Electrostatic multipoles

For the Lebesgue branches (a) and (b) assume Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), inherited from the exact measure interface (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)); the finite sums in branch (c) are defined and computed without Choice. Let the laboratory region, its chart, its spatial Euclidean structure and its units be those of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields and Framework-local SI units, the vacuum constants and the exact/measured status of their values, and let ${\delta }_{ij}$ be the Kronecker symbol in the $\delta$-orthonormal basis $(e_1,e_2,e_3)$ of the frame, with $\mathrm{\mid }y\mathrm{\mid }$ the Euclidean norm of The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$. This definition assigns source moments to a static charge distribution in the fixed-time reading of The static and pure electrostatic regimes and their fixed-time reading (d) and (e). Three branches are declared.

(a) [Absolutely continuous branch] Let $a>0$ and let $\rho \in L^1({\mathbb{R}}^3)$ be real (The class $L^1(\mu )$ of integrable functions) and vanish ${\lambda }_3$-almost everywhere outside the closed ball ${\bar{B}}_a(0)=\{y:\mathrm{\mid }y\mathrm{\mid }\le a\}$ (Open ball, closed ball and sphere in a metric space, Measure-null sets and almost-everywhere statements relative to a measure). The total charge $Q$, the dipole moment $p$, the degree-$l$ Cartesian moments $M_{i_1\dots i_l}$ and the traceless quadrupole $Q_{ij}$ are

$Q:={\int }_{{\mathbb{R}}^3}\rho d{\lambda }_3,p_i:={\int }_{{\mathbb{R}}^3}{y}_i\rho (y)d{\lambda }_3(y),M_{i_1\dots i_l}:={\int }_{{\mathbb{R}}^3}{y}_{i_1}\cdots y_{i_l}\rho (y)d{\lambda }_3(y)(l\ge 1),$

$M^{(0)}:=Q,M_i:=p_i,Q_{ij}:={\int }_{{\mathbb{R}}^3}(3y_i{y}_j-\mathrm{\mid }y{\mathrm{\mid }}^2{\delta }_{ij})\rho (y)d{\lambda }_3(y),$

with $l\ge 1$ and indices in $\{1,2,3\}$; units are C for $Q$, C m for $p$, C m${}^l$ for the degree-$l$ Cartesian moment and C m${}^2$ for $Q_{ij}$, and all integrals are in the sense of Integral over a measurable subset and Integrable real and complex functions, and their integrals. Each integral is absolutely convergent: since $\rho =0$ ${\lambda }_3$-a.e. off ${\bar{B}}_a(0)$, and the maps $y\mapsto y_{i_1}\cdots y_{i_l}$ and $y\mapsto 3y_i{y}_j-\mathrm{\mid }y{\mathrm{\mid }}^2{\delta }_{ij}$ are bounded ${\lambda }_3$-a.e. on that ball by $a^l$ and by $4a^2$ respectively, the integrands are dominated ${\lambda }_3$-a.e. by $a^l\mathrm{\mid }\rho \mathrm{\mid }$ and $4a^2\mathrm{\mid }\rho \mathrm{\mid }$, which lie in $L^1$ (The modulus of an integral is bounded by the integral of the modulus, Monotonicity and nonnegative homogeneity of the nonnegative integral). The traceless quadrupole is traceless,

${\sum }_{i=1}^3{Q}_{ii}=0,$

because ${\sum }_{i=1}^3(3y_i^2-\mathrm{\mid }y{\mathrm{\mid }}^2)=0$ pointwise, and summing the three integrable functions $3y_i^2\rho -\mathrm{\mid }y{\mathrm{\mid }}^2\rho$ and applying finite linearity of the integral on $L^1$ (The Lebesgue integral is linear on $L^1(\mu )$) evaluates the sum as the integral of the pointwise-zero function; replacing $\rho$ by an equal ${\lambda }_3$-a.e. density changes no moment (Two integrable functions are equal almost everywhere exactly when all of their indefinite integrals agree).

(b) [General locally integrable branch] If instead $\rho \in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1({\mathbb{R}}^3)$ is real and ${\int }_{{\mathbb{R}}^3}\mathrm{\mid }y{\mathrm{\mid }}^l\mathrm{\mid }\rho (y)\mathrm{\mid }d{\lambda }_3(y)<\mathrm{\infty }$ for a given integer $l\ge 0$, then the degree-$l$ Cartesian moments of the same formula are absolutely convergent and defined; if the absolute weighted moment ${\int }_{{\mathbb{R}}^3}\mathrm{\mid }y{\mathrm{\mid }}^L\mathrm{\mid }\rho (y)\mathrm{\mid }d{\lambda }_3(y)$ is finite for some real $L\ge 0$, then every integer Cartesian degree $0\le l\le L$ is defined, and in particular $Q=\int \rho d{\lambda }_3$ is absolutely convergent, so that $\rho \in L^1({\mathbb{R}}^3)$ is a conclusion of this branch and not an added hypothesis. Here $\rho$ is restricted to the closed unit ball, where it is integrable by local integrability, and outside it $\mathrm{\mid }y{\mathrm{\mid }}^l\le \mathrm{\mid }y{\mathrm{\mid }}^L$; splitting the nonnegative integral over these two measurable sets and applying monotonicity and additivity of the integral gives $\int \mathrm{\mid }y{\mathrm{\mid }}^l\mathrm{\mid }\rho \mathrm{\mid }\le {\int }_{{\bar{B}}_1(0)}\mathrm{\mid }\rho \mathrm{\mid }+{\int }_{\{\mathrm{\mid }y\mathrm{\mid }>1\}}\mathrm{\mid }y{\mathrm{\mid }}^L\mathrm{\mid }\rho \mathrm{\mid }<\mathrm{\infty }$ (Monotonicity and nonnegative homogeneity of the nonnegative integral, Open ball, closed ball and sphere in a metric space). The convention $\mathrm{\mid }y{\mathrm{\mid }}^0=1$ includes the origin, no compact support and no $L^1$-membership is assumed in this branch, and Cartesian monomials of degree $l$ have modulus at most $\mathrm{\mid }y{\mathrm{\mid }}^l$, so all lower moments, including $Q$, exist together with the degree-$l$ moment.

(c) [Finite system of stationary point charges] Let $N\ge 0$, let $x_1,\dots ,x_N\in {\mathbb{R}}^3$ and let $q_1,\dots ,q_N\in \mathbb{R}$ be charge parameters (Charge parameter, frame charge and current densities, volume charge and flux, and the four-current (a)), read as the stationary point charges $q_k{\delta }_{x_k}$ in the fixed-time reading of The static and pure electrostatic regimes and their fixed-time reading (e); for $N=0$ the lists are empty and every sum below is the empty sum $0$. Define

$Q:={\sum }_{k=1}^N{q}_k,p:={\sum }_{k=1}^N{q}_k{x}_k,M_{i_1\dots i_l}:={\sum }_{k=1}^N{q}_k{x}_{k,i_1}\cdots x_{k,i_l}(l\ge 1),M^{(0)}:=Q,$

$Q_{ij}:={\sum }_{k=1}^N{q}_k(3x_{k,i}{x}_{k,j}-\mathrm{\mid }{x}_k{\mathrm{\mid }}^2{\delta }_{ij}),$

finite sums in the sense of Finite sums and finite products, by recursion (Laws of finite sums and finite products); again

${\sum }_{i=1}^3{Q}_{ii}=0,$

by finite distributivity. The points need not be distinct; charges at a common point add in every sum.

(d) [Reference point and scope] Every moment of (a)-(c) is computed about the chart origin of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields; this origin is part of the definition. These are source moments of a prescribed static distribution, not momenta, rest masses, intrinsic spins, self-energies or dynamical quantities, and no field, force, equation of motion, equilibrium or empirical claim is made here; the static regime in which the source is read is the adopted idealisation of The static and pure electrostatic regimes and their fixed-time reading (f). Every Cartesian degree $l$ in this definition is a nonnegative integer (the number of Cartesian indices); only the absolute weighted exponent $L$ in (b) may be any nonnegative real number.

For the density-integral branch assume Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$) through the Lebesgue interface (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)); the finite-point branch of all identities below is choice-free. Let $a>0$ and let $\rho \in L^1({\mathbb{R}}^3)$ be real and vanish ${\lambda }_3$-a.e. outside ${\bar{B}}_a(0)$ (Open ball, closed ball and sphere in a metric space), or let $(x_k,q_k{)}_{k=1}^N$ be a finite system of stationary point charges, with the source moments of Total charge, dipole moment, Cartesian moments and traceless quadrupole of a static charge distribution branches (a) and (c) respectively. For $b\in {\mathbb{R}}^3$ define the moments about the reference point $b$ by replacing, in each integrand of Total charge, dipole moment, Cartesian moments and traceless quadrupole of a static charge distribution (a) and in each summand of (c), the vector $y$ by $y-b$ (respectively $x_k$ by $x_k-b$):

$Q^{(b)}:={\int }_{{\mathbb{R}}^3}\rho d{\lambda }_3,p^{(b)}:={\int }_{{\mathbb{R}}^3}(y-b)\rho (y)d{\lambda }_3(y),$

$M_{i_1\dots i_l}^{(b)}:={\int }_{{\mathbb{R}}^3}{\prod }_{r=1}^l(y-b{)}_{i_r}\rho (y)d{\lambda }_3(y)(l\ge 1),M^{(b,0)}:=Q^{(b)},$

$Q_{ij}^{(b)}:={\int }_{{\mathbb{R}}^3}(3(y-b{)}_i(y-b{)}_j-\mathrm{\mid }y-b{\mathrm{\mid }}^2{\delta }_{ij})\rho (y)d{\lambda }_3(y),$

and analogously with the finite sums $M_{i_1\dots i_l}^{(b)}={\sum }_{k=1}^N{q}_k{\prod }_{r=1}^l(x_k-b{)}_{i_r}$ $(l\ge 1)$ and $Q_{ij}^{(b)}={\sum }_{k=1}^N{q}_k(3(x_k-b{)}_i(x_k-b{)}_j-\mathrm{\mid }{x}_k-b{\mathrm{\mid }}^2{\delta }_{ij})$ in the point-charge branch; for $N=0$ the sums are the empty sum $0$. Every displayed integral is absolutely convergent: since $\rho =0$ ${\lambda }_3$-a.e. off ${\bar{B}}_a(0)$, the estimates $\mathrm{\mid }(y-b{)}_{i_1}\cdots (y-b{)}_{i_l}\mathrm{\mid }\le (\mathrm{\mid }y\mathrm{\mid }+\mathrm{\mid }b\mathrm{\mid }{)}^l\le (a+\mathrm{\mid }b\mathrm{\mid }{)}^l$ and $\mathrm{\mid }3(y-b{)}_i(y-b{)}_j-\mathrm{\mid }y-b{\mathrm{\mid }}^2{\delta }_{ij}\mathrm{\mid }\le 4(\mathrm{\mid }y\mathrm{\mid }+\mathrm{\mid }b\mathrm{\mid }{)}^2\le 4(a+\mathrm{\mid }b\mathrm{\mid }{)}^2$ hold ${\lambda }_3$-a.e. on the support, so each integrand is dominated ${\lambda }_3$-a.e. by a finite constant times $\mathrm{\mid }\rho \mathrm{\mid }\in L^1$ (Total charge, dipole moment, Cartesian moments and traceless quadrupole of a static charge distribution (a), The modulus of an integral is bounded by the integral of the modulus, Monotonicity and nonnegative homogeneity of the nonnegative integral, The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$). Then:

(a) $Q^{(b)}=Q$ for every $b$;

(b) $p^{(b)}=p-Qb$, so the dipole moment is independent of the reference point exactly when $Q=0$;

(c) for every $l\ge 0$,

$M_{i_1\dots i_l}^{(b)}={\sum }_{S\subseteq \{1,\dots ,l\}}(-1{)}^{\mathrm{\mid }S\mathrm{\mid }}({\prod }_{r\in S}{b}_{i_r})M_{S^c},$

where for a subset $S\subseteq \{1,\dots ,l\}$ we write $S^c:=\{1,\dots ,l\}\setminus S$ and $M_{S^c}$ for the Cartesian moment whose index list is $(i_r{)}_{r\in S^c}$ in increasing order of $r$, with the empty index list read as $M_{\mathrm{\varnothing }}:=Q$; in particular

$M_{ij}^{(b)}=M_{ij}-b_i{p}_j-b_j{p}_i+b_i{b}_{j}Q;$

(d) $Q_{ij}^{(b)}=Q_{ij}-3(b_i{p}_j+b_j{p}_i)+2(b\cdot p){\delta }_{ij}+(3b_i{b}_j-\mathrm{\mid }b{\mathrm{\mid }}^2{\delta }_{ij})Q$; consequently $Q_{ij}^{(b)}=Q_{ij}$ for every $b$ and all $i,j$ if and only if $Q=0$ and $p=0$;

(e) [Scope] this is an exact identity between the declared source moments of Total charge, dipole moment, Cartesian moments and traceless quadrupole of a static charge distribution; it asserts nothing about fields, forces, dynamics or nature, and the reference point is a choice of coordinate origin, not an operation performed on the source.

Assume the Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), inherited from the cited Lebesgue and Newtonian interfaces (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Let $a>0$, let $\rho \in L^1({\mathbb{R}}^3)$ be real and vanish ${\lambda }_3$-a.e. outside the closed ball ${\bar{B}}_a(0)$ (Open ball, closed ball and sphere in a metric space, Measure-null sets and almost-everywhere statements relative to a measure), let $Q$ and $p$ be the total charge and dipole moment of $\rho$ about the chart origin (Total charge, dipole moment, Cartesian moments and traceless quadrupole of a static charge distribution (a)), let $\mathrm{\parallel }\rho {\mathrm{\parallel }}_1:={\int }_{{\mathbb{R}}^3}\mathrm{\mid }\rho \mathrm{\mid }d{\lambda }_3$, and let $\phi :={\epsilon }_0^{-1}N\rho$, where $N\rho$ is the Newtonian potential of Newtonian potential of compactly supported data and ${\epsilon }_0$ is the constant of Framework-local SI units, the vacuum constants and the exact/measured status of their values. Then:

(a) [Uniform monopole-dipole expansion] for every $x$ with $r:=\mathrm{\mid }x\mathrm{\mid }\ge 2a$ the integral ${\int }_{{\mathbb{R}}^3}\rho (y)\mathrm{\mid }x-y{\mathrm{\mid }}^{-1}d{\lambda }_3(y)$ is absolutely convergent and equals $4\pi N\rho (x)$, and

$\phi (x)=\frac{Q}{4\pi {\epsilon }_{0}r}+\frac{p\cdot x}{4\pi {\epsilon }_0{r}^3}+R(x),\mathrm{\mid }R(x)\mathrm{\mid }\le \frac{16a^2\mathrm{\parallel }\rho {\mathrm{\parallel }}_1}{4\pi {\epsilon }_0{r}^3},$

the bound being uniform over the whole exterior region $\{x:\mathrm{\mid }x\mathrm{\mid }\ge 2a\}$;

(b) [Leading-term form] consequently, for $r\ge 2a$,

$\mid \phi (x)-\frac{Q}{4\pi {\epsilon }_{0}r}\mid \le \frac{\mathrm{\mid }p\mathrm{\mid }}{4\pi {\epsilon }_0{r}^2}+\frac{16a^2\mathrm{\parallel }\rho {\mathrm{\parallel }}_1}{4\pi {\epsilon }_0{r}^3},$

and if $Q=0$ and $p\mathrm{\ne }0$, then for every unit direction $n\in {\mathbb{R}}^3$, $\mathrm{\mid }n\mathrm{\mid }=1$, with $p\cdot n\mathrm{\ne }0$ and every $r\ge \max (2a,32a^2\mathrm{\parallel }\rho {\mathrm{\parallel }}_1\mathrm{/}\mathrm{\mid }p\cdot n\mathrm{\mid })$ the dipole term is the leading term in the sense that $\phi (rn)=\frac{p\cdot n}{4\pi {\epsilon }_0{r}^2}(1+\delta )$ with $\mathrm{\mid }\delta \mathrm{\mid }\le \frac{1}{2}$, so $\mathrm{\mid }\phi (rn)\mathrm{\mid }\ge \frac{\mathrm{\mid }p\cdot n\mathrm{\mid }}{8\pi {\epsilon }_0{r}^2}$ and the remainder bound decays one power of $r$ faster than the dipole term;

(c) [Physical reading] by The localized Newtonian potential and the ideal point-charge Coulomb field (c) the function $\phi$ is finite ${\lambda }_3$-a.e., locally integrable, of class $C^{\mathrm{\infty }}$ and harmonic off ${\bar{B}}_a(0)$, and satisfies $-\mathrm{\Delta }{T}_{\phi }=T_{\rho \mathrm{/}{\epsilon }_0}$ in ${\mathcal{D}}^{\mathrm{\prime }}({\mathbb{R}}^3)$; the displayed expansion is therefore the exterior expansion of the potential of the compactly supported source $\rho$ in the fixed-time reading of The static and pure electrostatic regimes and their fixed-time reading (e), and in the smooth branch (The localized Newtonian potential and the ideal point-charge Coulomb field (a)) the field $E:=-\mathrm{\nabla }\phi$ of The electric and magnetic fields as fixed-time spatial vector fields realises the pure electrostatic regime of The static and pure electrostatic regimes and their fixed-time reading (c) with source $\rho$;

(d) [Scope and exclusions] the hypotheses are a real compactly supported $L^1$ density, the region $r\ge 2a$, and the framework's constants; $\rho$ need not be continuous or bounded; the statement concerns $\phi$ itself, and no derivative estimate for $R$, no expansion of the field $E$, and no assertion on $\mathrm{\mid }x\mathrm{\mid }<2a$ or on the support of $\rho$ is claimed.

Assume the Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), inherited from the Lebesgue interface (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Let $a>0$ and let $\rho \in L^1({\mathbb{R}}^3)$ be real and vanish ${\lambda }_3$-a.e. outside ${\bar{B}}_a(0)$; let $E_{\mathrm{e}\mathrm{x}\mathrm{t}}:{\mathbb{R}}^3\to {\mathbb{R}}^3$ be a prescribed external vector-field datum, with electric-field units of Framework-local SI units, the vacuum constants and the exact/measured status of their values, that is continuous on ${\bar{B}}_a(0)$ and Lipschitz there with constant $L\ge 0$, i.e. $\mathrm{\mid }{E}_{\mathrm{e}\mathrm{x}\mathrm{t}}(x)-E_{\mathrm{e}\mathrm{x}\mathrm{t}}(x^{\mathrm{\prime }})\mathrm{\mid }\le L\mathrm{\mid }x-x^{\mathrm{\prime }}\mathrm{\mid }$ for all $x,x^{\mathrm{\prime }}\in {\bar{B}}_a(0)$. Consider the matter model in which the prescribed static distribution $\rho$ is carried by a body at rest in the laboratory frame and adopt additionally the prescribed external force model $f_{\mathrm{e}\mathrm{x}\mathrm{t}}=\rho E_{\mathrm{e}\mathrm{x}\mathrm{t}}$. This is a declared extension to these locally integrable charge data and Lipschitz fields, motivated by the smooth continuum coupling of The Lorentz coupling: external force on a test charge and the continuum force and power densities (ii); that postulate covers smooth sources and does not itself establish this extension. The only force counted by the present model is $f_{\mathrm{e}\mathrm{x}\mathrm{t}}$; the self-field of $\rho$, all other fields, mechanical support and back-reaction are excluded by this model. Define the total external force and the total external torque about the origin by the componentwise Lebesgue integrals

$F:={\int }_{{\mathbb{R}}^3}\rho (y)E_{\mathrm{e}\mathrm{x}\mathrm{t}}(y)d{\lambda }_3(y)\in {\mathbb{R}}^3,\tau :={\int }_{{\mathbb{R}}^3}\rho (y)(y\times E_{\mathrm{e}\mathrm{x}\mathrm{t}}(y))d{\lambda }_3(y)\in {\mathbb{R}}^3,$