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Regularization of self inductance
arXiv:1804.11098v1 [math.DG] 30 Apr 2018
Jun O’Hara ∗ May 1, 2018
Abstract
We introduce several methods to define the self-inductance of a single loop as the regu- larization of divergent integrals which we obtain by applying Neumann (or Weber) formula for the mutual inductance of a pair of loops to the case when two loops are identical. Keywords: self-inductance, Neumann’s formula, knot 2010 Mathematics Subject Classification: 53C65, 53C40, 46T30
1 Introduction
Inductance is an important notion in classical electrodynamics. Suppose two closed circuits Γ 1 and Γ 2 , mutually disjoint, carry currents I 1 and I 2 respectively. The magnetic field energy U 1 2 of the system is given by U 1 2 = I 1 I 2 L 1 2 , where the coefficient L 1 2 is called the mutual inductance. According to Neumann formula ([N]), the mutual inductance can be expressed by L N (Γ 1 , Γ 2 ) := µ 0 4π Γ 1 Γ 2 dx 1 · dx 2 |x 1 − x 2 | , (1.1) where µ 0 = 4π × 10 − 7 H/m is the permeability of free space. There is another expression of the mutual inductance, Weber’s formula; by L W (Γ 1 , Γ 2 ) := µ 0 4π Γ 1 Γ 2 (ˆr 1 2 · dx 1 ) (ˆr 1 2 · dx 2 ) |x 1 − x 2 | , (1.2) where r 1 2 := x 2 − x 1 and ˆr 1 2 := r 1 2 / |r 1 2 | . It is known that these two expressions are equivalent. The reader is referred to [Da] p.230 for the proof. The equivalence also follows from (2.11). If we put Γ 1 = Γ 2 in (1.1) (or (1.2)) the integral diverges in logarithmic order because of the contribution of a neighbourhood of the diagonal set. There have been some studies to avoid this divergence difficulty. Bueno and Assis ([BA]) used higher dimensional objects, i.e., a surface or a 3-dimensional wire thicked around a core loop Γ instead of Γ itself. They also showed that self-inductances obtained from Neumann formula and Weber formula are equivalent. Dengler ([D]) replace a neighbourhood of a point on the loop by a short straight wire segment to avoid the divergence of self-energy. In this article, we propose a new definition of the self-inductance of a single loop by regular- izing L N (Γ, Γ) (and L W (Γ, Γ)) without approximating the loop by higher dimensional objects ∗ Supported by JSPS KAKENHI Grant Number 16K05136. 1
(Theorem 2.1). To avoid the divergence of self-energy, we use two methods studied in the theory of generalized functions, Hadamard regularization and regularization via analytic continuation (see, for example, [GS] or [Z ]). The same methods have been used to define the M¨obius energy of a knot by regularizing Γ Γ |x − y| − 2 dx dy in [ O] and [B] respectively, and in general, the Riesz energy of a manifold M by regularizing M M |x − y| α dx dy in [OS2]. There is no difference between making a choice of two methods, but unlike in the result by Bueno and Assis, the difference between making a choice of Neumann and Weber formulae is equal to (µ 0 / 2π) times the length of the loop. We then show that our regularized self-inductance coincides with the regularization of the mutual inductance of a pair of close parallel loops as the distance between them tends to 0 (up to addition by a constant depending on the length of the loop) (Theorem 2.7). The same method has been used in [OS1] (see Subsection 3.2). Finally we study regularized self-inductance of solenoids (i.e. coils that wind cylinders) (Theorem 2.8). If we expand it in a series in the number n of turns per unit length, the dominant term is of the order of n 2 , and the coeffcient does not depend on the choice of Neumann or Weber formula to start with. This coefficient, when the length ℓ of the cylinder goes to infinity, is asymptotic to µ 0 π r 2 times ℓ plus constant. Thus, by taking the asymptotics as n and ℓ go to infinity, we can deduce that the dominat term of self-inductance of a solenoid is equal to µ 0 π r 2 n 2 ℓ which fits a well-known formula, which implies the validity of our definition of regularized self-inductance.
2 Regularization of self-inductance
2.1 Regularization with a single loop
We start with explaining the idea of Hadamard regularization of a divergent integral in a general setting. Take an “ε-neibourhood” of the set where the integrand blows up, restrict the integration to the complement of it, expand the result in a Laurent series in ε (possibly with a log term or terms with non-integer powers), and take the constant term. The constant term is called Hadamard’s finite part of the integral. The terms with negative powers (and a log term if exists) are called the counter terms. Let Γ be a smooth 1 simple loop in R 3 with length L parametrized by γ(s) by the arc-length.
Theorem 2.1 The self-inductance can be regularized in the following way.
(1) Hadamard regularization of L N (Γ, Γ) and L W (Γ, Γ) can be carried out as follows. Let ∆ ε
be an “ε-neighbourhood” of the diagonal set with respect to the distance in R 3 :
∆ ε := {(x 1 , x 2 ) ∈ R 3 × R 3 : |x 1 − x 2 | ≤ ε}.
There exist the limits
There exist the limits H N (Γ) := lim ε→0 + µ 0 4π ( Γ×Γ)\∆ ε dx 1 · dx 2 |x 1 − x 2 | + µ 0 L 2π log ε , (2.1) H W (Γ) := lim ε→0 + µ 0 4π ( Γ×Γ)\∆ ε (ˆr 1 2 · dx 1 ) (ˆr 1 2 · dx 2 ) |x 1 − x 2 | + µ 0 L 2π log ε . (2.2) 1 We assume the smoothness for the sake of simplicity. In fact, Γ being C 1 is enough. 2
(2) Fix the loop Γ and consider
F N (z) := µ 0 4π Γ ×Γ |x 1 − x 2 | z dx 1 · dx 2 , (2.3) F W (z) := µ 0 4π Γ ×Γ |x 1 − x 2 | z (ˆr 1 2 · dx 1 ) (ˆr 1 2 · dx 2 ) (2.4)
as functions of a complex variable z. Then both F N (z) and F W (z) are well-defined and
as functions of a complex variable z. Then both F N (z) and F W (z) are well-defined and holomorphic on {z ∈ C : ℜe z > −1}. The domains of F N (z) and F W (z) can be extended to the whole complex plane C by analytic continuation to make meromorphic functions with
possible simple poles at negative odd integers. Let them be denoted by the same symbols F N (z) and F W (z). The first residues are given by
Res(F N , −1) = Res(F W , −1) = µ 0 L 2π . A N (Γ) := lim z→−1 F N (z) − µ 0 L 2π(z + 1) ,
There exist the limits
A N (Γ) := lim z→−1 F N (z) − µ 0 L 2π(z + 1) , A W (Γ) := lim z→−1 F W (z) − µ 0 L 2π(z + 1) .
(3) In each case starting with Neumann formula or Weber formula, there is no difference
between making a choice of two methods of regularization, namely, H N (Γ) = A N (Γ) and
between making a c H W (Γ) = A W (Γ).
(4) The difference between making a choice of Neumann formula and Weber formula is equal
to (µ 0 /2π) times the length of the loop, namely,
H W (Γ) = H N (Γ) + µ 0 2π L. Definition 2.2 Let us call H N (Γ) = A N (Γ) and H W (Γ) = A W (Γ) the regularized self-inductances of Γ in the sense of Neumann and Weber respectively. The proof of Theorem 2.1 is divided in several steps. Assertions (1),(2) and (3) follow from the argument in Section 3 of [OS2] with slight modi- fication. Let t x j be the unit tangent vector to Γ at x j and θ j (j = 1, 2) be the angle between t x j and r 1 2 . We remark that the numerator of integrand of (2.2) can be experssed as (ˆr 1 2 · dx 1 ) (ˆr 1 2 · dx 2 ) = cos θ 1 cos θ 2 dx 1 d x 2 . Let us give the relation between the arc-length parameter and the chord length (the distance in R 3 ) of a nearby point from a fixed point on the knot explicitly. 3
Lemma 2.3 Let x 1 be a point on Γ. Suppose a nearby point x 2 on Γ is expressed by the arc-
Lemma 2.3 Let x 1 be a point on Γ. Suppose a nearby point x 2 on Γ is expressed by the arc- length parameter s from x 1 . Namely, x 1 = γ(s 1 ) for some s 1 and x 2 = γ(s 1 + s) (−L/2 < s < 3 2
L/2). Let t be the chord length (the distance in R 3 ) between x 1 and x 2 . Then 2
t = |s| 1 − κ 2 24 s 2 − κκ ′ 24 s 3 + O(s 4 ) , (2.5) s = t + κ 2 24 t 3 + κκ ′ 24 t 4 + O(t 5 ) (s ≥ 0), −t − κ 2 24 t 3 + κκ ′ 24 t 4 + O(t 5 ) (s < 0), (2.6) where κ and κ ′ are the curvature and its derivative with respect to s at point x 1 , i.e., at s = s 1 .
Furthermore the numerators of integrands of (2.1) and (2.2) can be estimated by
t x 1 · t x 2 = 1 − κ 2 2 s 2 − κκ ′ 2 s 3 + O(s 4 ) = 1 − κ 2 2 t 2 − κκ ′ 2 t 3 + O(t 4 ) (s ≥ 0) 1 − κ 2 2 t 2 + κκ ′ 2 t 3 + O(t 4 ) (s < 0), (2.7) (ˆr 1 2 · t x 1 )(ˆr 1 2 · t x 2 ) = 1 − κ 2 4 s 2 + O(s 3 ) = 1 − κ 2 4 t 2 + O(t 3 ). (2.8) Proof. Computation using Frenet-Serret’s formula implies that γ(s 1 + s) can be expressed with respect to the Frenet-Serret frame at γ(s 1 ) as ξ = s − κ 2 6 s 3 − κκ ′ 8 s 4 + O(s 5 ) η = κ 2 s 2 + κ ′ 6 s 3 − κ 2 + κτ 2 − κ ′ ′ 24 s 4 + O(s 5 ) ζ = κτ 6 s 3 + 2κ ′ τ + κτ ′ 24 s 4 + O(s 5 ), where τ means the torsion. Then, (2.5) can be obtained by substituting the above series expan- sion to t = |γ(s 1 + s) − γ(s)| = [(γ(s 1 + s) − γ(s)) · (γ(s 1 + s) − γ(s))] 1 2 . With this frame ˆr 1 2 is expressed as ˆr 1 2 = 1 − κ 2 8 s 2 , κ 2 s + κ ′ 6 s 2 , κτ 6 s 2 + O(s 3 ).
✷
Next lemma is needed for the proof of (2). Let B t ( x) denote the 3-ball with center x and radius t. Let d be the diameter of Γ. 2 The last terms of (2.5), (2.6) and (2.7) are not necessary in this article. We put them here to illustrate ψ ( 4) x 1 (0) = 0 in (2.10) in Lemma 2.4. 4
Lemma 2.4 Fix x 1 on Γ and put
n Γ and put ψ x 1 (t) := Γ ∩B t ( x 1 ) t x 1 · t x 2 dx 2 (0 ≤ t ≤ d).
Then ψ x 1 (t) extends to a smooth function on (−d, d), denoted by the same symbol, with ψ x 1 (−t) =
Then ψ x 1 (t) extends to a smooth function on (−d, d), denoted by the same symbol, with ψ x 1 (−t) = −ψ x 1 (t). Put ϕ(t) := ψ ′ x 1 (t). Then ϕ ( 2j−1) (0) = 0 for j ∈ N. To be precise, ϕ(t) can be expanded in a series in t as ϕ(t) = 2 − 3κ 2 4 t 2 + O(t 4 ). (2.9) Proof. The first half follows from Proposition 3.1 of [OS2] by putting ρ(x 1 , x 2 ) = t x 1 ·t x 2 . We remark that the assertion ψ ( 2j) x 1 (0) = 0 (j ∈ N) can be illustrated by the fact that δ − δ u 2 j−1 du = 0. We prove (2.9) in what follows. The equalities (2.6) and (2.7) imply ψ x 1 (t) = t 0 1 − κ 2 2 u 2 − κκ ′ 2 u 3 + O(u 4 ) 1 + κ 2 8 u 2 + κκ ′ 6 u 3 + O(u 4 ) du + 0 t 1 − κ 2 2 u 2 + κκ ′ 2 u 3 + O(u 4 ) − 1 − κ 2 8 u 2 + κκ ′ 6 u 3 + O(u 4 ) du = 2t − κ 2 4 t 3 + O(t 5 ), (2.10) and hence ϕ(t) = 2 − 3κ 2 4 t 2 + O(t 4 ), which completes the proof. ✷ Finally, we give a lemma which is needed for the proof of (4) of Theorem 2.1. Lemma 2.5 Let ω 1 be the 1-form on Γ given by ω 1 = dx 1 · ˆr 1 2 . Then we have dx 1 · dx 2 |x 1 − x 2 | α = α (ˆr 1 2 · dx 1 ) (ˆr 1 2 · dx 2 ) |x 1 − x 2 | α − d ω 1 |x 1 − x 2 | α −1 . (2.11) Proof. It can be obtained by modifying the proof of Proposition 4.11 of [OS1] (cf. Proposition 5 of [BP]). Suppose (x 1 , x 2 ) belongs to Γ × Γ \ ∆, where ∆ is the diagonal ∆ = {(x, x) : x ∈ R 3 } (the case when (x 1 , x 2 ) ∈ Γ 1 × Γ 2 can be proved in the same way). Let τ be the angle between the two oriented planes containing the line x 1 x 2 tangent to Γ at x 1 and x 2 respectively. Let {e 1 , e 2 , e 3 } be an orthonormal moving frame (along Γ) with e 1 = ˆr 1 2 , and e 3 ⊥ T x 1 Γ. Let ω i = dx 1 · e i and ω i j = de i · e j . Then there hold ω 2 = sin θ 1 dx 1 , ω 3 = 0, ω 1 2 = cos τ sin θ 2 dx 2 |x 2 − x 1 | , which implies dω 1 = ω 1 2 ∧ ω 2 + ω 1 2 ∧ ω 3 = − cos τ sin θ 1 sin θ 2 dx 1 ∧ dx 2 |x 2 − x 1 | . (2.12) 5
(1) The case α ̸= 1. We have cos θ 1 cos θ 2 dx 1 ∧ dx 2 |x 2 − x 1 | α = (ˆr 1 2 · dx 1 ) (ˆr 1 2 · dx 2 ) |x 1 − x 2 | α = − d(|x 2 − x 1 | ) ∧ ω 1 |x 2 − x 1 | α = 1 α − 1 d 1 |x 2 − x 1 | α −1 ∧ ω 1 = 1 α − 1 d ω 1 |x 2 − x 1 | α −1 − 1 α − 1 · 1 |x 2 − x 1 | α −1 dω 1 = 1 α − 1 · cos τ sin θ 1 sin θ 2 dx 1 ∧ dx 2 |x 2 − x 1 | α + 1 α − 1 d ω 1 |x 2 − x 1 | α −1 , and therefore cos τ sin θ 1 sin θ 2 dx 1 ∧ dx 2 |x 2 − x 1 | α = (α − 1) cos θ 1 cos θ 2 dx 1 ∧ dx 2 |x 2 − x 1 | α − d ω 1 |x 2 − x 1 | α −1 . Since dx 1 · dx 2 |x 2 − x 1 | α = (cos θ 1 cos θ 2 + cos τ sin θ 1 sin θ 2 ) dx 1 ∧ dx 2 |x 2 − x 1 | α , (2.13) (2.11) follows. (2) The case α = 1 follows from (2.12) and (2.13).
✷
Proof of Theorem 2.1. We prove only in the case of Neumann formula, as the argument goes parallel for Weber formula. We drop off the coefficient µ 0 / (4π) in the proof to make formulae shorter and simpler. (1) Since ( Γ×Γ)\∆ ε dx 1 · dx 2 |x 1 − x 2 | = Γ Γ \B ε ( x 1 ) t x 1 · t x 2 |x 1 − x 2 | dx 2 dx 1 the regularization process can be reduced to that of Γ |x 1 − x 2 | − 1 (t x 1 · t x 2 ) dx 2 . The assertion follows from (2.5) and (2.7) since the integrand can b e estimated by t x 1 · t x 2 |x 1 − x 2 | = 1 |s| + O(s). (2) Since Γ ×Γ |x 1 − x 2 | z dx 1 · dx 2 = Γ Γ |x 1 − x 2 | z (t x 1 · t x 2 ) dx 2 dx 1 , the regularization of F N (z) can be reduced to that of Γ |x 1 − x 2 | z (t x 1 · t x 2 ) dx 2 . By the coarea formula (a kind of geometric version of Fubini’s theorem), we have Γ |x 1 − x 2 | z (t x 1 · t x 2 ) dx 2 = d 0 t z ψ ′ x 1 (t) dt = d 0 t z ϕ(t) dt. 6
Put f (z) := d 0 t z ϕ(t) dt, which converges for ℜz > −1. Let k be any natural number, and consider the right hand side 3 of d 0 t z ϕ(t) dt = d 1 t z ϕ(t) dt + 1 0 t z ϕ (t) − ϕ(0) − ϕ ′ (0)t − · · · − ϕ ( k−1) (0) (k − 1)! t k −1 dt + k j=1 1 0 ϕ ( j−1) (0) (j − 1)! t z +j−1 dt. (2.14) The first term is a holomorphic function of z. The integrand of second term can be estimated by t z +k , hence the integral converges for ℜz > −k − 1. Since 1 0 ϕ ( j−1) (0) (j − 1)! t z +j−1 dt = ϕ ( j−1) (0) (j − 1)! (z + j) (z ̸= −j) f (z) is a meromorphic function on ℜz > −k − 1 possibly with simple poles at z = −1, . . . , −k with residues given by Res(f, −j) = ϕ ( j−1) (0) (j − 1)! (j = 1, . . . , k) (2.15) ([GS] Ch.1, 3.2). Since k ∈ N is arbitrary as ϕ is smooth, and ϕ ( 2i) (0) = 0 (i = 0, 1, 2, . . . ) by Lemma 2.4, this proves that F N (z) is a meromorphic function with possible simple poles at negative odd integers. Since ϕ(0) = 2 by (2.9), the residue at z = −1 is given by Res(F N , −1) = µ 0 4π Γ ϕ(0) dx 1 = µ 0 L 2π . (3) The above argument can be paraphrased into Hadamard regularization as follows. Putting k = 1 and z = −1 in (2.14), one obtains Γ \B ε ( x 1 ) t x 1 · t x 2 |x 1 − x 2 | dx 2 = d ε t −1 ϕ(t) dt = d 1 t −1 ϕ(t) dt + 1 ε t −1 [ϕ(t) − ϕ(0)] dt + ϕ(0) 1 ε t −1 dt, (2.16) Since 1 0 t z dt = 1 z +1 (z ̸= −1), the residue of z → 1 0 t z dt at z = −1 is equal to 1. Camparing (2.14) with k = 1 and (2.16), the first two terms in the right hand sides coincide since the integrals converge. The equality H N (Γ) = A N (Γ) follows from lim z→−1 1 0 t z dt − 1 z +1 = lim ε→0 + 1 ε dt t + log ε . 3 Here we divide the domain of integration at 1. When d < 1 we can take any d 0 with 0 < d 0 < d instead of 1. The argument goes parallel. 7
(4) For a sufficiently small ε > 0, B ε ( x) ∩ Γ consists of a single curve segment for any x ∈ Γ. We remark that the condition is satisfied if ε is smaller than the thickness of Γ (see, for exmaple, [LSDR] for the definition of thickness of a knot). Let x ε ,+ and x ε ,− be the endpoints of the curve B ε ( x) ∩ Γ, where we assume x ε ,+ is a little bit ahead of x with respect to the orientation of Γ. Then the boundary of Γ × Γ \ ∆ ε consists of two disjoint curves on Γ × Γ given by Γ ε ,± := {(x, x ε ,± ) : x ∈ Γ}. Suppose Γ ε ,± are endowed with the same orientation as Γ. Then the boundary of Γ × Γ \ ∆ ε is given by ∂ (Γ × Γ \ ∆ ε ) = Γ ε ,+ ∪ (−Γ ε ,− ) , where −Γ ε ,− is endowed with the reverse orientation. Since on Γ ε ,+ and Γ ε ,− ω 1 = dx 1 · x 2 − x 1 |x 2 − x 1 | = 1 + O(ε 2 ) dx 1 , we have ( Γ×Γ)\∆ ε (ˆr 1 2 · dx 1 ) (ˆr 1 2 · dx 2 ) |x 1 − x 2 | − dx 1 · dx 2 |x 1 − x 2 | = ( Γ×Γ)\∆ ε dω 1 = Γ ε ,+ ∪ (−Γ ε ,− ) ω 1 = 2L + O(ε 2 ). □ We close this subsection with giving a property and an example of the self-inductance.
Proposition 2.6 The regularized self-inductance behaves under homothety as follows.
H N (λΓ) = λH N (Γ) + µ 0 L 2π · λ (log λ) (λ > 0). This is immediate from the definition (2.1). Example 2.1 The regularized self-inductance of a unit circle Γ ◦ is given by H N (Γ ◦ ) = 2π lim ε→0 + µ 0 4π 2π−ε ε cos θ 2 sin θ 2 dθ + µ 0 2π log ε = (−1 + log 2) µ 0 π .
2.2 Regularization with parallel loops Theorem 2.7 Let Γ be a smooth 4 simple loop with non-vanishing curvature. Let Γ δ be a δ- parellel curve given by Γ δ = {x + δn(x) : x ∈ Γ}, where n is the unit principal normal vector
parellel curve given by Γ δ = {x + δn(x) : x ∈ Γ}, where n is the unit principal normal vector to Γ. Then lim δ→0 + L N (Γ, Γ δ ) + µ 0 L 2π log δ = H N (Γ) + (log 2) µ 0 L 2π . (2.17) 4 We assume the smoothness for the sake of simplicity. In this case we need C 4 . 8
We remark that if δ is smaller than the thickness of Γ then Γ δ ∩ Γ = ∅. Proof. We drop off the coefficient µ 0 / (4π) in the proof to make formulae shorter and simpler. The proof of Proposition 4.18 of [OS1] goes parallel with slight modification. We estimate, fixing x 1 on Γ, Γ δ ∩B ε ( x 1 ) t x 1 · t x 2 |x 1 − x 2 | dx 2 (2.18) for 0 < δ ≪ ε ≪ 1. Suppose Γ is parametrized by the arc-length as Γ = {γ(s)} 0 ≤s≤L . Then Γ δ can be expressed as Γ δ = {γ δ (s)} 0 ≤s≤L , where γ δ (s) = γ(s)+δκ(s) − 1 γ ′′ (s). Let x 1 = γ(s 1 ) and κ be the curvature of Γ at x 1 . We have γ ′ (s 1 ) · γ ′ δ (s 1 + s) = (1 − κδ) + O(1)δs + O(s 2 ), |γ δ (s 1 + s) − γ(s 1 ) | 2 = δ 2 + (1 − κδ)s 2 1 + O(1)s 2 . Let s 1 ± s ± be the values of parameter when Γ δ passes through the sphere ∂B ε ( x 1 ) . Then s ± = ± ε 2 − δ 2 1 − κδ + O(ε 3 ). The above equalities imply that (2.18) can be estimated by ε 2 − δ 2 1−κδ +O(ε 3 ) − ε 2 − δ 2 1−κδ +O(ε 3 ) ( 1 − κδ) + O(1)δs + O(1)s 2 1 + O(1)s 2 δ 2 + (1 − κδ)s 2 ds =2 ε 2 − δ 2 1−κδ 0 1 − κδ δ 2 + (1 − κδ)s 2 ds + O(ε 2 ) + o(δ) = 2 √ 1 − κδ log ε + √ ε 2 − δ 2 δ + O(ε 2 ) + o(δ) = 2 log 2 + 2 log ε − 2 log δ + O(ε). Snce δ ≪ ε, it follows that Γ Γ δ dx 1 · dx 2 |x 1 − x 2 | = Γ Γ δ ∩B ε ( x 1 ) t x 1 · t x 2 |x 1 − x 2 | dx 2 + Γ δ \B ε ( x 1 ) t x 1 · t x 2 |x 1 − x 2 | dx 2 dx 1 = Γ 2 log 2 + 2 log ε − 2 log δ + Γ \B ε ( x 1 ) t x 1 · t x 2 |x 1 − x 2 | dx 2 dx 1 + O(ε) = ( Γ×Γ)\∆ ε dx 1 · dx 2 |x 1 − x 2 | + 2L log ε + 2(log 2)L − 2L log δ + O(ε), which means Γ Γ δ dx 1 · dx 2 |x 1 − x 2 | + 2L log δ − 2(log 2)L = ( Γ×Γ)\∆ ε dx 1 · dx 2 |x 1 − x 2 | + 2L log ε + O(ε). Taking the limit as ε goes down to 0 and multiplying the both sides by µ 0 / (4π), we obtain (2.17). ✷ 9
2.3 Hadamard regularization in terms of the arc-length parameter
We introduce formulae of regularized self-inductance obtained by Hadamard regularization in terms of the arc-length parameter. They are more suitable for numerical experiments, and one of them will be used in the next subsection. In (2.1) and (2.2) we used the chord-length i.e. the distance in R 3 to define an “ε-neighbourhood ” ∆ ε of the diagonal set ∆ of Γ × Γ, which we use in Hadamard regularization. However, since it is easier to express a curve by the arc-length than by the chord-length, it is useful to have the formulae of Hadamard regularization in terms of the arc-length parametrization. In this setting, ∆ ε should be replaced by ∆ ε := {(x 1 , x 2 ) ∈ Γ × Γ : d Γ ( x 1 , x 2 ) ≤ ε}, where d Γ is the arc-length along Γ. Nevertheless, the counter terms and Hadamard’s finite parts do not change, which follows from the equalities (2.5) – (2.8), namely, H N (Γ) = lim ε→0 + µ 0 4π ( Γ×Γ)\∆ ε dx 1 · dx 2 |x 1 − x 2 | + µ 0 L 2π log ε , H W (Γ) = lim ε→0 + µ 0 4π ( Γ×Γ)\∆ ε (ˆr 1 2 · dx 1 ) (ˆr 1 2 · dx 2 ) |x 1 − x 2 | + µ 0 L 2π log ε . We remark that such a phenomenon can be observed when the power of denominator in the integrand is smaller than 3, as was explained for a similar functional in Remark 2.2.1 of [O]. Next, to have faster convergence in numerical experments, it is better to increase the number of counter terms. The series expansion in ε, after dropping off the constant µ 0 / (4π), is given by ( Γ×Γ)\∆ ε dx 1 · dx 2 |x 1 − x 2 | = 2L log 1 ε + 11 24 Γ κ(x) 2 dx ε 2 + O(ε 4 ), (2.19) ( Γ×Γ)\∆ ε (ˆr 1 2 · dx 1 ) (ˆr 1 2 · dx 2 ) |x 1 − x 2 | = 2L log 1 ε + 5 24 Γ κ(x) 2 dx ε 2 + O(ε 4 ). Let us show that (2.19) follows from (2.5) and (2.7). Fix a point x 1 . For a small positive number b and ε w ith 0 < ε < b, ε ≤d Γ ( x 1 , x 2 ) ≤b t x 1 · t x 2 |x 1 − x 2 | dx 2 = b ε 1 − κ 2 2 s 2 − κκ ′ 2 s 3 + O(s 4 ) s 1 − κ 2 24 s 2 − κκ ′ 24 s 3 + O(s 4 ) − 1 ds + −ε − b 1 − κ 2 2 s 2 − κκ ′ 2 s 3 + O(s 4 ) − s 1 − κ 2 24 s 2 − κκ ′ 24 s 3 + O(s 4 ) − 1 ds = O(1) − 2 log ε + 11 24 κ 2 · ε 2 + O(ε 4 ).
2.4 The self-inductance of solenoids
Let us consider a solenoid Γ n = Γ r ,ℓ,n that winds a cylinder of radius r and length ℓ with a number of turns per unit length n carrying a current I. Define the regularized self-inductance of Γ in the sense of Neumann and of Weber by (2.1) and (2.2). 10
Theorem 2.8 The asymptotic of the regularized self-inductances as n goes to infinity is given by
by lim n→∞ H N (Γ r ,ℓ,n ) n 2 = lim n→∞ H W (Γ r ,ℓ,n ) n 2 = 8µ 0 3 − r 3 + 1 8 − ℓ ℓ 2 − 4r 2 E − 4r 2 ℓ 2 + ℓ ℓ 2 + 4r 2 K − 4r 2 ℓ 2 , (2.20) where K(k) and E(k) are the complete elliptic integrals of the first and second kind respectively:
E(k) = π 2 0 1 − k sin 2 (t) dt, K(k) = π 2 0 1 1 − k sin 2 (t) dt. Let (2.20) be denoted by L r ,ℓ . The asymptotic of L r ,ℓ as ℓ go es to infinity is given by
r ,ℓ r ,ℓ lim ℓ→∞ L r ,ℓ − µ 0 π r 2 ℓ − 8 3 r 3 = 0. (2.21) Thus the asymptotics of regularized self-inductance as n and ℓ go to infinity is given by H N (Γ r ,ℓ,n ) ∼ H W (Γ r ,ℓ,n ) ∼ µ 0 π r 2 n 2 ℓ (n, ℓ → ∞), as is expected. Remark 2.9 The right hand side of (2.20) multiplied by n 2 has already appeared in [BAB] and [DO] (and in [L] according to [BAB]) as the limit of the mutual inductance of coaxial solenoids as they overlap. Proof. (1) The first equality of (2.20) follows from Theorem 2.1 (4) since L = ℓ ( 2πnr) 2 + 1 = O(n) . (2) The second equality of (2.20) can be proved as follows. Let M = M r ,ℓ be a cylinder with radius r and length l parametrized by cylindrical coordinates; p : [0, 2πr] × [−ℓ/2, ℓ/2] ∋ (t, z) → (r cos(t/r), r sin(t/r), z). Let v x be the unit tangent vector to the “meridean circle” of M r ,ℓ through x which is given by v x = p t = (∂p/∂t). We show lim n→∞ H N (Γ n ) n 2 = µ 0 4π M ×M v x 1 · v x 2 |x 1 − x 2 | dx 1 d x 2 (2.22) by the following steps. Let ε 0 be any posotive number. (i) For any positive number δ there exists a natural number n 0 such that for nay point x in M there holds ( Γ n × Γ n ) \N δ ( x ) dx 1 · dx 2 |x 1 − x 2 | − ( M ×M )\N δ ( x ) v x 1 · v x 2 |x 1 − x 2 | dx 1 d x 2 < ε 0 , where N δ (x) is a “curved square neighbourhood” of x = p(t, z) in M given by N δ (x) = p ((t − δ, t + δ) × ((z − δ, z + δ) ∩ [−ℓ/2, ℓ/2])) , 11
where we assume that the coordinate θ is considered modulo 2π. (ii) There exists a positive number δ 2 such that if 0 < δ ≤ δ 2 then for any x in M , ( M ×M ) ∩N δ ( x ) v x 1 · v x 2 |x 1 − x 2 | dx 1 d x 2 < ε 0 . (iii) There is a natural number n 1 such tha if n ≥ n 1 then for any positive number δ with 0 < δ ≤ δ 2 , 1 n lim ε→0 + [ −δ,−ε]∪[ε,δ] p t ( t, 0) · p t ( 0, 0) |p(t, 0) − p(0, 0)| dt + 2 log ε + 2 ⌊δ/n⌋ i=1 δ − δ p t ( t, i/n) · p t ( 0, 0) |p(t, i/n) − p(0, 0)| dt < 2ε 0 , where ⌊u⌋ is the floor function that gives the greatest integer less than or equal to u. (iv) There is a natural number n 2 with n 2 ≥ n 1 such that for any positive number δ with 0 < δ ≤ δ 2 and for any point x in M , 1 n lim ε→0 + ( Γ n ∩ N δ ( x ) )\∆ ε t x · t x 2 |x − x 2 | dx 2 + 2 log ε < 3ε 0 . Since the right hand side of (2.22) is given by µ 0 4π 2π 0 ℓ 0 2π 0 ℓ 0 cos(θ 1 − θ 2 ) 4 r 2 sin 2 θ 1 − θ 2 2 + (z 1 − z 2 ) 2 r 2 dz 1 d θ 1 d z 2 θ 2 = 2µ 0 r 3 π 2 0 ℓ r 0 ℓ r 0 1 − 2 sin 2 θ 4 sin 2 θ + (z 1 − z 2 ) 2 dz 1 d z 2 d θ. Some computation shows that it is equal to the right hand side of (2.20). (3) The equality (2.21) follows from the following expansion; 1 3 − ℓ ℓ 2 − 4 E − 4 ℓ 2 + ℓ ℓ 2 + 4 K − 4 ℓ 2 = πℓ + π 2 · 1 ℓ + O 1 ℓ 3 . ✷
3 Appendix
3.1 The second residues of F N and F W
We can proceed the analysis of F N (z) and F W a bit more.
Proposition 3.1 The second residues of F N and F W defined by (2.3) and (2.4) are given by
n 3.1 The second residues of F N and F W defined by (2.3) and (2.4) are Res(F N , −3) = − 3µ 0 16π Γ κ(x) 2 dx, Res(F W , −3) = − µ 0 16π Γ κ(x) 2 dx,
where κ denotes the curvature.
Proof. The first equality follows from (2.9) and (2.15). The second equality can be proved similarly using (2.8). 12
✷ 3.2 When the power of the denominator is 2
If we change the power of the denominator of (1.1) from 1 to 2, we obtain “average linking with random circles” ([OS1]). Let S(1, 3) be the set of oriented circles in R 3 , dC a measure on S(1, 3) which is invariant under M¨obius transformations, and lk(C, Γ i ) the linking number of Γ and an oriented circle C. Then we have Γ 1 Γ 2 dx 1 · dx 2 |x 1 − x 2 | 2 =2 Γ 1 Γ 2 (ˆr 1 2 · dx 1 ) (ˆr 1 2 · dx 2 ) |x 1 − x 2 | 2 = 3π 4 S (1,3) lk(C, Γ 1 ) · lk(C, Γ 2 ) dC. When Γ 1 and Γ 2 coincide, the above integrals blow up, and they can be regularized as lim ε→0 + ( Γ×Γ)\∆ ε dx 1 · dx 2 |x 1 − x 2 | 2 − 2L ε = lim ε→0 + 3π 4 S ε ( 1,3) lk(C, Γ) 2 dC − 3πL 2ε , where S ε ( 1, 3) is the set of oriented circles with radius r ≥ ε. The reader is referred to [OS1] for the details.
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