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Sliding-bar sliding-contact model: segmentwise emf, force integral and power balance

Physical scope

A hypothetical conditional sliding-contact model in a uniform static external test field: four separate segment velocities at sliding contacts, the direct segmentwise emf and flux computation, an adopted finite-volume bar force integral and the signed instantaneous mechanical/electrical power balance. Self-fields, radiation, contact resistance and dynamics are neglected; no measurement, no bar equation of motion and no total Maxwell or matter-dynamical solution is asserted.

Statement

Assume ${A C}_{ω}$ for the explicitly used Lebesgue bar-volume interfaces (The Axiom of Countable Choice ( ${A C}_{ω}$ )). 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 $Ω = R^{3}$ and an open laboratory time interval containing $J$ . Fix $l > 0$ , $B_{0} \geq 0$ , 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 $∣ x^{'} (t) ∣ < c$ . The fixed rails have centerlines $(s, \pm l / 2, 0)$ , $a_{0} \leq s \leq a_{1}$ , and the left closing side is $(a_{0}, w, 0)$ , $∣ w ∣ \leq l / 2$ . At time $t$ the ideal oriented contour $C (t)$ comprises only the rail portions $a_{0} \leq s \leq x (t)$ , 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 $v_{b} (t) = x^{'} (t) e_{1}$ . 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 $\sum_{γ} \int_{γ} (E + v_{γ} \times B) \cdot d r$ 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 $I : J \to R$ , 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 $(X, A, μ)$ , a measurable function $f : X \to R$ and $K \in A$ , write $\int_{K} f d μ : = \int (f χ_{K}) d μ$ using the signed Lebesgue integral of Integrable real and complex functions, and their integrals: $(f χ_{K})^{+} = \max {f χ_{K}, 0}$ and $(f χ_{K})^{-} = \max {- f χ_{K}, 0}$ , and the integral is their integral difference when both are finite. For a measurable vector map $G : X \to R^{3}$ with all three components integrable on $K$ , define $\int_{K} G d μ : = (\int_{K} G_{0} d μ, \int_{K} G_{1} d μ, \int_{K} G_{2} d μ)$ 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 $A = d h$ and set $K_{t} = [x (t) - d / 2, x (t) + d / 2] \times [- l / 2, l / 2] \times [- h / 2, h / 2]$ , so $λ_{3} (K_{t}) = A l$ . For each frozen time prescribe the uniform current density $j_{t} = (I (t) / A) e_{2}$ 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 $Φ (t) = B_{0} l (x (t) - a_{0})$ and the segmentwise emf is $e m f (t) = - B_{0} l x^{'} (t) = - Φ^{'} (t)$ . Its magnitude is $B_{0} l ∣ x^{'} ∣$ , including either motion direction and $B_{0} = 0$ .

(b) The finite-volume force is $F (t) = \int_{K_{t}} j_{t} \times B d λ_{3} = I (t) B_{0} l e_{1} .$

(c) The signed instantaneous balance is $F (t) \cdot v_{b} (t) + e m f (t) I (t) = I (t) B_{0} l x^{'} (t) - B_{0} l x^{'} (t) I (t) = 0.$ 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.

Facts & Assumptions

Given: Assume ${A C}_{ω}$ , the laboratory frame and units, the constants $l > 0$ , $B_{0} \geq 0$ , $d, h > 0$ , $A = d h$ , the $C^{1}$ motion $x : J \to (a_{0}, a_{1})$ with $∣ x^{'} ∣ < c$ , the prescribed current $I : J \to R$ and the contour and bar data of the Statement.

[A1]

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

[F1]

The line integral of a continuous vector field along a piecewise $C^{1}$ path is the sum of the integrals $\int_{t_{i}}^{t_{i + 1}} ⟨ F (γ (t)), v_{i} (t) ⟩ d t$ over an admissible partition. (Scalar line integrals with respect to arc length and vector-field line integrals).

[F2]

The cross product in coordinates is $(u \times v)_{x} = u_{y} v_{z} - u_{z} v_{y}$ and cyclically; the standard basis satisfies $e_{1} \times e_{3} = - e_{2}$ . (The cross product in $R^{3}$ ).

[F3]

The Euclidean inner product is $⟨ x, y ⟩ = \sum_{k} x_{k} y_{k}$ , in particular $⟨ e_{2}, e_{2} ⟩ = 1$ . (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ).

[F4]

For a nonnegative measurable $f : X \to [0, + \infty]$ and measurable $E$ , $\int_{E} f d μ = \int f χ_{E} d μ$ . (Integral over a measurable subset).

[F5]

For a nonnegative simple measurable function $s = \sum_{j} c_{j} χ_{E_{j}}$ , the simple integral is $\sum_{j} c_{j} μ (E_{j})$ , and the nonnegative integral of a nonnegative simple function equals its simple integral. (The integral of a nonnegative simple function, The nonnegative integral agrees with the simple integral on simple functions).

[F6]

The Lebesgue integral is linear on $L^{1}$ . (The Lebesgue integral is linear on $L^{1} (μ)$ ).

[F7]

Every set between the open and closed box forms is Lebesgue measurable with $λ_{n} (R) = \prod_{i} (b_{i} - a_{i})$ ; in particular closed boxes have the product volume and their faces are null when some side degenerates. (A box in $R^{n}$ with parameters $a_{i} \leq b_{i}$ is Lebesgue measurable of measure $\prod_{i < n} (b_{i} - a_{i})$ , whichever of its faces are included).

[F8]

If $f, g : D \to R$ are differentiable at a limit point $c \in D$ and $α \in R$ , then $f + g$ , $α f$ and $f g$ are differentiable at $c$ with the sum, scalar and product rules. (Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ ).

[F9]

Fields are maps $Ω \times I \to R^{3}$ ; $E$ carries V m $^{- 1}$ , $B$ carries T. (The electric and magnetic fields as fixed-time spatial vector fields).

[F10]

The framework adopts the SI constants including $c > 0$ and the units V, T; the emf unit is the volt. (Framework-local SI units, the vacuum constants and the exact/measured status of their values).

[F11]

The adopted continuum matter coupling gives the force density $ρ E + J \times B$ (N m $^{- 3}$ ) for matter interacting with the field through an explicitly stated matter model. (The Lorentz coupling: external force on a test charge and the continuum force and power densities).

[F12]

A moving material circuit of the strict branch is a $C^{2}$ embedded sheet with a single-valued $C^{1}$ material velocity; the strict theorem on this page applies only to such sheets. (C2 moving material circuit: swept patch, single-valued material velocity, flux and electromotive force).

[F13]

The laboratory region, its chart and the spatial frame are those of the laboratory geometry; laboratory fields are maps on $Ω \times I$ with the spatial coordinates of the frame. (Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields).

[F14]

For a measurable real function $f$ , put $f^{+} = \max {f, 0}$ and $f^{-} = \max {- f, 0}$ . It is integrable when $\int ∣ f ∣ < \infty$ , equivalently both part integrals are finite; then $\int f = \int f^{+} - \int f^{-}$ . (Integrable real and complex functions, and their integrals).

Proof

technique · direct

1.1F1F2F3algebra

Fix $t \in J$ . By [F1] the contact-model emf is the sum of the four segment integrals of $E + v_{γ} \times B$ with the segmentwise velocities. On the two rail portions and the left closing side the material velocity is zero and $E = 0$ , so the integrand vanishes identically and each of those three integrals is $0$ . On the bar, parametrized by $s \mapsto (x (t), s, 0)$ with $s$ from $- l / 2$ to $l / 2$ in the $+ e_{2}$ direction, the velocity is $v_{b} = x^{'} (t) e_{1}$ , so [F2] gives $v_{b} \times B = x^{'} (t) e_{1} \times B_{0} e_{3} = - B_{0} x^{'} (t) e_{2}$ and [F3] gives $⟨ v_{b} \times B, e_{2} ⟩ = - B_{0} x^{'} (t)$ ; the bar integral is therefore $\int_{- l / 2}^{l / 2} (- B_{0} x^{'} (t)) d s = - B_{0} l x^{'} (t)$ . Hence $e m f (t) = - B_{0} l x^{'} (t)$ for every $t \in J$ .

1.2F2F4F5F6F7F11F14algebra

For the force choose the constant extension of $j_{t} = (I (t) / A) e_{2}$ to $K_{t}$ and zero outside $K_{t}$ . This boundary convention adds no contact force; the faces are null by [F7]. Put $q_{t} : = I (t) B_{0} / A \in R$ . On the interior the adopted coupling [F11] gives $j_{t} \times B_{0} e_{3} = q_{t} e_{1}$ by [F2], so its volume-integrand extension has coordinate tuple $(q_{t} χ_{K_{t}}, 0, 0)$ in the Statement's convention. By [F7], $K_{t}$ is measurable and $λ_{3} (K_{t}) = d l h = A l < \infty$ . By [F4] and [F5], $\int χ_{K_{t}} = A l$ and $\int ∣ q_{t} χ_{K_{t}} ∣ = ∣ q_{t} ∣ A l < \infty$ , so [F14] gives scalar integrability for every sign of $I (t)$ . Applying [F6] to this scalar multiple of the integrable indicator and then the componentwise vector-integral definition yields $F (t) = q_{t} A l e_{1} = I (t) B_{0} l e_{1}$ . Thus the vector integral is finite component by component. This is the prescribed local matter coupling applied to the declared bar model; it asserts no force on an arbitrary source without such a matter model.

2.1F3F7F8step 1.1algebra

Put $D_{t} = [a_{0}, x (t)] \times [- l / 2, l / 2] \subseteq R^{2}$ and $ψ (s, w) = (s, w, 0)$ . Then $ψ_{s} \times ψ_{w} = e_{3}$ , so the parametrized rectangular flux is $Φ (t) = \int_{D_{t}} ⟨ B_{0} e_{3}, e_{3} ⟩ d λ_{2} = B_{0} λ_{2} (D_{t}) = B_{0} l (x (t) - a_{0})$ by [F7]. Thus $λ_{2}$ is used on the two-dimensional parameter rectangle; its pushforward by $ψ$ is the plane-area convention. With the prescribed $C^{1}$ neighbourhood motion, [F8] gives $Φ^{'} (t) = B_{0} l x^{'} (t)$ , with one-sided endpoint values and the specified extension jet at a singleton. Step 1.1 then gives $e m f = - Φ^{'}$ and $∣ e m f ∣ = B_{0} l ∣ x^{'} ∣$ .

2.2F3step 1.1step 1.2algebra

The signed power balance is immediate from steps 1.1 and 1.2 and the Euclidean pairing: with $v_{b} = x^{'} (t) e_{1}$ and $⟨ e_{1}, e_{1} ⟩ = 1$ by [F3], $F (t) \cdot v_{b} (t) = I (t) B_{0} l x^{'} (t)$ and $e m f (t) I (t) = - B_{0} l x^{'} (t) I (t)$ , so the sum is $0$ for every sign of $I (t)$ and of $x^{'} (t)$ , including the zero values.

3.1A1F9F10F12F13step 1.1step 1.2step 2.1step 2.2∎

The three displayed conclusions are established by steps 1.1, 1.2, 2.1 and 2.2. The prescribed constant fields are $C^{\infty}$ , and their types and units are those of [F9] and [F10]; the model is the hypothetical sliding-contact model of the Statement, not a single-valued smooth material sheet of [F12], and its strict theorem is not used; the laboratory reading is that of [F13]. No measurement, self-field, radiation, dissipative or dynamical conclusion is asserted, and ${A C}_{ω}$ is used only through the explicitly cited box-measure and integral interfaces.

Source notes

Griffiths §7.1.3, printed pp. 294-296, presents the primitive generator: a loop pulled at speed $v$ through a uniform field, emf $= v B h$ , the flux rule $E = - d Φ / d t$ for motional emf and the work/power discussion; Tong §4.1.1 and §4.1.3, printed pp. 69-77, gives the sliding bar $E = v B d$ , the Lorentz force $F = I B d$ 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.

Sliding-bar sliding-contact model: segmentwise emf, force integral and power balance

Statement

Facts & Assumptions

Proof

Source notes

Depends on

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Sources