Motional emf for a moving material circuit: emf = -flux derivative

Given: Assume ${\mathrm{A}\mathrm{C}}_{\omega }$, the laboratory region $U\cong I\times \mathrm{\Omega }$, constants $c>0,{\epsilon }_0>0,{\mu }_0>0$, fields $E,B$ jointly $C^2$ and sources $\rho ,J$ of class $C^1$ satisfying the smooth branch of the vacuum Maxwell equations pointwise on $\mathrm{\Omega }\times I$, a moving material circuit $(D,r,v)$ with time interval $I$, and a compact subinterval $K\subseteq I$ with the swept set $r(D\times K)$ compactly contained in $\mathrm{\Omega }$.

Countable Choice, written ${\mathrm{A}\mathrm{C}}_{\omega }$, says every sequence of nonempty sets has a choice function. (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)).

A moving material circuit is the data $(D,r,v)$: $D$ a compact finite elementary Green region with its positive boundary chain, $r:D\times I\to \mathrm{\Omega }$ extending to a $C^2$ map on a neighbourhood of $D\times I$ with $r_u\times r_v\mathrm{\ne }0$ on the interior of $D$ and $r(\cdot ,t)$ injective on $D$ for every $t\in I$, and $v$ a $C^1$ single-valued velocity field with $v(r(u,w,t),t)={\mathrm{\partial }}_{t}r(u,w,t)$ on a neighbourhood of the swept set. (C2 moving material circuit: swept patch, single-valued material velocity, flux and electromotive force).

The electromotive force of the circuit at $t$ is the vector line integral $\mathrm{e}\mathrm{m}\mathrm{f}(t)={\oint }_{{\gamma }_t}⟨E(\cdot ,t)+v(\cdot ,t)\times B(\cdot ,t),d\mathbf{r}⟩$ over the oriented boundary chain ${\gamma }_t$ with trace $C(t)=r(\mathrm{\partial }D,t)$, in the piecewise $C^1$ convention, with unit V; the definition asserts no relation between emf and ${\mathrm{\Phi }}_B^{\mathrm{\prime }}$. (C2 moving material circuit: swept patch, single-valued material velocity, flux and electromotive force).

The transport formula needed here is established in step 1.1 from the explicitly typed circuit data and the calculus and Stokes suppliers below. Transport of flux through a moving patch supplies the general parametrized formula; the present argument independently derives the identity for the explicitly typed material circuit.

A patch test geometry in $\mathrm{\Omega }$ is a regular $C^2$ surface patch $(D,r)$ over a finite elementary Green region: $D\subseteq {\mathbb{R}}^2$ finite elementary Green region, $r$ of class $C^2$ on an open neighbourhood of $D$, $r_u\times r_v\mathrm{\ne }0$ on the interior of $D$, and no interior parameter point shares its image with a distinct point of $D$; its induced boundary chain is $r(\mathrm{\partial }D)$ with the circulation of the piecewise $C^1$ convention, and its oriented flux is ${\int }_{S}F\cdot n_{S}dS={\int }_D(F\circ r)\cdot (r_u\times r_v)dudv$. (Fixed volume and patch test geometries, outward normals and induced orientations).

Assume the smooth branch of the vacuum Maxwell postulate holds pointwise with $E,B$ jointly $C^2$ and $\rho ,J$ of class $C^1$. Then for every patch test geometry $(D,r)$ the flux functions $t\mapsto {\int }_{S}B(\cdot ,t)\cdot n_{S}dS$ and $t\mapsto {\int }_{S}E(\cdot ,t)\cdot n_{S}dS$ are differentiable on $I$ and, for every $t\in I$, ${\oint }_{\mathrm{\partial }S}E(\cdot ,t)\cdot d\mathbf{r}=-\frac{d}{dt}{\int }_{S}B(\cdot ,t)\cdot n_{S}dS$. (The integral form of the vacuum Maxwell equations and its equivalence with the smooth differential form).

A field of class $C^k$ is a map $\mathrm{\Omega }\times I\to {\mathbb{R}}^3$ of class $C^k$; $E$ carries V m${}^{-1}$ and $B$ carries T. (The electric and magnetic fields as fixed-time spatial vector fields).

The framework adopts the SI constants $c>0$, ${\mu }_0>0$, ${\epsilon }_0=({\mu }_0{c}^2{)}^{-1}$, with time in seconds and the emf unit V; no numerical value of ${\mu }_0$ is asserted. (Framework-local SI units, the vacuum constants and the exact/measured status of their values).

The source fields $\rho$ (C m${}^{-3}$) and $J$ (A m${}^{-2}$) are maps on $\mathrm{\Omega }\times I$. (Charge parameter, frame charge and current densities, volume charge and flux, and the four-current).

In the smooth branch, jointly $C^1$ fields with the additional derivatives required by each equation existing and continuous and continuous sources satisfy $\mathrm{div}E=\rho \mathrm{/}{\epsilon }_0$, $\mathrm{div}B=0$, $\mathrm{curl}E=-{\mathrm{\partial }}_{t}B$, $\mathrm{curl}B={\mu }_{0}J+c^{-2}{\mathrm{\partial }}_{t}E$ pointwise. (The microscopic vacuum Maxwell equations: smooth and weak branches).

The adopted coupling gives the continuum force density $\rho E+J\times B$ for matter interacting with the field through an explicitly stated matter model, and the covariant per-charge equation of motion for a test charge; the nonrelativistic coordinate statement is not adopted as exact. (The Lorentz coupling: external force on a test charge and the continuum force and power densities).

The Euclidean inner product is additive in each argument: $⟨x+y,z⟩=⟨x,z⟩+⟨y,z⟩$. (The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$).

For $a<b$ and integrable $f,g$ on $[a,b]$, $f+g$ is integrable and ${\int }_a^b(f+g)={\int }_a^{b}f+{\int }_a^{b}g$. (Integrable functions on $[a,b]$ form a set closed under sums and scalar multiples, and ${\int }_a^b(\lambda f+\mu g)=\lambda {\int }_a^{b}f+\mu {\int }_a^{b}g$).

For a piecewise $C^1$ path $\gamma :[a,b]\to {\mathbb{R}}^n$ with $a<b$ and a continuous vector field $F$ on a set containing the trace, ${\int }_{\gamma }F\cdot d\mathbf{r}:={\sum }_i{\int }_{t_i}^{t_{i+1}}⟨F(\gamma (t)),v_i(t)⟩dt$; the value is independent of the admissible partition. (Scalar line integrals with respect to arc length and vector-field line integrals).

For a compact measurable parameter set and an integrand with continuous time derivative on its neighbourhood, differentiation passes through its Lebesgue integral on a compact nondegenerate time interval (Differentiation under the integral sign over a compact measurable set and over a compact $C^1$ surface). Under ${\mathrm{A}\mathrm{C}}_{\omega }$, bounded Borel Riemann-integrable functions on a rectangle have equal Riemann and Lebesgue integrals; this includes zero extensions of continuous integrands from compact Jordan regions (Riemann–Lebesgue comparison for distribution test integrands).

Stokes on each admissible regular $C^2$ elementary patch gives the curl flux as the circulation of its induced boundary chain (The classical Stokes theorem for a $C^2$ patch over a finite elementary Green region). Internal pairs of arcs cancel in the positive finite Green boundary chain (Positive orientation of elementary-region boundaries).

The chain, product and mixed-partial rules apply to the stated $C^2$ map; for $C^1$ spatial fields, $\mathrm{curl}(v\times B)=v\mathrm{div}B-B\mathrm{div}v+(D_{x}v)B-(D_{x}B)v$ (The chain rule for total derivatives: $D(g\circ f)(a)=Dg(f(a))\circ Df(a)$, Sums, scalar multiples, products and quotients: $(f+g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)+g^{\mathrm{\prime }}(c)$, $(\alpha f{)}^{\mathrm{\prime }}(c)=\alpha f^{\mathrm{\prime }}(c)$, $(fg{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)g(c)+f(c)g^{\mathrm{\prime }}(c)$, and $(f\mathrm{/}g{)}^{\mathrm{\prime }}(c)=(f^{\mathrm{\prime }}(c)g(c)-f(c)g^{\mathrm{\prime }}(c))\mathrm{/}g(c{)}^2$ when $g(c)\mathrm{\ne }0$, Clairaut--Schwarz theorem for continuous second partial derivatives, The divergence and curl of a cross product).