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Reciprocal linkage coefficients and magnetic energy of smooth compact conserved currents

Physical scope

Fixed compact smooth conserved current profiles of the adopted framework and their Coulomb-gauge Newtonian vector potential: finite reciprocal linkage coefficients, the linear linkage identity, the exact positive-semidefinite quadratic magnetic energy and the vanishing/positive-definiteness criterion. It asserts no thin-wire, sheet or singular self-inductance, no time-dependent, moving, radiating or material configuration and no uniqueness of the vector potential; the identification of the coefficients with the lumped circuit inductance matrix is the separately adopted circuit model.

Statement

Assume Countable Choice ( ${A C}_{ω}$ ), with the exact meaning of The Axiom of Countable Choice ( ${A C}_{ω}$ ). Let $μ_{0} > 0$ , $c > 0$ , $ε_{0} = (μ_{0} c^{2})^{- 1}$ be the constants of Framework-local SI units, the vacuum constants and the exact/measured status of their values, and let $j_{1}, \dots, j_{n} \in C_{c}^{\infty} (Ω; R^{3})$ with $div j_{α} = 0$ be fixed compact smooth conserved current profiles of the class of The steady-current (magnetostatic) regime, the compact smooth current class and the declared ideal noncompact models(b), extended by zero to $R^{3}$ . Fix amplitudes $I \in R^{n}$ , label them by $I_{α} : = I (α - 1)$ for $1 \leq α \leq n$ in the zero-based convention of The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , and put $J_{I} : = \sum_{α = 1}^{n} I_{α} j_{α}$ , let $A_{J_{I}}$ be the Coulomb-gauge Newtonian vector potential of Coulomb-gauge Newtonian vector potential for a compact smooth conserved current applied to the conserved current $J_{I}$ , and let $B_{J_{I}} = curl A_{J_{I}}$ . Define the linkage coefficients $L_{α β} : = \frac{μ_{0}}{4 π} \int_{R^{3}} \int_{R^{3}} \frac{⟨ j_{α} (x), j_{β} (y) ⟩}{∣ x - y ∣} d λ_{3} (y) d λ_{3} (x), α, β = 1, \dots, n .$ The amplitudes $I_{α}$ carry amperes and the profiles $j_{α}$ are shapes per unit current, with units $m^{- 2}$ ; their numerical SI maps belong to the stated smooth class. Consequently $J_{I}$ has unit $A m^{- 2}$ , $A_{J_{I}}$ has $T m$ , $L_{α β}$ has H, the linkage has Wb and $U$ has joules. $A_{j_{α}}$ denotes the linear potential response per ampere. Then:

(a) [Finiteness and self-energy] every $L_{α β}$ is a finite real number: the defining double integral converges absolutely, and in particular every self-coefficient $L_{α α}$ is finite, so the compact smooth profiles carry a finite self-energy. This is a statement about the smooth profiles only; no finite value is assigned to a zero-radius thin-wire self-inductance.

(b) [Reciprocity and flux linkage] $L_{α β} = L_{β α}$ for all $α, β$ ; $A_{J_{I}} = \sum_{α} I_{α} A_{j_{α}}$ componentwise; and for every $β$ the linkage integral is $\int_{R^{3}} ⟨ j_{β}, A_{J_{I}} ⟩ d λ_{3} = \sum_{α = 1}^{n} L_{β α} I_{α} .$ Thus the numbers $L_{α β}$ are the linkage coefficients of the profiles: the linkage of profile $β$ with the field of the superposition is the linear combination $\sum_{α} L_{β α} I_{α}$ .

(c) [Magnetic energy and positivity] with $U (I) : = \frac{1}{2} \int_{R^{3}} ⟨ J_{I}, A_{J_{I}} ⟩ d λ_{3}$ , $U (I) = \frac{1}{2} \sum_{α, β = 1}^{n} L_{α β} I_{α} I_{β} = \frac{1}{2 μ_{0}} \int_{R^{3}} ∣ B_{J_{I}} ∣^{2} d λ_{3} \geq 0,$ all quantities finite; moreover $U (I) = 0$ holds precisely when $J_{I} = 0$ on $R^{3}$ . Identify these profile-labelled coefficients with the standard matrix by $L (i, j) : = L_{i + 1, j + 1}$ for $0 \leq i, j < n$ . Consequently the symmetric matrix $L$ is positive semidefinite, and it is positive definite exactly when the profiles are pointwise linearly independent, i.e. when $\sum_{α} I_{α} j_{α} \equiv 0$ forces $I = 0$ .

(d) [Scope] the theorem concerns fixed compact smooth conserved current profiles and their Coulomb-gauge fields. It asserts no thin-wire, sheet or singular self-inductance; no time-dependent, moving, radiating or material configuration; and no uniqueness of the vector potential. The identification of these coefficients with the inductance matrix of the adopted lumped circuit model, and of the linkage derivatives with terminal voltages, is adopted in Lumped circuit approximation: branch data, terminal laws and the declared quasistatic regime; the strict material-loop theorem applies separately only to an actual moving sheet and compatible smooth Maxwell fields satisfying its hypotheses. No terminal-voltage law for arbitrary varying frozen profiles is inferred from the present magnetostatic computation.

Facts & Assumptions

Given: Assume ${A C}_{ω}$ , the constants $μ_{0} > 0$ , $c > 0$ , $ε_{0} = (μ_{0} c^{2})^{- 1}$ , compact smooth divergence-free profiles $j_{α}$ , amplitudes $I \in R^{n}$ with $I_{α} = I (α - 1)$ and matrix entries $L (i, j) = L_{i + 1, j + 1}$ , the current $J_{I} = \sum_{α} I_{α} j_{α}$ , and the Coulomb-gauge field $A_{J_{I}}$ of the magnetostatic potential theorem.

[A1]

Countable Choice, written ${A C}_{ω}$ , says every sequence of nonempty sets has a choice function. (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

For $J \in C_{c}^{\infty} (Ω; R^{3})$ with $div J = 0$ , the Coulomb-gauge Newtonian potential $A = μ_{0} N (J)$ is $C^{\infty}$ with $- Δ A = μ_{0} J$ and $div A = 0$ pointwise, and $B = curl A$ is $C^{\infty}$ with $div B = 0$ , $curl B = μ_{0} J$ pointwise. (Coulomb-gauge Newtonian vector potential for a compact smooth conserved current).

[F2]

For $u, v \in C_{c}^{\infty} (R^{3}; R^{3})$ the three integrals $\int ⟨ u, N v ⟩$ , $\int ⟨ N u, v ⟩$ and $\frac{1}{4 π} \iint \frac{⟨ u (x), v (y) ⟩}{∣ x - y ∣}$ converge absolutely and have a common value; the pairing is symmetric and bilinear. (Symmetry and Fubini representation of the Newtonian linkage pairing).

[F3]

For $J \in C_{c}^{\infty} (R^{3}; R^{3})$ with $div J = 0$ and $A = μ_{0} N (J)$ : $\int ∣ curl A ∣^{2} = μ_{0} \int ⟨ A, J ⟩$ and $\frac{1}{2} \int ⟨ A, J ⟩ = \frac{1}{2 μ_{0}} \int ∣ curl A ∣^{2} \geq 0$ , the integrals finite, and $curl A \in L^{2}$ . (Curl-field energy identity for Newtonian potentials of divergence-free compact currents).

[F4]

The compact smooth current class consists of the currents $J \in C_{c}^{\infty} (Ω; R^{3})$ with $div J = 0$ , extended by zero to $R^{3}$ . (The steady-current (magnetostatic) regime, the compact smooth current class and the declared ideal noncompact models).

[F5]

The framework constants are $μ_{0} > 0$ , $c > 0$ and $ε_{0} = (μ_{0} c^{2})^{- 1}$ . (Framework-local SI units, the vacuum constants and the exact/measured status of their values).

[F6]

The lumped circuit model of Lumped circuit approximation: branch data, terminal laws and the declared quasistatic regime is an adopted reduced model: its matrix $L$ is declared model data, the flux linkage is $L I$ , and its terminal law is a branch relation of the model, not a derivation from arbitrary time-dependent Maxwell fields. (Lumped circuit approximation: branch data, terminal laws and the declared quasistatic regime).

[F7]

The Newtonian potential is $N f (x) = \int_{R^{n}} Φ (x - y) f (y) d y$ at each $x$ where the integral is absolutely finite. (Newtonian potential of compactly supported data).

[F8]

The Lebesgue integral is linear on $L^{1}$ . (The Lebesgue integral is linear on $L^{1} (μ)$ ).

[F9]

For a $C^{2}$ field $F$ on open $U \subseteq R^{3}$ , $curl curl F = \nabla div F - Δ F$ . (The curl of a curl is the gradient of the divergence minus the Laplacian).

[F10]

For a $C^{2}$ map $Δ F$ is componentwise and the Laplacian is $div \nabla$ . (The Laplacian of a $C^{2}$ function and of a $C^{2}$ vector field).

[F11]

The divergence and curl are defined pointwise from the first partial derivatives. (Divergence and curl of a $C^{1}$ vector field).

[F12]

The Euclidean inner product is symmetric and real bilinear. (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ).

[F13]

A nonnegative measurable function has zero integral exactly when it vanishes almost everywhere. (A nonnegative measurable function has integral $0$ exactly when it vanishes almost everywhere).

[F14]

Under Countable Choice every continuous map $R^{n} \to R^{m}$ is Borel measurable. (Continuous functions on Euclidean spaces are Borel measurable).

[F15]

For measurable $f, g : X \to [0, + \infty]$ with $f \leq g$ , $\int f \leq \int g$ . (Monotonicity and nonnegative homogeneity of the nonnegative integral).

[F16]

For $n \geq 1$ , $x \in R^{n}$ , $r > 0$ the ball satisfies $0 < λ (B (x, r)) < \infty$ . (Euclidean balls have positive finite Lebesgue measure).

[F17]

On a measure space $(X, A, μ)$ , for measurable $f : X \to [0, + \infty]$ and $E \in A$ , $\int_{E} f d μ : = \int f χ_{E} d μ$ , with the nonnegative Lebesgue integral on the right. (Integral over a measurable subset).

Proof

technique · direct

1.1F1F4F7F8F10F11algebra

For each $α$ the profile $j_{α}$ lies in the class of [F4], and so does $J_{I}$ ; by [F1] applied to $J_{I}$ the field $A_{J_{I}}$ is $C^{\infty}$ with $- Δ A_{J_{I}} = μ_{0} J_{I}$ and $div A_{J_{I}} = 0$ pointwise, and $B_{J_{I}} = curl A_{J_{I}}$ is $C^{\infty}$ with $curl B_{J_{I}} = μ_{0} J_{I}$ , the Laplacian here being the componentwise operator of [F10]. Since each $A_{j_{α}} = μ_{0} N (j_{α})$ is the Newtonian potential of [F7] and $N$ is componentwise linear by [F8], $A_{J_{I}} = \sum_{α} I_{α} A_{j_{α}}$ componentwise, and $B_{J_{I}} = \sum_{α} I_{α} B_{j_{α}}$ by linearity of the pointwise curl [F11].

1.2F2F3algebra

By [F2] applied to $(J_{I}, J_{I})$ and the definitions of $U$ and the coefficients, $U (I) = \frac{1}{2} \int ⟨ J_{I}, A_{J_{I}} ⟩ = \frac{μ_{0}}{2} \frac{1}{4 π} \iint \frac{⟨ J_{I} (x), J_{I} (y) ⟩}{∣ x - y ∣} = \frac{1}{2} \sum_{α, β} L_{α β} I_{α} I_{β}$ , all finite. On the other hand [F3] applied to the divergence-free current $J_{I}$ with parameter $μ_{0}$ gives $\int ⟨ J_{I}, A_{J_{I}} ⟩ = \frac{1}{μ_{0}} \int ∣ B_{J_{I}} ∣^{2}$ , hence $U (I) = \frac{1}{2 μ_{0}} \int ∣ B_{J_{I}} ∣^{2} \geq 0$ ; this is the energy identity of (c).

2.1F2F8F12step 1.1algebra

Apply [F2] to the pair $(j_{α}, j_{β})$ : the double integral defining $L_{α β}$ (with the prefactor $μ_{0}$ ) converges absolutely, so $L_{α β}$ is a finite real number, and exchanging the two fields and the two integration variables leaves it unchanged, so $L_{α β} = L_{β α}$ ; this is (a) and the reciprocity of (b). Applying [F2] to $(j_{β}, J_{I})$ and using the symmetry of the Euclidean pairing [F12] and the linearity of the integrals [F8] gives $\int ⟨ j_{β}, A_{J_{I}} ⟩ = μ_{0} \frac{1}{4 π} \iint \frac{⟨ j_{β} (x), J_{I} (y) ⟩}{∣ x - y ∣} = \sum_{α} I_{α} L_{β α}$ ; this is the linkage identity of (b).

2.2F9F13F14F15F16F17step 1.1step 1.2algebra

If $U (I) = 0$ then $\int ∣ B_{J_{I}} ∣^{2} = 0$ by step 1.2. The function $∣ B_{J_{I}} ∣^{2}$ is nonnegative, continuous by step 1.1 and Borel by [F14], so [F13] gives $∣ B_{J_{I}} ∣ = 0$ almost everywhere; were $B_{J_{I}} (x_{0}) \neq 0$ for some $x_{0}$ , continuity would give $∣ B_{J_{I}} ∣^{2} \geq ε > 0$ on a ball $B (x_{0}, r)$ of finite positive measure [F16], and [F15] with [F17] would give $\int ∣ B_{J_{I}} ∣^{2} \geq ε λ_{3} (B (x_{0}, r)) > 0$ , a contradiction; hence $B_{J_{I}} = 0$ everywhere. Then $curl curl A_{J_{I}} = \nabla div A_{J_{I}} - Δ A_{J_{I}} = μ_{0} J_{I}$ by [F9] and step 1.1, while $curl B_{J_{I}} = 0$ ; so $μ_{0} J_{I} = 0$ and $J_{I} = 0$ . Conversely $J_{I} = 0$ gives $A_{J_{I}} = 0$ componentwise and $U (I) = 0$ .

3.1step 2.1step 1.2step 2.2algebra

By step 1.2, $U$ is the quadratic form $I \mapsto \frac{1}{2} I^{T} L I$ of the symmetric matrix $L$ of step 2.1, and $U \geq 0$ ; hence $L$ is positive semidefinite. By step 2.2, $U (I) = 0$ exactly when $\sum_{α} I_{α} j_{α} \equiv 0$ ; therefore $L$ is positive definite exactly when that vanishing forces $I = 0$ , that is, exactly when the profiles are pointwise linearly independent.

4.1A1F5F6step 1.1step 2.1step 1.2step 2.2step 3.1∎

The theorem establishes (a)-(c) for the fixed compact smooth conserved profiles and their Coulomb-gauge fields, with no thin-wire, singular, moving, radiating or material configuration treated; the identification of $L$ with the lumped-model inductance matrix and of the linkage derivatives with terminal voltages is the adopted model content of [F6], not a consequence of this magnetostatic computation, and no terminal-voltage law for arbitrary varying frozen profiles is inferred. The constants used are those of [F5], and Countable Choice is inherited only through the named Newtonian, measure and vector-calculus interfaces.

Source notes

Griffiths §7.2.3, printed pp. 310-317, derives $M_{21} = M_{12}$ and the Neumann formula; Fitzpatrick §7.2, printed pp. 216-218, treats stationary-loop linkage and reciprocity in (7.3)-(7.10); §7.5, printed pp. 225-231, treats coil energy and the mutual term $M I_{1} I_{2}$ in (7.37)-(7.71); Tong §4.1.2, printed pp. 71-73, defines the inductance as flux per unit current and writes the magnetostatic energy $\frac{1}{2} L I^{2}$ . The present theorem replaces the thin-loop formulas by the explicitly defined smooth compact conserved profiles, makes the reciprocity an instance of the symmetry of the Newtonian linkage pairing, and identifies the energy with the frozen curl-field energy of the Newtonian construction; the thin-wire self-inductance and every singular limit remain outside the statement.

Reciprocal linkage coefficients and magnetic energy of smooth compact conserved currents

Statement

Facts & Assumptions

Proof

Source notes

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