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Fixed-charge and scaled small-body limits

Physical scope

Algebraic consequences of the conditional static spherical exterior-energy bound for a fixed-charge point limit and a scaled family with charge and mass tending to zero. No existence of a material family, experimental process, or universal point-particle limit is asserted.

Example

Assume the static Coulomb-exterior support-radius hypotheses of Coulomb-exterior support-radius energy bound, so every member obeys $ℓ \geq q^{2} / (8 π ε_{0} M c^{2})$ .

(a) For fixed $q \neq 0$ and fixed finite $M > 0$ , the lower bound is the positive constant $q^{2} / (8 π ε_{0} M c^{2})$ . Thus no family satisfying these hypotheses can have $ℓ \to 0$ while keeping both $q$ and $M$ fixed.

(b) For $q (λ) = λ q_{0}$ , $M (λ) = λ M_{0}$ , and $ℓ (λ) = λ ℓ_{0}$ with $λ > 0$ , $q_{0} \neq 0$ coulombs, $M_{0} > 0$ kilograms, and $ℓ_{0} > 0$ metres, the same necessary condition reduces to $ℓ_{0} \geq q_{0}^{2} / (8 π ε_{0} M_{0} c^{2})$ , independent of $λ$ . The inequality therefore does not rule out this scaling, although it does not construct a body family or prove that its interior energy remains admissible.

Facts & Assumptions

Given: The static spherical exterior-energy bound, a fixed-charge comparison with $q \neq 0$ , and positive scaling parameter $λ$ with fixed $q_{0} \neq 0$ , $M_{0} > 0$ .

[A1]

Case (a) holds $q \neq 0$ and finite $M > 0$ fixed; case (b) sets $q = λ q_{0}$ , $M = λ M_{0}$ , and $ℓ = λ ℓ_{0}$ with $λ > 0$ .

[F1]

Every body satisfying the stated spherical exterior field and nonnegative energy-accounting assumptions obeys $ℓ \geq q^{2} / (8 π ε_{0} M c^{2})$ . Coulomb-exterior support-radius energy bound

Verification

technique · direct

1.1givenA1F1algebra

In case (a) of [A1], [F1] gives $ℓ \geq q^{2} / (8 π ε_{0} M c^{2})$ . The numerator is strictly positive and the denominator is finite positive, so this lower bound is positive and independent of $ℓ$ ; consequently $ℓ$ cannot tend to zero under the fixed- $q, M$ assumptions.

2.1givenA1F1algebra∎

In case (b) of [A1], substitute $q = λ q_{0}$ , $M = λ M_{0}$ , and $ℓ = λ ℓ_{0}$ into [F1]. Since $λ > 0$ , division by $λ$ gives $ℓ_{0} \geq q_{0}^{2} / (8 π ε_{0} M_{0} c^{2})$ . Thus the necessary size ratio is scale-independent. The argument is only a necessary condition on members that already satisfy the static model; it supplies no construction, stability result, or empirical inference.

Remarks

This comparison separates a fixed nonzero charge and fixed mass limit from a scaling in which both charge and mass vanish. It does not establish that any such family solves a closed matter or Maxwell evolution model.

Depends on

Coulomb-exterior support-radius energy bound