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Biot-Savart representation for a compact smooth conserved current and its declared thin-loop limit

Physical scope

Biot-Savart representation of the magnetostatic field of a compact smooth conserved current, together with the declared thin-loop limit of that representation. Both legs are conditional on the framework's adopted equations and on the declared ideal models; the smooth-current formula holds even on its support, whereas no pointwise value on the ideal loop is asserted; nothing is claimed about self-energy, radiation, forces or noncompact currents, and no measurement, apparatus or uncertainty is involved.

Statement

Assume Countable Choice ( ${A C}_{ω}$ ), with the exact meaning of The Axiom of Countable Choice ( ${A C}_{ω}$ ). Let $J \in C_{c}^{\infty} (Ω; R^{3})$ with $div J = 0$ be a compact smooth current of The steady-current (magnetostatic) regime, the compact smooth current class and the declared ideal noncompact models(b), extended by zero to $R^{3}$ , let $A$ be the vector potential of Coulomb-gauge Newtonian vector potential for a compact smooth conserved current and $B = curl A$ its magnetostatic field, with the constants of Framework-local SI units, the vacuum constants and the exact/measured status of their values. Then for every $x \in R^{3}$ the absolutely convergent Biot-Savart integral represents the field, $B (x) = \frac{μ_{0}}{4 π} \int_{R^{3}} \frac{J (y) \times (x - y)}{∣ x - y ∣^{3}} d y .$ Moreover the same formula holds in the declared thin-loop limit, constructed in the all-space laboratory $Ω = R^{3}$ : if $γ : R / L Z \to R^{3}$ is a closed embedded $C^{2}$ curve parametrized by arc length ( $∣ γ^{'} ∣ \equiv 1$ ), $I \in R$ , and $j_{ε} = j * η_{ε}$ are the mollified divergence-free compact currents of Thin closed current: divergence-free distribution and convergence of its mollified Biot-Savart fields associated with the tangent line current of $γ$ , then their fields (given by the displayed formula with $J : = j_{ε}$ ) converge, with every derivative uniformly on every compact $K$ disjoint from $γ$ , to $B_{γ} (x) = \frac{μ_{0} I}{4 π} \int_{0}^{L} \frac{γ^{'} (s) \times (x - γ (s))}{∣ x - γ (s) ∣^{3}} d s,$ a locally integrable global distributional field with $div B_{γ} = 0$ , $curl B_{γ} = μ_{0} J_{γ}$ , smoothness off the curve and decay at spatial infinity, hence the admitted field of the declared ideal thin loop under Physical adoption of the declared ideal magnetostatic loop, line and cylindrical-sheet models. Its almost-everywhere representative on $R^{3}$ is supplied by Thin closed current: divergence-free distribution and convergence of its mollified Biot-Savart fields, independently of the away-from-curve convergence. Nothing is asserted on the wire, about line self-energy, about the infinite-wire or sheet models, or for currents outside the compact smooth class and its declared thin-loop limit.

Facts & Assumptions

Given: Assume ${A C}_{ω}$ ; let $J \in C_{c}^{\infty} (Ω; R^{3})$ with $div J = 0$ , extended by zero to $R^{3}$ ; let $A$ , $B = curl A$ be as in Coulomb-gauge Newtonian vector potential for a compact smooth conserved current with the constants of Framework-local SI units, the vacuum constants and the exact/measured status of their values.

[F1]

The vector potential satisfies $A_{i} = μ_{0} N (J_{i})$ componentwise and $B = curl A$ is $C^{\infty}$ with $div B = 0$ and $curl B = μ_{0} J$ pointwise on $R^{3}$ (Coulomb-gauge Newtonian vector potential for a compact smooth conserved current, clauses (a)-(c)).

[F2]

For $f \in C_{c}^{\infty} (R^{3}; R^{3})$ the curl of the Newtonian potential is $curl N (f) (x) = \frac{1}{4 π} \int_{R^{3}} \frac{f (y) \times (x - y)}{∣ x - y ∣^{3}} d y$ at every $x \in R^{3}$ , the integral being absolutely convergent, and the estimate $∣ f (y) \times (x - y) ∣ \leq ∥ f ∥_{\infty} ∣ x - y ∣$ with $\int_{B_{R} (x)} ∣ x - y ∣^{- 2} d λ_{3} < + \infty$ underlies it (Divergence and curl of the Newtonian potential of a smooth compact vector field).

[F3]

For a closed embedded $C^{2}$ arc-length curve $γ$ and $I \in R$ , the tangent line current $j$ has $div j = 0$ distributionally; for any family of nonnegative smooth radial unit-mass kernels supported in $B_{ε} (0)$ the mollified fields $j_{ε} = j * η_{ε}$ are smooth, compactly supported and divergence-free; their Biot-Savart fields converge with every derivative uniformly on every compact $K$ disjoint from $γ$ to $B_{γ} (x) = \frac{μ_{0} I}{4 π} \int_{0}^{L} \frac{γ^{'} (s) \times (x - γ (s))}{∣ x - γ (s) ∣^{3}} d s$ ; and the same field has a locally integrable representative on $R^{3}$ , is smooth off $γ$ , satisfies $div T_{B_{γ}} = 0$ and $curl T_{B_{γ}} = μ_{0} j$ globally, and decays with $∣ B_{γ} (x) ∣ \leq \frac{μ_{0} ∣ I ∣ L}{4 π (∣ x ∣ - a)^{2}}$ for $∣ x ∣ > a$ when $γ \subset B_{a} (0)$ (Thin closed current: divergence-free distribution and convergence of its mollified Biot-Savart fields).

[F4]

The compact smooth current class is The steady-current (magnetostatic) regime, the compact smooth current class and the declared ideal noncompact models(b); the ideal closed thin loop is The steady-current (magnetostatic) regime, the compact smooth current class and the declared ideal noncompact models(c1); and a field is admitted for it exactly when the predicate (c4) holds, which includes local integrability, $C^{1}$ regularity and vanishing divergence and curl off the source, the two global distributional identities, the model symmetry and the decay requirement (The steady-current (magnetostatic) regime, the compact smooth current class and the declared ideal noncompact models).

[F5]

The physical reading of the declared ideal models is the separate adoption Physical adoption of the declared ideal magnetostatic loop, line and cylindrical-sheet models; it asserts no existence, uniqueness, finite self-energy, force, self-force or apparatus accuracy, and it is an adopted assumption rather than a deduction.

[F6]

The cross product is that of The cross product in $R^{3}$ , volume integrals are over measurable sets in the sense of Integral over a measurable subset, and the modulus of an integral is bounded by the integral of the modulus (The modulus of an integral is bounded by the integral of the modulus).

[F7]

${A C}_{ω}$ is the countable-choice principle assumed here and inherited by the cited measure, regular-distribution and Newtonian interfaces (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Proof

technique · direct; the differentiated-kernel identity for the compact class, then the declared thin-current limit with its supplier-supplied global properties

1.1givenF1F2F6

By [F1], $B = curl A$ with $A_{i} = μ_{0} N (J_{i})$ . Applying [F2] to $f : = J$ and multiplying by $μ_{0}$ gives, at every $x \in R^{3}$ , $B (x) = \frac{μ_{0}}{4 π} \int_{R^{3}} \frac{J (y) \times (x - y)}{∣ x - y ∣^{3}} d y$ , with the integral absolutely convergent by the estimate $∣ J (y) \times (x - y) ∣ \leq ∥ J ∥_{\infty} ∣ x - y ∣$ and $\int_{B_{R} (x)} ∣ x - y ∣^{- 2} d λ_{3} < + \infty$ for a ball containing $supp J$ ; this proves the first assertion at every point, including points of the support of $J$ .

2.1step 1.1F3F4

By [F3], the tangent line current $j$ of the declared loop is divergence-free, its mollified currents $j_{ε}$ are smooth, compactly supported and divergence-free, so, taking $Ω = R^{3}$ for this loop construction, each $j_{ε}$ is a compact smooth conserved current of the class F4; step 1.1 applied with $J : = j_{ε}$ identifies the field of $j_{ε}$ with its displayed Biot-Savart integral. The supplier's every-derivative uniform convergence on every compact $K$ disjoint from $γ$ therefore identifies the limit field with $B_{γ} (x) = \frac{μ_{0} I}{4 π} \int_{0}^{L} \frac{γ^{'} (s) \times (x - γ (s))}{∣ x - γ (s) ∣^{3}} d s$ on the complement of $γ$ .

3.1step 2.1F3F4F5

The same supplier [F3] supplies, independently of the away-from-curve convergence, the globally locally integrable representative of $B_{γ}$ on $R^{3}$ , its smoothness off $γ$ , the global distributional identities $div T_{B_{γ}} = 0$ and $curl T_{B_{γ}} = μ_{0} j$ , and the decay bound. Together with the symmetry requirement (vacuous for the loop clause) these are exactly the clauses of the admissibility predicate F4, so $B_{γ}$ is an admitted field of the declared ideal thin loop; the physical reading of that declaration is the separate adopted assumption [F5], which asserts no existence, uniqueness, self-energy or force content.

4.1step 1.1step 2.1step 3.1F3F7∎

Clauses (a) and (b) of the Statement are steps 1.1 and 2.1-3.1; the scope clause records that no formula is asserted for noncompact currents (the infinite-wire model is verified separately on the companion examples page), and that no near-wire, self-energy, radiation or force claim is made. Countable Choice is inherited only through the cited measure, regular-distribution and Newtonian supplier interfaces [F7]; no stronger choice principle occurs.

Source notes

Griffiths, Section 5.2.2, printed pp. 215-219, states the Biot-Savart law for line, surface and volume currents and evaluates the straight wire (5.36) and circular loop (5.38); Section 5.3.2, printed pp. 222-225, derives $curl B = μ_{0} J$ from the volume integral. Tong, Section 3.2.3, printed pp. 49-52, gives (3.15) and its thin-wire form (3.16); Chew, Lecture 5 Section 5.1, printed pp. 49-52, derives the law from the vector potential, and Landau-Lifshitz, Sections 43-44, printed pp. 101-104, give the corresponding constant-field and Biot-Savart formulas, re-expressed here in SI. The thin-loop leg of this item is conditional on its mathematical supplier Thin closed current: divergence-free distribution and convergence of its mollified Biot-Savart fields for the global distributional identities and the local integrability, which are not consequences of away-from-curve convergence alone.

Biot-Savart representation for a compact smooth conserved current and its declared thin-loop limit

Statement

Facts & Assumptions

Proof

Source notes

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