ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedverified 2026-10-15 (gpt-6-luna, gpt-6.1-sol, and delegated Codex review/repair agents; see completed run evidence)

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Uniform electric field from rest

Physical scope

A symbolic specialization of the conditional constant-electric-field test-particle theorem with chosen SI model parameters. It is not a measurement or empirical claim.

Example

Choose $m = 1 k g$ , $q = 1 C$ , $E_{0} = 2 N C^{- 1}$ , $B = 0$ , and initial data $x_{0} = 0 m$ , $p_{0} = 0 k g m s^{- 1}$ at $t = 0$ in Exact relativistic orbit in a uniform electric field. Coordinate time $t$ is in seconds. Then $F_{0} = 2 N$ , $p_{1} (t) = F_{0} t$ , and for every finite $t$ the model orbit is

$x^{1} (t) = \frac{c}{F_{0}} (\sqrt{m^{2} c^{2} + F_{0}^{2} t^{2}} - m c), x^{2} (t) = x^{3} (t) = 0,$

with $v_{1} (t) = c F_{0} t / \sqrt{m^{2} c^{2} + F_{0}^{2} t^{2}}$ . This is a symbolic model calculation with chosen parameters, not a report of measured field or particle data.

Facts & Assumptions

Given: The chosen $m, q, E_{0}$ , zero initial position and momentum, and the adopted model assumptions in the theorem.

[F1]

Exact relativistic orbit in a uniform electric field: for $F_{0} = q E_{0} \neq 0$ , $Π = \sqrt{m^{2} c^{2} + ∣ p_{⊥} ∣^{2}}$ , $p_{1} = p_{01} + F_{0} t$ , and the exact position and velocity are the formulas stated there.

Verification

1.1givenF1algebra

Here $F_{0} = q E_{0} = (1 C) (2 N C^{- 1}) = 2 N \neq 0$ , $p_{01} = 0$ , $p_{⊥} = 0$ , and $Π = m c$ . Inserting these values into [F1] gives $p_{1} (t) = F_{0} t$ , zero transverse coordinates, and the displayed $x^{1} (t)$ .

2.1step 1.1F1algebra∎

Substitution in the velocity formula of [F1] gives $v_{1} = c F_{0} t / \sqrt{m^{2} c^{2} + F_{0}^{2} t^{2}}$ and $v_{2} = v_{3} = 0$ . At $t = 0$ , $x^{1} = 0$ and $v_{1} = 0$ ; for finite $t$ , $∣ v_{1} ∣ < c$ because the denominator is strictly larger than $∣ F_{0} t ∣$ .

Depends on

Exact relativistic orbit in a uniform electric field

Uniform electric field from rest

Example

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources