# 210. Two-Charge Interaction as Cross-Stress Transfer This appendix derives the interaction of two charged modes directly from the same compact toroidal closure used in appendix 209. The static interaction is first obtained as the exact compact-limit cross energy of two bounded toroidal closures. The Coulomb law then follows by taking the gradient of that interaction potential. The moving interaction is obtained by applying the compact-limit Lorentz theorem of appendix 209 to each mode in the field generated by the other. No effective charge density or current density is introduced anywhere in the derivation. ## 210.1 Two Compact Toroidal Charged Modes Let $$ K_{1,\varepsilon_1}, \qquad K_{2,\varepsilon_2} $$ be two disjoint coherent toroidal charged modes of sizes $$ \varepsilon_1,\qquad \varepsilon_2, $$ centered at $$ \mathbf X_1,\qquad \mathbf X_2. $$ Write their signed through-hole flux classes as $$ q_1,\qquad q_2. $$ Assume the two compact modes are separated by a distance $$ d:=|\mathbf X_1-\mathbf X_2| $$ with $$ d\gg \varepsilon_1+\varepsilon_2. $$ In the compact toroidal limit of chapter 10 and appendix 209, the exterior field of each mode has the asymptotic form $$ \mathbf E_a(\mathbf r) = \frac{q_a}{4\pi\varepsilon_0} \frac{\mathbf r-\mathbf X_a}{|\mathbf r-\mathbf X_a|^3} + \mathbf e_{a,\mathrm{rem}}(\mathbf r), $$ $$ \mathbf B_a(\mathbf r) = \mathbf b_{a,\mathrm{rem}}(\mathbf r), $$ with bounds $$ |\mathbf e_{a,\mathrm{rem}}(\mathbf r)| \le C_a\frac{\varepsilon_a}{|\mathbf r-\mathbf X_a|^3}, $$ $$ |\mathbf b_{a,\mathrm{rem}}(\mathbf r)| \le C'_a\frac{\varepsilon_a}{|\mathbf r-\mathbf X_a|^3}, $$ for every point outside the toroidal core. At static leading order, the magnetic terms are higher multipoles and the interaction is governed by the electric cross energy. ## 210.2 Exact Static Cross-Energy in the Compact Limit Work first in a frame in which the two toroidal centers are instantaneously at rest and retain only the static leading interaction. Outside the two toroidal cores the fields are source-free and static, so $$ \nabla\times\mathbf E_a=0, \qquad \nabla\cdot\mathbf E_a=0. $$ Therefore there exist harmonic exterior potentials $$ \phi_1,\qquad \phi_2 $$ such that $$ \mathbf E_a=-\nabla\phi_a $$ outside the cores, with asymptotics $$ \phi_a(\mathbf r) = \frac{q_a}{4\pi\varepsilon_0|\mathbf r-\mathbf X_a|} + \phi_{a,\mathrm{rem}}(\mathbf r), $$ $$ |\phi_{a,\mathrm{rem}}(\mathbf r)| \le C''_a\frac{\varepsilon_a}{|\mathbf r-\mathbf X_a|^2}. $$ Fix radii $$ \rho_1,\rho_2,R $$ such that $$ 2\varepsilon_1<\rho_1\ll d, \qquad 2\varepsilon_2<\rho_2\ll d, \qquad R\gg d. $$ Let $$ \Omega_{R,\rho_1,\rho_2} := B_R(0)\setminus \bigl(B_{\rho_1}(\mathbf X_1)\cup B_{\rho_2}(\mathbf X_2)\bigr), $$ where $$ B_\rho(\mathbf X):=\{\,\mathbf r:|\mathbf r-\mathbf X|<\rho\,\}. $$ Define the static cross energy by $$ U_\times(R,\rho_1,\rho_2) := \varepsilon_0 \int_{\Omega_{R,\rho_1,\rho_2}} \mathbf E_1\cdot\mathbf E_2\,dV. $$ Since $$ \mathbf E_a=-\nabla\phi_a, $$ this becomes $$ U_\times(R,\rho_1,\rho_2) = \varepsilon_0 \int_{\Omega_{R,\rho_1,\rho_2}} \nabla\phi_1\cdot\nabla\phi_2\,dV. $$ Because $$ \Delta\phi_2=0 $$ throughout $\Omega_{R,\rho_1,\rho_2}$, Green's identity gives $$ U_\times(R,\rho_1,\rho_2) = \varepsilon_0 \int_{\partial\Omega_{R,\rho_1,\rho_2}} \phi_1\,\partial_n\phi_2\,dA. $$ We now evaluate the three boundary pieces. ### 210.2.1 Outer Boundary On the outer sphere $|\mathbf r|=R$, $$ \phi_1=O(R^{-1}), \qquad \partial_n\phi_2=O(R^{-2}), $$ so $$ \int_{|\mathbf r|=R}\phi_1\,\partial_n\phi_2\,dA = O(R^{-1}). $$ ### 210.2.2 Inner Boundary Around $\mathbf X_1$ Near $\mathbf X_1$, the potential $\phi_2$ is smooth because the second mode is disjoint from the first. Hence $$ \partial_n\phi_2=O(1) $$ on $|\mathbf r-\mathbf X_1|=\rho_1$, while $$ \phi_1=O(\rho_1^{-1}). $$ Therefore $$ \int_{|\mathbf r-\mathbf X_1|=\rho_1}\phi_1\,\partial_n\phi_2\,dA = O(\rho_1). $$ ### 210.2.3 Inner Boundary Around $\mathbf X_2$ On the sphere $$ S_{2,\rho_2}:=\{\,\mathbf r:|\mathbf r-\mathbf X_2|=\rho_2\,\}, $$ write $$ \mathbf r-\mathbf X_2=\rho_2\,\mathbf n_2, \qquad |\mathbf n_2|=1. $$ The outward normal of the punctured domain $\Omega_{R,\rho_1,\rho_2}$ on this inner boundary is $$ \mathbf n=-\mathbf n_2. $$ Hence $$ \partial_n\phi_2 = \frac{q_2}{4\pi\varepsilon_0\rho_2^2} + O\!\left(\frac{\varepsilon_2}{\rho_2^3}\right). $$ Since $\phi_1$ is smooth near $\mathbf X_2$, $$ \phi_1(\mathbf X_2+\rho_2\mathbf n_2) = \phi_1(\mathbf X_2)+O(\rho_2). $$ Therefore $$ \varepsilon_0 \int_{S_{2,\rho_2}} \phi_1\,\partial_n\phi_2\,dA = q_2\,\phi_1(\mathbf X_2) + O(\rho_2) + O\!\left(\frac{\varepsilon_2}{\rho_2}\right). $$ Collecting the three boundary pieces, we obtain $$ U_\times(R,\rho_1,\rho_2) = q_2\,\phi_1(\mathbf X_2) + O(R^{-1}) + O(\rho_1) + O(\rho_2) + O\!\left(\frac{\varepsilon_2}{\rho_2}\right). $$ Now let $$ R\to\infty, \qquad \rho_1\to 0, \qquad \rho_2\to 0, \qquad \frac{\varepsilon_2}{\rho_2}\to 0. $$ Then $$ \boxed{ U_\times = q_2\,\phi_1(\mathbf X_2) }. $$ By symmetry the same argument also gives $$ \boxed{ U_\times = q_1\,\phi_2(\mathbf X_1) }. $$ Using the compact monopole asymptotic of $\phi_1$ at $\mathbf X_2$, $$ \phi_1(\mathbf X_2) = \frac{q_1}{4\pi\varepsilon_0 d} + O\!\left(\frac{\varepsilon_1}{d^2}\right), $$ the compact-limit interaction potential is $$ \boxed{ U_\times = \frac{q_1q_2}{4\pi\varepsilon_0 d} }. $$ So the Coulomb potential is the exact compact-limit cross energy of two toroidal charged closures. ## 210.3 Exact Static Force from the Interaction Potential The force on the first toroidal mode is the negative gradient of the interaction energy with respect to its center: $$ \mathbf F_{1\leftarrow 2} := -\nabla_{\mathbf X_1}U_\times. $$ Using $$ U_\times=q_1\,\phi_2(\mathbf X_1), $$ we obtain $$ \mathbf F_{1\leftarrow 2} = -q_1\nabla\phi_2(\mathbf X_1) = q_1\,\mathbf E_2(\mathbf X_1). $$ Similarly, $$ \mathbf F_{2\leftarrow 1} = q_2\,\mathbf E_1(\mathbf X_2). $$ In the compact monopole limit, $$ \boxed{ \mathbf F_{1\leftarrow 2} = \frac{q_1q_2}{4\pi\varepsilon_0} \frac{\mathbf X_1-\mathbf X_2}{|\mathbf X_1-\mathbf X_2|^3} }, $$ $$ \boxed{ \mathbf F_{2\leftarrow 1} = \frac{q_1q_2}{4\pi\varepsilon_0} \frac{\mathbf X_2-\mathbf X_1}{|\mathbf X_2-\mathbf X_1|^3} = -\mathbf F_{1\leftarrow 2} }. $$ So the static Coulomb law is recovered exactly as the gradient of the compact toroidal cross-energy potential. ## 210.4 Moving Compact Modes Now allow the two compact toroidal modes to move. Appendix 209 already proved that a compact toroidal charged mode in a smooth external Maxwell field satisfies $$ \frac{d\mathbf p}{dt} = q\bigl(\mathbf E+\mathbf v\times\mathbf B\bigr) $$ exactly in the compact limit. Apply that theorem to each mode in the field generated by the other. Then $$ \boxed{ \mathbf F_{1\leftarrow 2} = q_1\bigl( \mathbf E_2(\mathbf X_1,t) + \mathbf v_1\times\mathbf B_2(\mathbf X_1,t) \bigr) }, $$ $$ \boxed{ \mathbf F_{2\leftarrow 1} = q_2\bigl( \mathbf E_1(\mathbf X_2,t) + \mathbf v_2\times\mathbf B_1(\mathbf X_2,t) \bigr) }. $$ So the moving two-torus interaction is not a new law layered on top of the static Coulomb potential. It is the compact-limit same-field interaction of each toroidal closure with the propagated field of the other. ## 210.5 Interpretation The result can now be read in the same monist way as appendix 209. - each charged body is a bounded toroidal closure, - its signed through-hole flux class appears externally as the monopole coefficient $q$, - the static interaction is the cross energy of the two closures, - the force is the gradient of that cross energy, - the moving interaction is the same compact toroidal coupling carried through the transported Maxwell field. So neither Coulomb nor Lorentz interaction is primitive action-at-a-distance. Both are compact-limit expressions of energy transfer and momentum transfer in one common electromagnetic substrate. ## 210.6 Summary For two compact toroidal charged modes, the static cross energy is $$ U_\times = \varepsilon_0\int \mathbf E_1\cdot\mathbf E_2\,dV. $$ In the compact limit this is exactly $$ U_\times = \frac{q_1q_2}{4\pi\varepsilon_0|\mathbf X_1-\mathbf X_2|}. $$ Taking the gradient gives the exact static interaction $$ \mathbf F_{1\leftarrow 2} = \frac{q_1q_2}{4\pi\varepsilon_0} \frac{\mathbf X_1-\mathbf X_2}{|\mathbf X_1-\mathbf X_2|^3}. $$ For moving compact modes, appendix 209 then gives $$ \mathbf F_{1\leftarrow 2} = q_1\bigl( \mathbf E_2(\mathbf X_1,t) + \mathbf v_1\times\mathbf B_2(\mathbf X_1,t) \bigr), $$ and similarly for mode 2. Thus the two-charge interaction is derived here directly from compact toroidal closure, cross energy, and transported same-field stress.