Induction and circuit approximations — Examples

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ for the explicitly used Lebesgue bar-volume interfaces (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Work in the laboratory frame and units of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields and Framework-local SI units, the vacuum constants and the exact/measured status of their values, choosing $\mathrm{\Omega }={\mathbb{R}}^3$ and an open laboratory time interval containing $J$. Fix $l>0$, $B_0\ge 0$, a nonempty interval $J$ with a specified $C^1$ neighbourhood extension for motion and its velocity jets when $J$ is not open (in particular singleton data are read through that supplied extension), and a prescribed $C^1$ motion $x:J\to (a_0,a_1)$ with $a_0<a_1$ and $\mathrm{\mid }{x}^{\mathrm{\prime }}(t)\mathrm{\mid }<c$. The fixed rails have centerlines $(s,\pm l\mathrm{/}2,0)$, $a_0\le s\le a_1$, and the left closing side is $(a_0,w,0)$, $\mathrm{\mid }w\mathrm{\mid }\le l\mathrm{/}2$. At time $t$ the ideal oriented contour $C(t)$ comprises only the rail portions $a_0\le s\le x(t)$, that left side, and the sliding bar centerline $(x(t),w,0)$. Give it normal $e_3$, hence traverse the bar in the $+e_2$ direction. The rails and closing side have material velocity zero; the bar velocity is $v_b(t)=x^{\mathrm{\prime }}(t)e_1$. These are separate segment velocities with sliding contacts; endpoint values at the contacts are taken independently on each segment. Define the contact-model emf as the sum of its four segment integrals ${\sum }_{\gamma }{\int }_{\gamma }(E+v_{\gamma }\times B)\cdot d\mathbf{r}$ in Scalar line integrals with respect to arc length and vector-field line integrals. This is not a single-valued smooth moving material sheet of C2 moving material circuit: swept patch, single-valued material velocity, flux and electromotive force. Prescribe an arbitrary finite-valued signed current $I:J\to \mathbb{R}$, in amperes; no time regularity of $I$ is needed for these instantaneous identities. The external test fields are $E(y,t)=0$, $B(y,t)=B_0{e}_3$ on the laboratory region; all self-fields are neglected.

For a measure space $(X,\mathcal{A},\mu )$, a measurable function $f:X\to \mathbb{R}$ and $K\in \mathcal{A}$, write ${\int }_{K}fd\mu :=\int (f{\chi }_K)d\mu$ using the signed Lebesgue integral of Integrable real and complex functions, and their integrals: $(f{\chi }_K{)}^{+}=\max \{f{\chi }_K,0\}$ and $(f{\chi }_K{)}^{-}=\max \{-f{\chi }_K,0\}$, and the integral is their integral difference when both are finite. For a measurable vector map $G:X\to {\mathbb{R}}^3$ with all three components integrable on $K$, define ${\int }_{K}Gd\mu :=({\int }_K{G}_{0}d\mu ,{\int }_K{G}_{1}d\mu ,{\int }_K{G}_{2}d\mu )$ in the ordered spatial coordinates of the laboratory frame; the stored coordinate indices are $0,1,2$ and the corresponding physical basis labels are $e_1,e_2,e_3$.

For force bookkeeping choose an actual finite-volume homogeneous bar, separate from the ideal centerline: fix $d,h>0$ with cross-section area $A=dh$ and set $K_t=[x(t)-d\mathrm{/}2,x(t)+d\mathrm{/}2]\times [-l\mathrm{/}2,l\mathrm{/}2]\times [-h\mathrm{/}2,h\mathrm{/}2]$, so ${\lambda }_3(K_t)=Al$. For each frozen time prescribe the uniform current density $j_t=(I(t)\mathrm{/}A)e_2$ on its interior, with the ideal local conductor-matter coupling of The Lorentz coupling: external force on a test charge and the continuum force and power densities(ii) applied there. Its extension to the boundary is used only for the volume integral; contacts/end faces have zero volume and no additional contact force is included. The same signed lumped current runs around the contour. The local force-law adoption and prescribed current do not assert a total Maxwell or matter-dynamical solution.

(a) The rectangular flux is $\mathrm{\Phi }(t)=B_{0}l(x(t)-a_0)$ and the segmentwise emf is $\mathrm{e}\mathrm{m}\mathrm{f}(t)=-B_{0}l{x}^{\mathrm{\prime }}(t)=-{\mathrm{\Phi }}^{\mathrm{\prime }}(t)$. Its magnitude is $B_{0}l\mathrm{\mid }{x}^{\mathrm{\prime }}\mathrm{\mid }$, including either motion direction and $B_0=0$.

(b) The finite-volume force is $F(t)={\int }_{K_t}{j}_t\times Bd{\lambda }_3=I(t)B_{0}l{e}_1\mathrm{.}$

(c) The signed instantaneous balance is $F(t)\cdot v_b(t)+\mathrm{e}\mathrm{m}\mathrm{f}(t)I(t)=I(t)B_{0}l{x}^{\mathrm{\prime }}(t)-B_{0}l{x}^{\mathrm{\prime }}(t)I(t)=0.$ It is ideal mechanical/electrical transfer bookkeeping with prescribed motion and current; no dissipative, radiation or field-storage term is included. A support enforcing the motion may supply the opposite mechanical power.

(d) This is a hypothetical conditional sliding-contact model with no measurement. Self-fields, radiation, friction, contact resistance, deformation and constitutive details are neglected; no bar equation of motion, physical realization, or total Maxwell solution is asserted. The strict $C^2$ material-sheet theorem remains a separate result under its own hypotheses.

Griffiths §7.1.3, printed pp. 294-296, presents the primitive generator: a loop pulled at speed $v$ through a uniform field, emf $=vBh$, the flux rule $\mathcal{E}=-\mathrm{d}\mathrm{\Phi }\mathrm{/}\mathrm{d}t$ for motional emf and the work/power discussion; Tong §4.1.1 and §4.1.3, printed pp. 69-77, gives the sliding bar $\mathcal{E}=vBd$, the Lorentz force $F=IBd$ on the bar and the coupled motion with the resistor. The present thought experiment extracts only the instantaneous idealised statements: the segmentwise emf, the flux function and its derivative, the adopted finite-volume force integral and the signed power balance. The coupled bar dynamics and the Joule-heating energetics of the sources are not reproduced, because they require a mechanical mass model and a dynamical circuit solution that this item does not adopt.

Assume the lumped circuit model of Lumped circuit approximation: branch data, terminal laws and the declared quasistatic regime with the units of Framework-local SI units, the vacuum constants and the exact/measured status of their values, and let $L=\left(\begin{array}{cc}{L}_1& M\\ M& L_2\end{array}\right)$ be a constant symmetric block of its inductance matrix with $L_1,L_2>0$. Let $I_1,I_2:J\to \mathbb{R}$ be $C^1$ currents on a nonempty interval $J$ (each extending to a $C^1$ map on an open interval containing $J$, as in the model's conventions) with terminal voltages $V_{\alpha }={\sum }_{\beta }{L}_{\alpha \beta }{\dot{I}}_{\beta }$. Assume Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$, The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)) for the realization and bounded-linking clauses (b), inherited from the linkage and energy suppliers.

(a) [Energy and instantaneous power] the stored inductive bookkeeping energy is $W(t)=\frac{1}{2}{L}_1{I}_1(t{)}^2+\frac{1}{2}{L}_2{I}_2(t{)}^2+M{I}_1(t)I_2(t),$ and $V_1{I}_1+V_2{I}_2=\dot{W}$ by part (a) of Restricted inductor energy LI^2/2 and its identification with the frozen magnetic field energy; the mutual term increases the energy for currents of the same sign and decreases it for opposite signs (as the sign of the product $I_1{I}_2$ indicates when $M>0$).

(b) [Perfect linkage and the bound] if the two currents are realised by fixed compact smooth conserved profiles of the class of The steady-current (magnetostatic) regime, the compact smooth current class and the declared ideal noncompact models(b) with linkage matrix $L$ as in Reciprocal linkage coefficients and magnetic energy of smooth compact conserved currents, then $W(t)\ge 0$ for all currents, and evaluating the quadratic form on both $(\sqrt{L_2},-\sqrt{L_1})$ and $(\sqrt{L_2},+\sqrt{L_1})$ gives the bound $\mathrm{\mid }M\mathrm{\mid }\le \sqrt{L_1{L}_2};$ moreover if $M=s\sqrt{L_1{L}_2}$ with $s\in \{+1,-1\}$ (perfect linkage), the null direction $I=(\sqrt{L_2},-s\sqrt{L_1})$ has $J_I=0$ pointwise by the vanishing clause of Reciprocal linkage coefficients and magnetic energy of smooth compact conserved currents(c).

(c) [Ideal common-flux model] if $M=\sqrt{L_1{L}_2}$, choose any $\mathrm{\Lambda }>0$ in henries and set $N_1:=\sqrt{L_1\mathrm{/}\mathrm{\Lambda }}$, $N_2:=\sqrt{L_2\mathrm{/}\mathrm{\Lambda }}$; then $L_1=\mathrm{\Lambda }{N}_1^2$, $L_2=\mathrm{\Lambda }{N}_2^2$ and $M=\mathrm{\Lambda }{N}_1{N}_2$. Writing ${\mathrm{\Phi }}_c:=\mathrm{\Lambda }(N_1{I}_1+N_2{I}_2)$ for the common flux linkage, the terminal voltages are $V_1=N_1{\dot{\mathrm{\Phi }}}_c,V_2=N_2{\dot{\mathrm{\Phi }}}_c,\text{i.e.}\frac{V_1}{N_1}=\frac{V_2}{N_2}={\dot{\mathrm{\Phi }}}_c,$ and the stored energy is $W={\mathrm{\Phi }}_c^2\mathrm{/}(2\mathrm{\Lambda })$; equivalently the induced emfs $e_{\alpha }:=-V_{\alpha }$ satisfy $e_1\mathrm{/}{N}_1=e_2\mathrm{/}{N}_2=-{\dot{\mathrm{\Phi }}}_c$. Here $N_1,N_2>0$ are dimensionless real effective winding factors, rather than asserted integer turn counts; ${\mathrm{\Phi }}_c$ has unit Wb. This is the explicitly adopted common-linkage ('ideal transformer') model with effective ratio $N_1:N_2$.

(d) [Limitations] for an imperfect realized linkage, $\mathrm{\mid }M\mathrm{\mid }<\sqrt{L_1{L}_2}$, the perfect common-flux voltage ratio is not an identity for all current histories; a particular history can nevertheless satisfy that ratio accidentally (including both voltages zero). Reversing the second winding retains $s=-1$: with $M=s\sqrt{L_1{L}_2}$ put ${\mathrm{\Phi }}_c=\mathrm{\Lambda }(N_1{I}_1+s{N}_2{I}_2)$ and obtain $V_1\mathrm{/}{N}_1=V_2\mathrm{/}(s{N}_2)={\dot{\mathrm{\Phi }}}_c$, with the same energy ${\mathrm{\Phi }}_c^2\mathrm{/}(2\mathrm{\Lambda })$. The positive-orientation model (c) remains as stated. Winding resistance, hysteresis and core loss, stray capacitance, magnetising current and finite permeability are outside the declared model and are not inferred from it. No measurement is involved; the statements are conditional computations in the ideal model. Regularity on a non-open interval uses the stated $C^1$ neighbourhood extension; if the interval is a singleton its terminal-voltage jets are those of a specified extension, not derivatives inferred from its value restriction.

Fitzpatrick §7.4, printed pp. 222-225, computes the two-solenoid mutual inductance and records perfect linkage $M=\sqrt{L_1{L}_2}$ with the leakage warning; Griffiths §7.2.3, printed pp. 310-317, derives reciprocity and the Neumann formula; Tong §4.1.2, printed pp. 71-73, gives the single-coil energy and its field-energy representation; the mutual term here is obtained by local expansion of the linkage quadratic form. The example separates what is proved for the declared smooth compact profiles (the PSD bound and the null direction) from what is adopted as the common-flux ideal-transformer model, and it proves the failure of a universal voltage ratio under strict linkage by exhibiting the forced row proportionality and its contradiction; no physical transformer behaviour, loss mechanism or measurement is asserted.

In the RL branch of Restricted RL decay and LC oscillation under the adopted circuit model with the declared units of Framework-local SI units, the vacuum constants and the exact/measured status of their values, let the source voltage be the constant $V_0\in \mathbb{R}$, let $R>0$ and $L>0$, let $t_0\in J$ and $I_0\in \mathbb{R}$, with $J$ an interval and derivatives read in the model's domain-relative convention, through the stated $C^1$ open-neighbourhood extension for a non-open $J$. For nondegenerate $J$ the unique $C^1$ solution of $L\dot{I}+RI=V_0$, $I(t_0)=I_0$, is $I(t)=\frac{V_0}{R}+(I_0-\frac{V_0}{R})\exp \text{\hspace{0.17em}⁣}(-\frac{R(t-t_0)}{L}),t\in J\mathrm{.}$ In particular $I(t)-V_0\mathrm{/}R=(I_0-V_0\mathrm{/}R)\exp (-R(t-t_0)\mathrm{/}L)$ and hence $\mathrm{\mid }I(t)-V_0\mathrm{/}R\mathrm{\mid }=\mathrm{\mid }{I}_0-V_0\mathrm{/}R\mathrm{\mid }\exp (-R(t-t_0)\mathrm{/}L)$ for every $t\in J$, with time constant $\tau =L\mathrm{/}R$; $I\equiv V_0\mathrm{/}R$ is the unique stationary solution (an extension with zero derivative, also in the singleton convention); if $J$ is unbounded above, then $I(t)\to V_0\mathrm{/}R$ as $t\to +\mathrm{\infty }$ along $J$; and the transient part is the free decay $\exp (-R(t-t_0)\mathrm{/}L)$ of the initial deviation $I_0-V_0\mathrm{/}R$. For $J=\{t_0\}$ the restriction is $I(t_0)=I_0$ and the jets are those of the same formula read through the stated extension, giving $\dot{I}(t_0)=(V_0-R{I}_0)\mathrm{/}L$. The computation is conditional on the adopted lumped model, constant elements and fixed geometry; no measurement is involved.