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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.

Force per unit length between two declared ideal parallel wires

Example

Facts & Assumptions

Verification

Source notes

Depends on

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