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The exact constant drift solution in subcritical crossed fields

Physical scope

A symbolic specialization of the subcritical crossed-field theorem. The chosen field parameters describe an exact model solution and are not measured values; general initial data still give inverse-boosted magnetic orbits.

Example

Let $B_{0} > 0$ be in tesla and set $E_{0} = c B_{0} / 2$ in newtons per coulomb, so $E = E_{0} e_{1}$ , $B = B_{0} e_{3}$ is subcritical. In Subcritical crossed fields and exact drift-frame orbits, the drift velocity and boost factor are $v_{d} = - (c / 2) e_{2}$ in metres per second and $γ_{d} = 2 / \sqrt{3}$ . The constant-velocity orbit $x (t) = x_{0} + v_{d} t$ in a chart with $x^{0} = c t$ solves the adopted model for any $m > 0$ and constant charge $q$ : its Lorentz force vanishes. This is one exact solution; other initial data yield boosted magnetic orbits.

Facts & Assumptions

Given: The chosen fields, positive $B_{0}$ , $m > 0$ , constant $q$ , and the subcritical external-field model of the theorem.

[F1]

Subcritical crossed fields and exact drift-frame orbits: for $∣ E_{0} ∣ < c ∣ B_{0} ∣$ , $v_{d} = - (E_{0} / B_{0}) e_{2}$ , $γ_{d} = (1 - ∣ v_{d} ∣^{2} / c^{2})^{- 1 / 2}$ , and $E + v_{d} \times B = 0$ ; general solutions are inverse boosts of magnetic orbits.

Verification

1.1givenF1algebra

The ratio $E_{0} / (c B_{0}) = 1 / 2$ verifies the strict subcritical hypothesis. Substituting $E_{0} = c B_{0} / 2$ in [F1] gives $v_{d} = - (c / 2) e_{2}$ and $γ_{d} = (1 - 1 / 4)^{- 1 / 2} = 2 / \sqrt{3}$ .

2.1step 1.1F1algebra∎

Directly, $v_{d} \times B = - (c B_{0} / 2) (e_{2} \times e_{3}) = - (c B_{0} / 2) e_{1} = - E$ . Thus [F1] makes the Lorentz force zero, momentum stays constant, and the straight worldline is an exact orbit. The theorem's inverse-boost construction applies to general initial data, so this example does not assert that every orbit has this constant velocity.

Depends on

Subcritical crossed fields and exact drift-frame orbits

The exact constant drift solution in subcritical crossed fields

Example

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources