# 12. Newton as Flux Accounting Newton's second law is the integrated continuity law for momentum in a nearly stable localized electromagnetic configuration. Chapter 7 identified the energy flow with the Maxwell realization $$ \mathbf{S}=\frac{1}{\mu_0}\,\mathbf{E}\times\mathbf{B}. $$ The same transport therefore carries momentum. The electromagnetic momentum density is $$ \mathbf{g}=\frac{\mathbf{S}}{c^2} = \epsilon_0\,\mathbf{E}\times\mathbf{B}. $$ To track how momentum moves through a surface, we use the Maxwell stress tensor: $$ T_{ij} = \epsilon_0\left(E_iE_j-\frac{1}{2}\delta_{ij}\mathbf{E}^2\right) + \frac{1}{\mu_0}\left(B_iB_j-\frac{1}{2}\delta_{ij}\mathbf{B}^2\right). $$ Differentiate $\mathbf{g}$ in time and substitute Maxwell's equations. The first result is $$ \partial_t\mathbf{g} = \frac{1}{\mu_0}(\nabla\times\mathbf{B})\times\mathbf{B} - \epsilon_0\,\mathbf{E}\times(\nabla\times\mathbf{E}). $$ Using the standard vector identity for $(\nabla\times\mathbf{A})\times \mathbf{A}$ and the source-free conditions $\nabla\cdot\mathbf{E}=0$, $\nabla\cdot\mathbf{B}=0$, this rearranges into the exact local momentum continuity law $$ \partial_t g_i + \partial_j T_{ij} = 0, $$ or, in vector form, $$ \partial_t\mathbf{g}+\nabla\cdot\mathbf{T}=0. $$ This is the momentum analogue of Poynting's theorem. Momentum does not appear or disappear. It changes only by flux through the boundary of a region. Integrate over a localized region $K$ containing a nearly stable bounded configuration: $$ \mathbf{P}_K=\int_K \mathbf{g}\,d^3x. $$ Then $$ \frac{d}{dt}\mathbf{P}_K = -\int_{\partial K}\mathbf{T}\cdot\mathbf{n}\,dA. $$ The right-hand side is what later language calls force. It is the net rate at which momentum crosses the boundary. To connect this with motion of the object as a whole, define the energy in the region and its center of energy: $$ E_K=\int_K u\,d^3x, \qquad \mathbf{X}_K=\frac{1}{E_K}\int_K \mathbf{x}\,u\,d^3x. $$ When boundary leakage is small and the mode remains coherent, $$ E_K\,\dot{\mathbf{X}}_K \approx \int_K \mathbf{S}\,d^3x, \qquad \mathbf{P}_K \approx \frac{E_K}{c^2}\dot{\mathbf{X}}_K. $$ For such a bounded configuration, define the effective inertial mass $$ m_K := \frac{E_K}{c^2}. $$ If the energy of the localized configuration is roughly constant, then $$ m_K\,\ddot{\mathbf{X}}_K \approx -\int_{\partial K}\mathbf{T}\cdot\mathbf{n}\,dA. $$ This is Newton's second law in its effective form for a stable bounded mode: $$ \mathbf{F}=\frac{d\mathbf{P}}{dt}. $$ It describes momentum bookkeeping for a bounded region of field. A mediating force field inserted between supposedly independent substrates does not rescue their independence. If interaction is real, a common structure is already present, and force is only the accounting of that coupling. Particles are localized regions. Forces are boundary integrals. Newton's second law is momentum continuity applied to a stable electromagnetic knot.