2\pi a^2{e}_3$. No measurement, apparatus or uncertainty enters either clause.

Griffiths, Section 5.4.3, printed pp. 242-246, introduces the line-current and volume-current magnetic moments for volume currents (eqs. (5.84) and (5.91)); Tong, Sections 3.3.1-3.3.2, printed pp. 54-57, gives $m=\frac{1}{2}\int d^3{r}^{\mathrm{\prime }}{r}^{\mathrm{\prime }}\times J(r^{\mathrm{\prime }})$ (eq. (3.24)); Landau-Lifshitz, Section 44, printed pp. 103-105, gives the corresponding charged-particle moment in Gaussian units, re-expressed here in SI. The definition keeps the moment a declared source quantity of the two explicit models and defers every far-field interpretation to the dipole theorem.