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Magnetostatics in vacuum — Examples

1 · Prerequisites

2 · Summary

The examples verify the declared ideal magnetostatic models explicitly. The infinite straight wire and the infinite cylindrical current sheet are exhibited with their exact fields, circulations, jumps and distributional source equations, and checked against the admissibility predicate of the steady-current definition; the axis field of the declared ideal circular loop is evaluated from the Biot-Savart line integral; and the force per unit length between two declared ideal parallel wires is computed from the external field of one wire under the adopted thin-wire coupling. Each computation is exact and conditional on its declared idealisation, with self-fields, support forces, dynamics and any measurement excluded.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

None yet.

5 · Examples, experiments and counterexamples

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Axis field of the declared ideal circular thin loop

Physical scope

An explicit ideal-model computation: the axis field of a declared ideal circular thin loop. It involves no apparatus, no sampling, no uncertainty and no empirical claim, and it is explicitly not the field of any physical finite-radius coil.

Example

Assume Countable Choice ( ${A C}_{ω}$ ), with the exact meaning of The Axiom of Countable Choice ( ${A C}_{ω}$ ). Let $a > 0$ , $I \in R$ , let $γ (s) = a (\cos (s / a), \sin (s / a), 0)$ , $s \in [0, 2 π a]$ , be the circle of radius $a$ in the plane $x_{3} = 0$ parametrized by arc length, let the declared ideal thin-loop current of amplitude $I$ run along $γ$ in the sense of The steady-current (magnetostatic) regime, the compact smooth current class and the declared ideal noncompact models(c1), with moment $m = I π a^{2} e_{3}$ (The magnetic dipole moment of a compact conserved current and of the declared ideal thin loop(b)), and let $μ_{0}$ , $c$ , $ε_{0}$ be the constants of Framework-local SI units, the vacuum constants and the exact/measured status of their values. Then for every point $x = z e_{3}$ on the axis ( $z \in R$ , $dist (x, γ) = \sqrt{a^{2} + z^{2}} > 0$ ) the field of the ideal loop is the absolutely convergent line integral $B (z e_{3}) = \frac{μ_{0} I}{4 π} \int_{0}^{2 π a} \frac{γ^{'} (s) \times (z e_{3} - γ (s))}{∣ z e_{3} - γ (s) ∣^{3}} d s = \frac{μ_{0} I a^{2}}{2 (a^{2} + z^{2})^{3 / 2}} e_{3},$ i.e. the Biot-Savart field of Biot-Savart representation for a compact smooth conserved current and its declared thin-loop limit and the declared thin-current limit of Thin closed current: divergence-free distribution and convergence of its mollified Biot-Savart fields agree on the axis, with the sign fixed by the positive (counterclockwise) orientation of $γ$ and by the sign of $I$ ; in particular $B (0) = \frac{μ_{0} I}{2 a} e_{3}$ . This is a property of the declared idealisation; it is not the field of any physical finite-radius coil, nothing is claimed at points off the axis or on the wire, and no measurement is involved. The explicit finite Riemann axis evaluation and circle moment calculation alone are choice-free. Countable Choice is inherited by the asserted global distributional model and thin-current supplier.

Facts & Assumptions

Given: Assume ${A C}_{ω}$ ; let $a > 0$ , $I \in R$ and the circle $γ (s) = a (\cos (s / a), \sin (s / a), 0)$ with $s \in [0, 2 π a]$ ; let the ideal loop model be the declared one of The steady-current (magnetostatic) regime, the compact smooth current class and the declared ideal noncompact models(c1) with physical adoption Physical adoption of the declared ideal magnetostatic loop, line and cylindrical-sheet models.

[F1]

The field of the declared ideal thin loop is $B_{γ} (x) = \frac{μ_{0} I}{4 π} \int_{0}^{L} \frac{γ^{'} (s) \times (x - γ (s))}{∣ x - γ (s) ∣^{3}} d s$ for $x$ off the curve: this is the Biot-Savart representation of the compact smooth class together with its declared thin-loop limit, and the ideal loop's admissibility is supplied by the strengthened thin-current supplier together with the declared adoption (Biot-Savart representation for a compact smooth conserved current and its declared thin-loop limit, Thin closed current: divergence-free distribution and convergence of its mollified Biot-Savart fields, The steady-current (magnetostatic) regime, the compact smooth current class and the declared ideal noncompact models, Physical adoption of the declared ideal magnetostatic loop, line and cylindrical-sheet models).

[F2]

The moment of clause (b) of the definition is $m = \frac{I}{2} \int_{0}^{L} γ \times γ^{'} d s$ ; for the present circle $γ \times γ^{'} = a e_{3}$ and the integral evaluates to $2 π a^{2} e_{3}$ , so $m = I π a^{2} e_{3}$ (The magnetic dipole moment of a compact conserved current and of the declared ideal thin loop).

[F3]

For $u, v \in R^{3}$ , $u \times v = (u_{2} v_{3} - u_{3} v_{2}, u_{3} v_{1} - u_{1} v_{3}, u_{1} v_{2} - u_{2} v_{1})$ (The cross product in $R^{3}$ ); components of continuous integrands on the compact interval $[0, 2 π a]$ 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).

[F4]

${A C}_{ω}$ is the countable-choice principle assumed here and inherited by the asserted global distributional model and thin-current supplier (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F5]

The sine and cosine have derivatives $\cos$ and $- \sin$ , and period $2 π$ (The derivatives of sine and cosine are cosine and minus sine, The zero sets of sine and cosine and the least positive common period 2 pi). The chain rule and FTC integrate their compositions on a compact interval (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , 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 substitution $s = a t$ is the $C^{1}$ monotone substitution of Substitution: if $φ$ is differentiable on $[c, d]$ with $φ^{'}$ integrable and $f$ is continuous on an interval containing $φ ([c, d])$ , then $\int_{φ (c)}^{φ (d)} f = \int_{c}^{d} (f \circ φ) φ^{'}$ .

Verification

technique · direct; identify the axis field with the declared line integral, evaluate the integral by the substitution $t=s/a$, and read off the center value and the moment

1.1givenF1

Every axis point $z e_{3}$ lies at distance $\sqrt{a^{2} + z^{2}} \geq a > 0$ from the circle, so it is off the curve; by [F1] the field of the declared ideal loop there is the absolutely convergent line integral $B (z e_{3}) = \frac{μ_{0} I}{4 π} \int_{0}^{2 π a} \frac{γ^{'} (s) \times (z e_{3} - γ (s))}{∣ z e_{3} - γ (s) ∣^{3}} d s$ , the same integral being the Biot-Savart field of the compact smooth currents in the declared thin-current limit.

1.2givenF3F5

Substituting $t = s / a$ , so that $d s = a d t$ and $γ^{'} (s) = (- \sin t, \cos t, 0)$ while $z e_{3} - γ (s) = (- a \cos t, - a \sin t, z)$ , one has $γ^{'} (s) \times (z e_{3} - γ (s)) = (z \cos t, z \sin t, a)$ by [F3] (the first two components are $z \cos t$ and $z \sin t$ respectively, and the third is $a \cos^{2} t + a \sin^{2} t = a$ ), and $∣ z e_{3} - γ (s) ∣ = (a^{2} + z^{2})^{1 / 2}$ is constant. Hence the component integrals are continuous functions of $t$ on the compact interval $[0, 2 π]$ .

2.1step 1.1step 1.2F3F5

By steps 1.1 and 1.2, $B (z e_{3}) = \frac{μ_{0} I}{4 π} \cdot \frac{a}{(a^{2} + z^{2})^{3 / 2}} \int_{0}^{2 π} (z \cos t, z \sin t, a) d t = \frac{μ_{0} I}{4 π} \cdot \frac{2 π a^{2}}{(a^{2} + z^{2})^{3 / 2}} e_{3} = \frac{μ_{0} I a^{2}}{2 (a^{2} + z^{2})^{3 / 2}} e_{3}$ , the first two components vanishing because $\cos$ and $\sin$ have vanishing integral over a full period. At $z = 0$ this gives $B (0) = \frac{μ_{0} I}{2 a} e_{3}$ , and the sign is the product of the orientation sign of $γ$ and the sign of $I$ .

3.1step 2.1F2F4∎

The computation also fixes the source moment: with $γ \times γ^{'} = a e_{3}$ and $\int_{0}^{2 π a} a e_{3} d s = 2 π a^{2} e_{3}$ , clause (b) of [F2] gives $m = I π a^{2} e_{3}$ , the value quoted in the Example. The whole verification is a conditional ideal-model computation on the axis only: nothing is claimed off the axis or on the wire, no apparatus, sampling model, uncertainty or empirical inference is involved, and Countable Choice enters only through the asserted global distributional model and thin-current supplier [F4].

Source notes

Griffiths, Example 5.6, printed pp. 217-218, evaluates the axis field (5.38) $μ_{0} I R^{2} / [2 (R^{2} + z^{2})^{3 / 2}]$ ; Tong, Section 3.2.3 and Section 3.3.1, printed pp. 49-56, gives the thin-wire Biot-Savart integral and the loop far field; Chew, Lecture 5 Section 5.1, printed pp. 49-52, gives the current-element form. The example verifies the declared ideal loop's axis field against those formulas; it is explicitly not a statement about any physical coil.

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Declared ideal infinite wire and cylindrical current sheet: explicit fields and their distributional equations

Physical scope

Explicit verification of the two declared ideal noncompact source models, the infinite straight wire and the infinite cylindrical current sheet (ideal solenoid). The models carry declared symmetry and decay data; they are not finite-energy localised-current theorems, nothing is claimed about self-energy, inductance or self-forces, and no measurement, apparatus or statistical inference is involved.

Example

Assume Countable Choice ( ${A C}_{ω}$ ), with the exact meaning of The Axiom of Countable Choice ( ${A C}_{ω}$ ).

(a) [Ideal infinite straight wire]. Let $γ (s) = (0, 0, s)$ , $s \in R$ , let $J_{w}$ be the declared ideal line current of amplitude $I$ along $γ$ (The steady-current (magnetostatic) regime, the compact smooth current class and the declared ideal noncompact models(c2)), and define, on $R^{3} ∖ γ$ , $B (x) : = \frac{μ_{0} I}{2 π ρ} e_{φ}, ρ = \sqrt{x_{1}^{2} + x_{2}^{2}}, e_{φ} = ρ^{- 1} (- x_{2}, x_{1}, 0) .$ Then $B$ is locally integrable on $R^{3}$ and of class $C^{\infty}$ off $γ$ ; $div B = 0$ and $curl B = 0$ pointwise off $γ$ ; $B$ is invariant under translations along $γ$ and rotations about it, and its transverse far-zone supremum tends to $0$ as $ρ \to \infty$ ; for every $r > 0$ and $h \in R$ , the circulation of $B$ along $γ_{r, h} (t) = (r \cos t, r \sin t, h)$ , $0 \leq t \leq 2 π$ , traversed once with increasing $t$ , is $μ_{0} I$ (Scalar line integrals with respect to arc length and vector-field line integrals); and $div B = 0$ , $curl B = μ_{0} J_{w}$ hold in the distributional sense of The steady-current (magnetostatic) regime, the compact smooth current class and the declared ideal noncompact models(c4). Hence $B$ is an admitted field of the declared wire model; the transverse decay excludes an added homogeneous longitudinal field, and no further selection, uniqueness or energy claim is made.

(b) [Ideal infinite cylindrical sheet]. Let $a > 0$ , $S = {ρ = a}$ , $K \in R$ , let $J_{S}$ be the declared ideal sheet current of amplitude $K$ along $S$ (The steady-current (magnetostatic) regime, the compact smooth current class and the declared ideal noncompact models(c3)), and define $B : = μ_{0} K e_{3}$ for $ρ < a$ and $B : = 0$ for $ρ > a$ . Then $B$ is locally integrable on $R^{3}$ and of class $C^{\infty}$ off $S$ ; $div B = 0$ and $curl B = 0$ pointwise off $S$ ; $B$ has the declared symmetry (translations along and rotations about the axis) and its transverse far-zone supremum is $0$ ; the one-sided jump across $S$ , with $n = e_{ρ}$ from the interior to the exterior and $[B] = B_{+} - B_{-}$ , is $n \times [B] = μ_{0} K e_{φ}$ and $n \cdot [B] = 0$ , the stationary tangential-sheet jump for this noncompact cylinder, established by the test-localized The piecewise constant cylinder field $e_{3} 1_{{ρ < a}}$ : distributional curl on the cylinder for the current density $K e_{φ}$ (computed directly as $e_{ρ} \times (0 - μ_{0} K e_{3}) = μ_{0} K e_{φ}$ ); and $div B = 0$ , $curl B = μ_{0} J_{S}$ hold in the distributional sense of The steady-current (magnetostatic) regime, the compact smooth current class and the declared ideal noncompact models(c4). Hence $B$ is an admitted field of the declared solenoid model.

(c) [Scope]. Both models are declared noncompact idealisations: they are not finite-energy localized-current theorems; nothing is asserted about self-energy, inductance, self-force, the fields on the sources, or the approximation of any physical apparatus; and the fields are selected within the declared symmetry and decay class, not shown to be the only solutions of the magnetostatic equations off the sources without that class. The external force between two such wires is treated in Force per unit length between two declared ideal parallel wires under the adopted thin-wire coupling.

Facts & Assumptions

Given: Assume ${A C}_{ω}$ ; let the ideal line current $J_{w}$ and the ideal sheet current $J_{S}$ be the declared objects of The steady-current (magnetostatic) regime, the compact smooth current class and the declared ideal noncompact models(c2) and (c3); let $B$ be the fields defined in (a) and (b) of the Example.

[F1]

The field $f : = ρ^{- 1} e_{φ}$ on $R^{3} ∖ γ$ is locally integrable, is $C^{\infty}$ off $γ$ with $div f = curl f = 0$ there, has circulation $2 π$ along each positively oriented circle $γ_{r, h}$ defined in (a), and satisfies $⟨ curl T_{f}, ψ ⟩ = 2 π A (ψ)$ with $A (ψ) = \int_{R} ψ_{3} (0, 0, x_{3}) d x_{3}$ and $⟨ div T_{f}, φ ⟩ = 0$ for all scalar and vector tests (The azimuthal line field $ρ^{- 1} e_{φ}$ : local integrability and distributional curl on the axis).

[F2]

The piecewise constant field $g = e_{3} 1_{{ρ < a}}$ extended by zero outside the cylinder satisfies $⟨ curl T_{g}, ψ ⟩ = \int_{S} e_{φ} \cdot ψ d S$ and $⟨ div T_{g}, φ ⟩ = 0$ for all tests, with the one-sided values satisfying $g_{+} - g_{-} = - e_{3}$ , $e_{ρ} \times (g_{+} - g_{-}) = e_{φ}$ and $e_{ρ} \cdot (g_{+} - g_{-}) = 0$ (The piecewise constant cylinder field $e_{3} 1_{{ρ < a}}$ : distributional curl on the cylinder).

[F3]

The declared line current acts as $⟨ J_{w}, ψ ⟩ = I \int_{R} ψ_{3} (0, 0, s) d s$ and the declared sheet current as $⟨ J_{S}, ψ ⟩ = K \int_{S} e_{φ} \cdot ψ d S$ (The steady-current (magnetostatic) regime, the compact smooth current class and the declared ideal noncompact models(c2)-(c3)), and the admissibility predicate (c4) requires local integrability, $C^{1}$ regularity and vanishing divergence and curl off the source, the two global distributional identities, the declared symmetry and the stated decay (The steady-current (magnetostatic) regime, the compact smooth current class and the declared ideal noncompact models).

[F4]

The cross product and the identities $e_{3} \times e_{ρ} = e_{φ}$ , $e_{ρ} \times e_{3} = - e_{φ}$ hold (The cross product in $R^{3}$ ); divergence and curl are the coordinate expressions of Divergence and curl of a $C^{1}$ vector field; the circulation is the vector line integral of Scalar line integrals with respect to arc length and vector-field line integrals; surface integrals over the cylinder are those of Surface integration on compact C1 hypersurfaces, with the compact-face area computed through the Riemann integral of The Riemann integral of a bounded function over a bounded Jordan measurable set and A continuous real function on a compact Jordan measurable set is Riemann integrable over that set.

[F5]

The physical reading of these declared ideal models is the separate adopted assumption Physical adoption of the declared ideal magnetostatic loop, line and cylindrical-sheet models, which asserts no existence, uniqueness, finite self-energy, force, self-force or apparatus accuracy (Physical adoption of the declared ideal magnetostatic loop, line and cylindrical-sheet models).

[F6]

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

Verification

technique · direct; the two explicit fields are scalar multiples of the fields of the two mathematical lemmas, so their pointwise, symmetry and distributional properties transport, and the admissibility predicate is then checked clause by clause

1.1givenF1

The field of (a) is $B = \frac{μ_{0} I}{2 π} f$ with $f = ρ^{- 1} e_{φ}$ ; by [F1] it is locally integrable on $R^{3}$ and $C^{\infty}$ off $γ$ , with $div B = curl B = 0$ pointwise off $γ$ , and all these properties are stable under multiplication by the constant $\frac{μ_{0} I}{2 π}$ .

1.2givenF2

The field of (b) is $B = μ_{0} K g$ with $g$ the piecewise constant cylinder field: it is bounded, hence locally integrable, constantly $μ_{0} K e_{3}$ for $ρ < a$ and $0$ for $ρ > a$ , so it is $C^{\infty}$ off $S$ with $div B = curl B = 0$ there; it is invariant under translations along and rotations about the third axis, and its supremum on any compact set of cross-sectional parameters is $0$ for every $ρ > a$ , hence its transverse far-zone supremum is $0$ .

2.1step 1.1F1F3F4

For the wire, $⟨ curl T_{B}, ψ ⟩ = \frac{μ_{0} I}{2 π} ⟨ curl T_{f}, ψ ⟩ = μ_{0} I A (ψ) = μ_{0} ⟨ J_{w}, ψ ⟩$ and $⟨ div T_{B}, φ ⟩ = 0$ for all tests by [F1] and [F3]; moreover $B$ depends only on $ρ$ through the scalar factor $\frac{μ_{0} I}{2 π ρ}$ multiplying $e_{φ}$ , so it is invariant under translations along $γ$ and rotations about it, and its transverse far-zone supremum is $\frac{μ_{0} ∣ I ∣}{2 π ρ} \to 0$ . The circulation of $B$ along the increasing- $t$ traversal $γ_{r, h}$ of (a) is $\frac{μ_{0} I}{2 π}$ times the circulation $2 π$ of $f$ , that is $μ_{0} I$ , by [F1] and the linearity of the line integral [F4].

2.2step 1.2F2F3F4

For the sheet, $⟨ curl T_{B}, ψ ⟩ = μ_{0} K ⟨ curl T_{g}, ψ ⟩ = μ_{0} K \int_{S} e_{φ} \cdot ψ d S = μ_{0} ⟨ J_{S}, ψ ⟩$ and $⟨ div T_{B}, φ ⟩ = 0$ for all tests by [F2] and [F3]. The one-sided values across $S$ are $B_{+} = 0$ and $B_{-} = μ_{0} K e_{3}$ in the direction from the interior to the exterior, so $[B] = B_{+} - B_{-} = - μ_{0} K e_{3}$ and, by [F4], $n \times [B] = e_{ρ} \times (- μ_{0} K e_{3}) = μ_{0} K e_{φ}$ and $n \cdot [B] = 0$ ; this is the jump displayed in (b), computed directly from the one-sided values of the declared field.

3.1step 1.1step 2.1F3

For the wire, every clause of the admissibility predicate F3 holds: local integrability and off-source $C^{\infty}$ regularity from step 1.1, the two pointwise vanishing identities off $γ$ from step 1.1, the two distributional identities and the line-measure value from step 2.1, the translation and rotation symmetry from step 2.1, and the transverse decay from step 2.1. Hence the field of (a) is admitted for the declared wire model; the decay requirement is what rules out adding a constant longitudinal field, and nothing else is claimed.

3.2step 1.2step 2.2F3

For the sheet, every clause of F3 holds: local integrability, off-source $C^{\infty}$ regularity, the two pointwise vanishing identities and the symmetry and the vanishing transverse far-zone supremum from step 1.2, and the two distributional identities from step 2.2, with the one-sided jump of step 2.2 as the interface content. Hence the field of (b) is admitted for the declared solenoid model.

4.1step 3.1step 3.2F3F5F6∎

Steps 3.1 and 3.2 prove that both explicit fields are admitted under the predicate (c4) of [F3]; the physical reading of the declared models is the separate adopted assumption [F5], which adds no existence, uniqueness, self-energy, inductance, force or apparatus claim. Both models are declared noncompact idealisations: no finite-energy or localized-current conclusion is drawn, nothing is asserted on the sources, and the fields are selected within the declared symmetry and decay class rather than shown to be the only solutions off the sources. Countable Choice is inherited only through the cited measure, surface, regular-distribution and Newtonian interfaces [F6]; no stronger choice principle occurs.

Source notes

Griffiths, Example 5.5 and Examples 5.7 and 5.9, printed pp. 215-217 and 226-228, evaluates the wire field $μ_{0} I / (2 π r)$ and the infinite-solenoid field $μ_{0} n I$ inside and zero outside; Tong, Sections 3.1.1-3.1.2, printed pp. 42-46, gives the wire field, the surface-current discontinuity (3.5)-(3.6) and the solenoid field (3.7); Chew, Lecture 4 Section 4.3.3-4.3.4, printed pp. 46-48, gives the current-sheet jump and the normal-field continuity. The verification above is the exact ideal-model check against the framework's admissibility predicate; no measurement, apparatus or uncertainty is involved, and the idealizations are not claimed to describe any real conductor or coil.

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Force per unit length between two declared ideal parallel wires

Physical scope

A local external-force comparison between two declared ideal parallel straight wires, restricted to the force per unit length that wire 1's external field exerts on wire 2 under the adopted neutral-thin-wire coupling. Self-fields and support forces of infinitely long wires are excluded; no dynamics, circuit model, material response or measurement is claimed.

Example

Assume Countable Choice ( ${A C}_{ω}$ ), with the exact meaning of The Axiom of Countable Choice ( ${A C}_{ω}$ ). Let $d > 0$ and $I_{1}, I_{2} \in R$ ; let wire 1 be the declared ideal line current of amplitude $I_{1}$ along the third coordinate axis and wire 2 the translate of that ideal line current by $d e_{1}$ , of amplitude $I_{2}$ (The steady-current (magnetostatic) regime, the compact smooth current class and the declared ideal noncompact models(c2)), and adopt for the force on wire 2 the thin-wire coupling (W) of Force and torque on a declared ideal thin-wire loop in a prescribed external field, restricted to the external field of wire 1. Then the force per unit length on wire 2 is $f = \frac{μ_{0} I_{1} I_{2}}{2 π d} e_{3} \times e_{2} = - \frac{μ_{0} I_{1} I_{2}}{2 π d} e_{1},$ i.e. attraction towards wire 1 (direction $- e_{1}$ ) for currents in the same direction ( $I_{1} I_{2} > 0$ ), repulsion for opposite directions, and magnitude $μ_{0} ∣ I_{1} I_{2} ∣ / (2 π d)$ per unit length. The comparison uses only the external field of wire 1 at the position of wire 2; the self-fields and the mechanical support forces of the two infinite wires are excluded, no claim is made about the total force on either infinite wire, and no measurement is involved.

Facts & Assumptions

Given: Assume ${A C}_{ω}$ ; let $d > 0$ , $I_{1}, I_{2} \in R$ ; wire 1 and wire 2 are the declared ideal line currents of The steady-current (magnetostatic) regime, the compact smooth current class and the declared ideal noncompact models(c2) along the third axis and its translate by $d e_{1}$ ; the coupling (W) of Force and torque on a declared ideal thin-wire loop in a prescribed external field is restricted to the external field of wire 1.

[F1]

The declared ideal wire 1 has the admitted field $B_{1} (x) = \frac{μ_{0} I_{1}}{2 π ρ} e_{φ}$ on $R^{3} ∖ γ$ , with $ρ = \sqrt{x_{1}^{2} + x_{2}^{2}}$ and $e_{φ} = ρ^{- 1} (- x_{2}, x_{1}, 0)$ , verified as an admitted field of the declared wire model (Declared ideal infinite wire and cylindrical current sheet: explicit fields and their distributional equations, The steady-current (magnetostatic) regime, the compact smooth current class and the declared ideal noncompact models, Physical adoption of the declared ideal magnetostatic loop, line and cylindrical-sheet models).

[F2]

Under the coupling (W) of the force theorem, the force on a parametrized wire segment $γ : [0, L] \to R^{3}$ of the declared model in a prescribed external field is $F = I \int_{0}^{L} γ^{'} (s) \times B (γ (s)) d s$ , a componentwise Riemann integral of a continuous integrand (Force and torque on a declared ideal thin-wire loop in a prescribed external field, clause (a)).

[F3]

For $u, v \in R^{3}$ , $u \times v = (u_{2} v_{3} - u_{3} v_{2}, u_{3} v_{1} - u_{1} v_{3}, u_{1} v_{2} - u_{2} v_{1})$ ; in particular $e_{3} \times e_{2} = - e_{1}$ . The Riemann integral on a compact interval is linear and the integral of a constant $c$ over $[- S, S]$ is $2 S c$ (The cross product in $R^{3}$ , Linearity, monotonicity, the absolute-value estimate and coordinate-slice additivity for the Riemann integral in $R^{m}$ ).

[F4]

${A C}_{ω}$ is the countable-choice principle assumed here and inherited by the asserted ideal-line model; the final constant-force computation is finite (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Verification

technique · direct; evaluate the external field of wire 1 on the line of wire 2, integrate the coupling over a compact straight segment, and divide by its length

1.1givenF1

On the points $(d, 0, x_{3})$ of wire 2 one has $ρ = d$ and $(- x_{2}, x_{1}, 0) = (0, d, 0)$ , so $e_{φ} = e_{2}$ and, by [F1], $B_{1} (d, 0, x_{3}) = \frac{μ_{0} I_{1}}{2 π d} e_{2}$ , constant along the whole line of wire 2.

2.1step 1.1F2F3

Fix $S > 0$ and let $γ (x_{3}) : = (d, 0, x_{3})$ , $x_{3} \in [- S, S]$ , a compact straight segment of wire 2 parametrized by arc length, so $γ^{'} = e_{3}$ and $∣ γ^{'} ∣ = 1$ . By [F2] the force on this segment under (W), restricted to the external field $B_{1}$ of wire 1, is $I_{2} \int_{- S}^{S} e_{3} \times B_{1} (d, 0, x_{3}) d x_{3} = I_{2} (\frac{μ_{0} I_{1}}{2 π d}) (e_{3} \times e_{2}) \int_{- S}^{S} 1 d x_{3} = 2 S \frac{μ_{0} I_{1} I_{2}}{2 π d} (e_{3} \times e_{2})$ , using that the integrand is constant and the integral of a constant over $[- S, S]$ is $2 S$ times it by [F3].

3.1step 2.1F2F4∎

Dividing the force of step 2.1 by the segment length $2 S$ gives the force per unit length $f = \frac{μ_{0} I_{1} I_{2}}{2 π d} (e_{3} \times e_{2})$ ; by [F3], $e_{3} \times e_{2} = - e_{1}$ , so $f = - \frac{μ_{0} I_{1} I_{2}}{2 π d} e_{1}$ , which for $I_{1} I_{2} > 0$ points from wire 2 towards wire 1 (attraction) and for $I_{1} I_{2} < 0$ away from it (repulsion), with magnitude $\frac{μ_{0} ∣ I_{1} I_{2} ∣}{2 π d}$ ; the result is independent of $S$ , so it is a force per unit length of the comparison. Only the external field of wire 1 is used: the self-field of wire 2 and the support forces of both infinite wires are excluded, the compact segment is a local part of a fed or closed circuit (the endpoint identity of Force and torque on a declared ideal thin-wire loop in a prescribed external field applies to the standalone segment), and no dynamics, circuit model, material response or measurement is claimed. Countable Choice enters only through the asserted ideal-line model [F4].

Source notes

Griffiths, Section 5.2.2, printed pp. 217-218, computes the attraction between two long parallel wires from the field $μ_{0} I_{1} / (2 π d)$ and the line-current force law; Tong, Section 3.4.1, printed pp. 57-59, gives $f = - (μ_{0} I_{1} I_{2} / 2 π d) \hat{x}$ (3.25) with attraction for equal directions and repulsion for opposite ones; Landau-Lifshitz, Section 42, printed pp. 100-101, describes the interaction of two systems through their fields, re-expressed here in SI. The computation above is the exact local external-force comparison for the declared ideal wires under the adopted coupling; it is not a solution of the two-wire dynamics.

Magnetostatics in vacuum — Examples

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

5 · Examples, experiments and counterexamples

Axis field of the declared ideal circular thin loop

Example

Facts & Assumptions

Verification

Source notes

Declared ideal infinite wire and cylindrical current sheet: explicit fields and their distributional equations

Example

Facts & Assumptions

Verification

Source notes

Force per unit length between two declared ideal parallel wires

Example

Facts & Assumptions

Verification

Source notes

Sources