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Force and torque on a declared ideal thin-wire loop in a prescribed external field

Physical scope

Force and torque on a declared ideal thin-wire loop in a prescribed external magnetic field, under an explicitly adopted neutral-wire Lorentz coupling. Only the external field acts: self-fields, radiation reaction, mechanical support, deformation and material failure are outside the statement; no dynamics, radiation or empirical measurement is claimed, and the result is conditional on the adopted coupling (W).

Statement

Let $γ : [0, L] \to R^{3}$ be a $C^{2}$ curve parametrized by arc length ( $∣ γ^{'} ∣ \equiv 1$ ) that is either closed ( $γ (0) = γ (L)$ with matching one-sided derivatives) or a wire segment with free endpoints, let $I \in R$ , let $m = \frac{I}{2} \int_{0}^{L} γ (s) \times γ^{'} (s) d s$ be the finite Riemann source moment defined by this displayed integral in the closed case. For an embedded closed loop this agrees with The magnetic dipole moment of a compact conserved current and of the declared ideal thin loop(b); embeddedness is not required for the present parametrized-curve calculation. Let $B$ be a prescribed external magnetic field, of class $C^{0}$ on a neighbourhood of the trace of $γ$ in part (a) and of class $C^{2}$ on an open set containing the closed ball $\overline{B_{a} (0)}$ in parts (b)-(d). Adopt the thin-wire coupling hypothesis (W): for a general closed parametrized curve use the stated direct neutral-wire Riemann coupling (with multiplicity of traversal retained); in the embedded closed-loop specialization its source interpretation is exactly The steady-current (magnetostatic) regime, the compact smooth current class and the declared ideal noncompact models(c1); for an open curve use a compact segment of a fed or closed neutral circuit, including a segment of the declared infinite wire of (c2). Adopt separately for that parametrized wire or local segment the direct external magnetic coupling $d F = I d γ \times B$ and $d N = γ \times d F$ , as an ideal extension motivated by the smooth continuum Lorentz law The Lorentz coupling: external force on a test charge and the continuum force and power densities(ii); that smooth-source postulate does not itself include singular currents. An isolated static neutral open segment is not asserted: its standalone current satisfies $div J_{s e g} = I (δ_{γ (0)} - δ_{γ (L)})$ , so its endpoint feed/return cannot be suppressed in a global model. Force and torque here concern only the specified local segment under this adopted coupling. Then:

(a) [Force and torque under (W)] $F : = I \int_{0}^{L} γ^{'} (s) \times B (γ (s)) d s, N : = I \int_{0}^{L} γ (s) \times (γ^{'} (s) \times B (γ (s))) d s,$ the componentwise Riemann integrals of continuous functions on the compact interval $[0, L]$ (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).

(b) [Constant field] If $γ$ is closed and $B$ is constant on a neighbourhood of the loop, then $F = 0$ and $N = m \times B$ exactly.

(c) [Slowly varying field] Let $γ$ be closed with $∣ γ (s) ∣ \leq a$ for all $s$ , let $B$ be $C^{2}$ on an open set containing $\overline{B_{a} (0)}$ with constants $M_{1}, M_{2}$ such that $∣ B (x) - B (y) ∣ \leq M_{1} ∣ x - y ∣$ and $∣ B (x) - B (0) - D B (0) x ∣ \leq \frac{1}{2} M_{2} ∣ x ∣^{2}$ for $∣ x ∣, ∣ y ∣ \leq a$ , and suppose $div B = 0$ on $B_{a} (0)$ (as holds for the external part of any smooth-branch vacuum field by The microscopic vacuum Maxwell equations: smooth and weak branches, Divergence and curl of a $C^{1}$ vector field). Then $F = \nabla (m \cdot B) (0) + R_{F}, N = m \times B (0) + R_{N},$ with $∣ R_{F} ∣ \leq \frac{1}{2} ∣ I ∣ L a^{2} M_{2}$ and $∣ R_{N} ∣ \leq ∣ I ∣ L a^{2} M_{1}$ , where $\nabla (m \cdot B) (0)$ is the gradient at $0$ of $x \mapsto m \cdot B (x)$ for the fixed vector $m$ (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ).

(d) [Sufficient condition] If the external field is the magnetostatic field of a compact smooth steady current $J_{e x t} \in C_{c}^{\infty} (Ω; R^{3})$ with $div J_{e x t} = 0$ (Coulomb-gauge Newtonian vector potential for a compact smooth conserved current, Biot-Savart representation for a compact smooth conserved current and its declared thin-loop limit), then it is $C^{\infty}$ on $R^{3}$ , satisfies $div B = 0$ , and meets the hypotheses of (c) on every ball $\overline{B_{a} (0)}$ with finite constants $M_{1}, M_{2}$ .

(e) [Scope] Only the external field acts on the loop: the loop's own field, self-force, radiation reaction, mechanical support and material response are outside the statement, and no claim is made about the loop's dynamics, deformation or failure. The coupling (W) is an explicit idealisation of a neutral thin wire at rest in the laboratory frame; it is adopted, not derived, and the theorem is conditional on it. No measurement or uncertainty is involved. Parts (a)-(c), conditional on the explicit coupling (W) and the supplied curve and external field, are finite Riemann-integral calculations and use no Choice axiom. Part (d), transporting the Newtonian-potential construction, assumes Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Facts & Assumptions

Given: The $C^{2}$ arc-length curve $γ$ , the real $I$ , the prescribed field $B$ , the coupling (W) of the Statement, and, in parts (b)-(d), the ball containment and regularity hypotheses stated there.

[F1]

Under the coupling (W), the force and torque on the parametrized wire are the componentwise Riemann integrals $F = I \int_{0}^{L} γ^{'} \times B (γ) d s$ and $N = I \int_{0}^{L} γ \times (γ^{'} \times B (γ)) d s$ of the Statement; continuous integrands on the compact interval are Riemann integrable (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, $C^{k}$ Euclidean maps and diffeomorphisms, The cross product in $R^{3}$ ).

[F2]

The cross product is $u \times v = (u_{2} v_{3} - u_{3} v_{2}, u_{3} v_{1} - u_{1} v_{3}, u_{1} v_{2} - u_{2} v_{1})$ with $(u \times v)_{i} = ε_{i j k} u_{j} v_{k}$ , the inner product is $⟨ u, v ⟩ = \sum_{i} u_{i} v_{i}$ , and the identities $a \times (b \times c) = b (a \cdot c) - c (a \cdot b)$ and $ε_{i j k} ε_{l j p} = δ_{i l} δ_{k p} - δ_{i p} δ_{k l}$ hold (The cross product in $R^{3}$ , The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ).

[F3]

The Riemann integral on a compact rectangle is linear, monotone, satisfies $∣ \int f ∣ \leq \int ∣ f ∣$ , and the integral of a constant $c$ over $[0, L]$ is $c L$ ; these properties are used for the remainder estimates (Linearity, monotonicity, the absolute-value estimate and coordinate-slice additivity for the Riemann integral in $R^{m}$ , The Riemann integral of a bounded function over a bounded Jordan measurable set).

[F4]

The bounds $∣ B (x) - B (y) ∣ \leq M_{1} ∣ x - y ∣$ and $∣ B (x) - B (0) - D B (0) x ∣ \leq \frac{1}{2} M_{2} ∣ x ∣^{2}$ on $\overline{B_{a} (0)}$ are the explicit hypotheses of (c). For (d), the coordinate FTC of [F9] applied twice to $s \mapsto B (s x)$ gives the componentwise finite-Riemann identity $B (x) - B (0) - D B (0) x = \int_{0}^{1} (1 - s) D^{2} B (s x) [x, x] d s$ . It does not require a common Lagrange point for all vector components.

[F5]

The moment of The magnetic dipole moment of a compact conserved current and of the declared ideal thin loop(b) for a closed embedded arc-length loop is exactly $m = \frac{I}{2} \int_{0}^{L} γ \times γ^{'} d s$ ; the declared ideal loop source is The steady-current (magnetostatic) regime, the compact smooth current class and the declared ideal noncompact models(c1) and the declared infinite wire is The steady-current (magnetostatic) regime, the compact smooth current class and the declared ideal noncompact models(c2) (The magnetic dipole moment of a compact conserved current and of the declared ideal thin loop, The steady-current (magnetostatic) regime, the compact smooth current class and the declared ideal noncompact models).

[F6]

The compact smooth current class is The steady-current (magnetostatic) regime, the compact smooth current class and the declared ideal noncompact models(b); its magnetostatic field construction is Coulomb-gauge Newtonian vector potential for a compact smooth conserved current and its Biot-Savart representation is Biot-Savart representation for a compact smooth conserved current and its declared thin-loop limit: the external field of a compact smooth conserved current is $C^{\infty}$ on $R^{3}$ with $div B = 0$ (Coulomb-gauge Newtonian vector potential for a compact smooth conserved current, Biot-Savart representation for a compact smooth conserved current and its declared thin-loop limit); the smooth-branch vacuum equations are The microscopic vacuum Maxwell equations: smooth and weak branches and the smooth continuum Lorentz coupling that motivates (W) is The Lorentz coupling: external force on a test charge and the continuum force and power densities(ii).

[F7]

Distributional derivatives act by signed test differentiation, $⟨ \partial^{α} u, χ ⟩ = (- 1)^{∣ α ∣} ⟨ u, \partial^{α} χ ⟩$ (Distributional derivative); a $C^{1}$ composition $φ \circ γ$ has derivative $(φ \circ γ)^{'} = (\nabla φ) (γ) \cdot γ^{'}$ (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ ) and satisfies the fundamental theorem $\int_{0}^{L} (φ \circ γ)^{'} d s = φ (γ (L)) - φ (γ (0))$ (The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ ); the Dirac distribution at a point $p$ acts by $δ_{p} (φ) = φ (p)$ (Dirac delta and its derivatives).

[F8]

${A C}_{ω}$ is the countable-choice principle assumed in part (d) and inherited by the Newtonian-potential suppliers (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F9]

Compact closed Euclidean balls are compact (Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line); continuous real functions on a nonempty compact metric set attain finite extrema (A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value). The fundamental theorem integrates continuous derivatives on compact intervals (The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ ). Finite-vector Cauchy-Schwarz gives $∣ u \cdot v ∣ \leq ∣ u ∣ ∣ v ∣$ (Cauchy-Schwarz $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥_{2} ∥ y ∥_{2}$ with its equality case, the triangle inequality for $∥ \cdot ∥_{2}$ , the parallelogram law and polarisation); testing a vector integral against its unit direction gives the corresponding integral norm bound.

Proof

technique · direct evaluation of the pairings under the coupling, the antisymmetric first-moment identities for a closed curve, one Taylor expansion with an explicit remainder, and the compact-current sufficient condition

1.1givenF1F2

Under the adopted coupling (W) the total force and torque of the parametrized wire are the integrals of the Statement: at arc-length parameter $s$ the element $d γ = γ^{'} (s) d s$ carries current $I$ , so $d F = I γ^{'} (s) \times B (γ (s)) d s$ and $d N = γ (s) \times d F$ . The integrands are compositions and products of continuous maps on the compact interval $[0, L]$ , hence continuous and Riemann integrable, so the componentwise integrals of [F1] are finite. This is (a).

1.2givenF1F2F5

Let $B$ be constant on a neighbourhood of the closed curve. Then $F = I (\int_{0}^{L} γ^{'} (s) d s) \times B = I (γ (L) - γ (0)) \times B = 0$ . For the torque, $F$ 's definition and [F2] give $γ \times (γ^{'} \times B) = (γ \cdot B) γ^{'} - (γ \cdot γ^{'}) B$ , so with the matrix $M_{j l} : = \int_{0}^{L} γ_{j} γ_{l}^{'} d s$ one has $N = I \int_{0}^{L} (γ \cdot B) γ^{'} d s - I (\int_{0}^{L} γ \cdot γ^{'} d s) B = I B_{j} M_{j l} e_{l}$ . The scalar $γ \cdot γ^{'} = \frac{1}{2} \frac{d}{d s} ∣ γ ∣^{2}$ integrates to $0$ over the closed curve, and integrating $\frac{d}{d s} (γ_{j} γ_{l})$ over the closed curve gives $M_{j l} + M_{l j} = 0$ , so $M$ is antisymmetric. For antisymmetric $M$ , the moment $m_{p} = \frac{I}{2} ε_{p a b} M_{a b}$ satisfies $I M_{j l} = ε_{j l p} m_{p}$ because $ε_{a b p} ε_{p c d} = δ_{a c} δ_{b d} - δ_{a d} δ_{b c}$ gives $ε_{a b p} m_{p} = \frac{I}{2} (M_{a b} - M_{b a}) = I M_{a b}$ . Substituting, $N = ε_{j l p} m_{p} B_{j} e_{l} = m \times B$ by the cyclic identity of the alternating symbol, and this is (b).

1.3givenF7

Let $J_{s e g}$ be the standalone open-segment current defined by $⟨ J_{s e g}, ψ ⟩ : = I \int_{0}^{L} γ^{'} (s) \cdot ψ (γ (s)) d s$ for vector tests $ψ$ . With the componentwise distributional divergence pairing of [F7], $⟨ div J_{s e g}, φ ⟩ = - I \int_{0}^{L} γ^{'} (s) \cdot \nabla φ (γ (s)) d s = - I \int_{0}^{L} (φ \circ γ)^{'} (s) d s = I (φ (γ (0)) - φ (γ (L))) = I (δ_{γ (0)} - δ_{γ (L)}) (φ)$ for every scalar test $φ$ , the chain rule and the fundamental theorem of [F7] giving the middle equality. This is the endpoint identity asserted in the Statement, and it shows why the segment is treated as a local part of a fed or closed circuit.

2.1step 1.1F4

Under the hypotheses of (c), step 1.1's formulas apply, and [F4] gives the decomposition $B (x) = B (0) + D B (0) x + R (x)$ with $∣ R (x) ∣ \leq \frac{1}{2} M_{2} ∣ x ∣^{2}$ for $∣ x ∣ \leq a$ , together with the Lipschitz bound $∣ B (x) - B (y) ∣ \leq M_{1} ∣ x - y ∣$ on the ball.

3.1step 1.2step 2.1F2F3

For the force in (c), the constant term vanishes as in step 1.2, and the linear term is $I ε_{i j k} \partial_{l} B_{k} (0) M_{l j} e_{i}$ with the matrix $M$ of step 1.2. Substituting $I M_{l j} = ε_{l j p} m_{p}$ from step 1.2 and contracting by [F2], $I ε_{i j k} \partial_{l} B_{k} (0) M_{l j} e_{i} = (δ_{i l} δ_{k p} - δ_{i p} δ_{k l}) \partial_{l} B_{k} (0) m_{p} e_{i} = (\partial_{i} (m \cdot B) (0) - m_{i} div B (0)) e_{i} = \nabla (m \cdot B) (0)$ , since $m$ is constant and $div B (0) = 0$ . By [F3] and step 2.1 the remainder obeys $∣ I \int_{0}^{L} γ^{'} \times R (γ) d s ∣ \leq ∣ I ∣ \int_{0}^{L} ∣ γ^{'} ∣ ∣ R (γ) ∣ d s \leq \frac{1}{2} ∣ I ∣ L a^{2} M_{2}$ , because $∣ γ^{'} ∣ \equiv 1$ and $∣ γ (s) ∣ \leq a$ . This proves the force assertion of (c) with $∣ R_{F} ∣ \leq \frac{1}{2} ∣ I ∣ L a^{2} M_{2}$ .

3.2step 1.2step 2.1F2F3

For the torque in (c), split $B (γ (s)) = B (0) + (B (γ (s)) - B (0))$ . The first part is the constant-field torque of step 1.2 with the constant field $B (0)$ , that is $m \times B (0)$ . For the remainder, [F2] gives $γ \times (γ^{'} \times (B (γ) - B (0))) = γ^{'} (γ \cdot (B (γ) - B (0))) - (B (γ) - B (0)) (γ \cdot γ^{'})$ and hence, by [F3] and step 2.1, $∣ I \int_{0}^{L} γ \times (γ^{'} \times (B (γ) - B (0))) d s ∣ \leq ∣ I ∣ \int_{0}^{L} ∣ γ ∣ ∣ γ^{'} ∣ ∣ B (γ) - B (0) ∣ d s \leq ∣ I ∣ a \cdot 1 \cdot M_{1} a \cdot L$ . This proves the torque assertion of (c) with $∣ R_{N} ∣ \leq ∣ I ∣ L a^{2} M_{1}$ .

3.3step 2.1F4F6F9

For (d): by [F6] the external field of a compact smooth conserved current satisfies $B = curl A$ , hence is $C^{\infty}$ on $R^{3}$ with $div B = 0$ ; on the compact ball $\overline{B_{a} (0)}$ let $C_{1}, C_{2}$ be the maxima of the absolute coordinate first and second partial derivatives, finite by [F9], and put $M_{1} : = 9 C_{1}$ , $M_{2} : = 27 C_{2}$ . The coordinate sum bounds give $∣ D B (x) v ∣ \leq M_{1} ∣ v ∣$ and $∣ D^{2} B (x) [v, w] ∣ \leq M_{2} ∣ v ∣ ∣ w ∣$ . Integrating the first derivative along the segment from $x$ to $y$ gives $∣ B (x) - B (y) ∣ \leq M_{1} ∣ x - y ∣$ ; integrating the identity of [F4] gives $∣ B (x) - B (0) - D B (0) x ∣ \leq \frac{1}{2} M_{2} ∣ x ∣^{2}$ . All segments lie in the convex closed ball Hence the hypotheses of (c) are met on every such ball with finite constants.

4.1step 1.1step 1.2step 1.3step 3.1step 3.2step 3.3F5F8∎

Parts (a), (b), (c), (d) are steps 1.1, 1.2, 3.1, 3.2 and 3.3; the endpoint identity quoted in the Statement is step 1.3. The scope clause (e) records that only the external field acts, that the coupling (W) is an adopted idealisation motivated by the smooth continuum law The Lorentz coupling: external force on a test charge and the continuum force and power densities(ii) rather than derived from it, and that no self-field, self-force, radiation reaction, mechanical support, deformation or dynamical claim is made. Parts (a)-(c) are finite Riemann-integral calculations and use no Choice axiom; part (d) inherits Countable Choice only through the Newtonian-potential suppliers [F8]. The moment identity is contracted without dividing by $I$ ; for $I = 0$ the moment, force, torque and both remainders vanish, and the algebra applies to every stated closed parametrized $C^{2}$ curve, including multiply traversed and self-intersecting ones, since only closure, unit speed and the finite moment integral are used.

Source notes

Griffiths, Section 5.1.3, printed pp. 208-214, gives the wire force $F = \int I d l \times B$ (5.17), and Section 6.1.2, printed pp. 255-260, gives $N = m \times B$ in a uniform field (6.1), the vanishing net force in a uniform field, and $F = \nabla (m \cdot B)$ for a small loop in a nonuniform field (6.3), with Problems 6.2 and 6.4; Tong, Sections 3.4.1-3.4.2, printed pp. 57-62, performs the Taylor expansion over a small localised current and the moment manipulations; Landau-Lifshitz, Section 45, printed pp. 105-106, gives the average moment $K = m \times H$ (45.1), re-expressed here in SI. The theorem above is the exact conditional version for the declared parametrized wire under the explicitly adopted coupling (W); the texts do not supply the neutral-wire hypothesis, which is stated here as an idealisation.

Force and torque on a declared ideal thin-wire loop in a prescribed external field

Statement

Facts & Assumptions

Proof

Source notes

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