C2 moving material circuit: swept patch, single-valued material velocity, flux and electromotive force

Let $\mathrm{\Omega }\subseteq {\mathbb{R}}^3$ be nonempty and open and $I\subseteq \mathbb{R}$ a nonempty open interval, with the laboratory frame, fields $E,B:\mathrm{\Omega }\times I\to {\mathbb{R}}^3$, units and $c>0$ of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields, The electric and magnetic fields as fixed-time spatial vector fields and Framework-local SI units, the vacuum constants and the exact/measured status of their values.

(a) A moving material circuit is data $(D,r,v)$. The nonempty compact parameter set $D=D_1\cup \cdots \cup D_N\subseteq {\mathbb{R}}^2$ has a supplied finite elementary Green decomposition, including the opposite orientations on shared arcs, as in Type I, Type II, and elementary regions for Green's theorem. Its positive boundary chain $\mathrm{\partial }D=({\sigma }_1,\dots ,{\sigma }_m)$ is the finite list of surviving oriented piecewise $C^1$ arcs after internal pairs cancel (Positive orientation of elementary-region boundaries). The map $r:D\times I\to \mathrm{\Omega }$ extends to a $C^2$ map on an open neighbourhood of $D\times I$, has $r_u\times r_w\mathrm{\ne }0$ on $D^{\circ }\times I$, and is injective on $D$ at each time. These conditions are the meaning of an embedded material sheet here. Its swept spacetime graph is $\mathcal{G}:=\{(r(p,t),t):p\in D,t\in I\}\subseteq \mathrm{\Omega }\times I$. A compatible material velocity is a $C^1$ map $v:W\to {\mathbb{R}}^3$ on an open $W\subseteq \mathrm{\Omega }\times I$ containing $\mathcal{G}$, satisfying $v(r(p,t),t)=r_t(p,t)$. Thus coincident images must have the same velocity. Material labels are the points $p=(u,w)\in D$.

(b) Put $S(t):=r(D,t)$ and ${\gamma }_t:=(r(\cdot ,t)\circ {\sigma }_k{)}_{k=1}^m$, a boundary chain, whose trace is $C(t):=r(\mathrm{\partial }D,t)$. We use $\mathrm{\partial }S(t)$ for this parametrized boundary chain, rather than the ambient topological boundary of the subset $S(t)\subseteq {\mathbb{R}}^3$. A finite Green region can have several components or holes, so ${\gamma }_t$ need not be one closed path. For $B$ jointly $C^1$ on a spacetime neighbourhood of $\mathcal{G}$, the magnetic flux is ${\mathrm{\Phi }}_B(t):={\int }_D⟨B(r(u,w,t),t),r_u(u,w,t)\times r_w(u,w,t)⟩dudw\mathrm{.}$ This is the parametrized moving-patch flux convention of Moving parametrized surface patches and their flux, specified here with its spacetime arguments explicit. Its unit is $\mathrm{W}\mathrm{b}=\mathrm{T}{\mathrm{m}}^2$.

(c) For $E,B$ continuous on a spacetime neighbourhood of $\mathcal{G}$, the electromotive force is the finite sum of vector line integrals $\mathrm{e}\mathrm{m}\mathrm{f}(t):={\oint }_{{\gamma }_t}(E(\cdot ,t)+v(\cdot ,t)\times B(\cdot ,t))\cdot d\mathbf{r}:={\sum }_{k=1}^m{\int }_{r(\cdot ,t)\circ {\sigma }_k}(E(\cdot ,t)+v(\cdot ,t)\times B(\cdot ,t))\cdot d\mathbf{r},$ with Scalar line integrals with respect to arc length and vector-field line integrals and The cross product in ${\mathbb{R}}^3$. Its unit is $\mathrm{V}=\mathrm{k}\mathrm{g}{\mathrm{m}}^2{\mathrm{s}}^{-3}{\mathrm{A}}^{-1}$. For the force interpretation, additionally adopt The Lorentz coupling: external force on a test charge and the continuum force and power densities(i): a constant $q\mathrm{\ne }0$, a mass $m>0$, a $C^2$ future timelike test-charge worldline and a prescribed external $C^1$ electromagnetic tensor. In the worldline conventions of Worldlines, four-velocity, chart readings and the frame/observer distinction and Norms of four-velocity and four-acceleration, and the observer rest-space decomposition(e), $U=\gamma (c,v)$, $d\mathrm{/}d\tau =\gamma d\mathrm{/}dt$ and $\gamma =(1-\mathrm{\mid }v{\mathrm{\mid }}^2\mathrm{/}{c}^2{)}^{-1\mathrm{/}2}$. The tensor convention The electromagnetic two-form, its observer decomposition and the four-current gives $F_{i0}=E_i\mathrm{/}c$, $F_{ij}={ϵ}_{ijk}{B}_k$ and $\mathrm{\Phi }(U{)}^i=F_{i\beta }{U}^{\beta }=\gamma (E+v\times B{)}^i$. For the spatial momentum $p:=m\gamma v$, the spatial components of $mdU\mathrm{/}d\tau =q\mathrm{\Phi }(U)$ therefore yield $\gamma dp\mathrm{/}dt=q\gamma (E+v\times B)$, hence $dp\mathrm{/}dt=q(E+v\times B)$. When the charge velocity agrees with the material velocity, the integrand is this lab momentum force divided by $q$. This interpretation adds those external-coupling hypotheses; the emf definition itself only names the integral.

(d) The stated field regularity must hold near every point of $\mathcal{G}$; singularity crossings are excluded. The force interpretation additionally requires $\mathrm{\mid }v\mathrm{\mid }<c$ along the material motion. No conservation law, flux-derivative relation, empirical assertion or parametrization-invariance assertion is made.

Well-definedness and conventions

Each $D_i$ is compact and Jordan measurable; its boundary has content zero. The finite union $D$ is Jordan measurable, and its continuous flux integrand is Riemann integrable by A continuous real function on a compact Jordan measurable set is Riemann integrable over that set. At a fixed time each elementary piece is an admissible regular $C^2$ patch: its interior is connected and dense in that piece, the parameter cross product is nonzero there, and injectivity prevents interior image identifications (Regular parametrized surface patches on compact Jordan parameter regions). All integrals over the full sheet mean the sum over the pieces, with internal boundary arcs cancelling. Neither connectedness nor a single boundary loop is assumed for their union.

Composition of each ${\sigma }_k$ with the $C^2$ slice gives a piecewise $C^1$ path. Its trace lies in the time-$t$ slice of $W$, where $E+v\times B$ is continuous, so each line integral exists. The chain sum is independent of its ordering. Spatial derivatives of two $C^2$ extensions agreeing on $D\times I$ agree first on $D^{\circ }\times I$ by difference quotients and then on its closure in each time slice by continuity; since $I$ is open the same argument gives the time derivative. The flux and chain therefore do not depend on the extension. Boundary rank failures remain allowed.

Tong §4.1–4.1.1, printed pp. 67–71, supplies the moving-loop flux and force-per-charge emf convention. The finite Green chain, spacetime domains and precise regularity above are the local formulation; the source is not claimed to impose them.