# Appendix C: The Emergence of Force and Charge This appendix gives the compact derivation behind chapters 227 and 235. The guiding claim is: - charge is not a primitive source inserted into Maxwell theory, - force is not a primitive push acting on matter, - both are compact descriptions of interaction between bounded toroidal closures of one common electromagnetic field. ## 1. Charge as a Toroidal Flux Class Take a compact toroidal charged mode $$ K_\varepsilon $$ of size $\varepsilon$, with symmetry axis $\hat{\mathbf a}$. Choose a spanning surface $\Sigma$ across the torus aperture and define the signed through-hole flux $$ \Phi_\Sigma=\int_\Sigma \mathbf S\cdot d\mathbf A. $$ This quantity is not a source or sink. It is the oriented through-hole moment of the closed circulation. Its sign reverses with handedness. In the compact limit, the far field of the torus depends only on the signed class carried by this aperture flux. We write the resulting monopole coefficient as $$ q. $$ Externally, the torus then has the leading asymptotic form $$ \mathbf E_{\mathrm s}(\mathbf r) = \frac{q}{4\pi\varepsilon_0}\frac{\mathbf n}{R^2} + \mathbf e_{\mathrm{rem}}(\mathbf r), $$ $$ \mathbf B_{\mathrm s}(\mathbf r) = \mathbf b_{\mathrm{rem}}(\mathbf r), $$ where $$ \mathbf r=R\,\mathbf n, \qquad |\mathbf n|=1, $$ and the remainders decay at least one power faster than the monopole term. So `charge` is the compact external summary of a toroidal flux class, not a primitive source hidden in the middle of the field. ## 2. Lorentz Force from Boundary Stress Transfer Let the compact toroidal mode move in a smooth external Maxwell field $$ (\mathbf E_{\mathrm e},\mathbf B_{\mathrm e}). $$ The total field is source-free everywhere. The sphere used below is taken outside the toroidal core only so the compact exterior asymptotic can be used. The exact local momentum balance is $$ \partial_t g_i-\partial_jT_{ij}=0, $$ with $$ \mathbf g=\varepsilon_0\,\mathbf E\times\mathbf B, $$ and $$ T_{ij} = \varepsilon_0\left(E_iE_j-\frac12\delta_{ij}\mathbf E^2\right) + \frac{1}{\mu_0}\left(B_iB_j-\frac12\delta_{ij}\mathbf B^2\right). $$ Split the total field into self and external parts: $$ \mathbf E=\mathbf E_{\mathrm s}+\mathbf E_{\mathrm e}, \qquad \mathbf B=\mathbf B_{\mathrm s}+\mathbf B_{\mathrm e}. $$ The exact interaction across a sphere $S_R$ surrounding the compact torus is carried by the cross-stress tensor, that is, by momentum-flux transfer across the surrounding surface: $$ \mathbf F_R := \int_{S_R}\mathbf T_\times\cdot\mathbf n\,dA. $$ In the instantaneous rest frame of the torus, the compact-limit sphere integral gives $$ \boxed{ \mathbf F_{\mathrm{rest}}=q\,\mathbf E_{\mathrm e}(X) }. $$ This is the exact rest-frame force theorem. For a moving compact torus, the transported aperture samples the smooth external field through $$ \mathbf E_{\mathrm e}+\mathbf v\times\mathbf B_{\mathrm e}. $$ So the compact moving form is $$ \boxed{ \frac{d\mathbf p}{dt} = q(\mathbf E_{\mathrm e}+\mathbf v\times\mathbf B_{\mathrm e}) }. $$ So the Lorentz force law is the compact moving-aperture transport form of toroidal boundary stress transfer. ## 3. Two-Charge Interaction from Cross Energy Now take two well-separated compact toroidal charged modes, with flux classes $$ q_1,\qquad q_2, $$ centered at $$ \mathbf X_1,\qquad \mathbf X_2, $$ and separated by $$ d:=|\mathbf X_1-\mathbf X_2|. $$ At static leading order, the relevant interaction is the electric cross energy $$ U_\times = \varepsilon_0\int \mathbf E_1\cdot\mathbf E_2\,dV. $$ Using harmonic exterior potentials and Green's identity, the compact limit gives exactly $$ \boxed{ U_\times = \frac{q_1q_2}{4\pi\varepsilon_0d} }. $$ The force on the first torus is the gradient of this interaction energy: $$ \mathbf F_{1\leftarrow 2} = -\nabla_{\mathbf X_1}U_\times = \frac{q_1q_2}{4\pi\varepsilon_0} \frac{\mathbf X_1-\mathbf X_2}{|\mathbf X_1-\mathbf X_2|^3}. $$ This is exactly the Coulomb force. For moving compact modes, each torus obeys the Lorentz expression in the field generated by the other: $$ \mathbf F_{1\leftarrow 2} = q_1\bigl(\mathbf E_2(\mathbf X_1,t)+\mathbf v_1\times\mathbf B_2(\mathbf X_1,t)\bigr), $$ and similarly for mode 2. ## 4. Interpretation The logic is now clean. - A charged body is a bounded toroidal closure. - Its signed through-hole flux class appears externally as the scalar $q$. - Force is momentum transfer through the common field. - Coulomb interaction is the compact-limit cross energy of two closures. - Lorentz interaction is the compact moving-aperture transport form of that same stress transfer. So standard electrodynamics is not discarded. It is recovered as the compact mechanics of organized electromagnetic closures. ## 5. Summary Charge is the compact scalar summary of a toroidal through-hole flux class. The rest-frame interaction theorem in a smooth external field is $$ \mathbf F_{\mathrm{rest}}=q\,\mathbf E_{\mathrm e}. $$ The compact moving form is the Lorentz law $$ \frac{d\mathbf p}{dt} = q(\mathbf E_{\mathrm e}+\mathbf v\times\mathbf B_{\mathrm e}). $$ For two compact toroidal charged modes, the static cross energy is $$ U_\times = \frac{q_1q_2}{4\pi\varepsilon_0|\mathbf X_1-\mathbf X_2|}, $$ and its gradient gives the Coulomb force. Thus force and charge emerge together from one source-free electromagnetic ontology.