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A circular magnetic orbit with zero axial momentum

Physical scope

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

Example

In Exact relativistic helix in a uniform magnetic field, choose $q B_{0} > 0$ , $p_{01} = p_{*} > 0$ with units kilogram metre per second, $p_{02} = p_{03} = 0$ , and arbitrary $m > 0$ kilograms. Then $S_{0} = \sqrt{m^{2} c^{2} + p_{*}^{2}}$ has momentum units, $γ_{0} = S_{0} / (m c)$ is dimensionless and $Ω = q B_{0} / (γ_{0} m) > 0$ has units inverse seconds. The orbit is the circle

$x^{1} = x_{0}^{1} + \frac{c p_{*}}{S_{0} Ω} \sin (Ω t), x^{2} = x_{0}^{2} - \frac{c p_{*}}{S_{0} Ω} (1 - \cos (Ω t)), x^{3} = x_{0}^{3},$

of radius $p_{*} / (q B_{0})$ , with constant energy $c S_{0}$ . At $t = π / (2 Ω)$ its momentum is $(0, - p_{*}, 0)$ . All values are model parameters, not observations.

Facts & Assumptions

Given: The parameter restrictions above and the adopted magnetic test-particle model in the theorem.

[F1]

Exact relativistic helix in a uniform magnetic field: the momentum and position formulas have signed angular frequency $Ω = q B_{0} / (γ_{0} m)$ , constant energy $c S_{0}$ , and transverse radius $∣ p_{⊥} ∣ / ∣ q B_{0} ∣$ .

Verification

1.1givenF1algebra

With $p_{02} = p_{03} = 0$ and $Ω > 0$ , substitution into [F1] gives $p_{1} = p_{*} \cos (Ω t)$ , $p_{2} = - p_{*} \sin (Ω t)$ and $p_{3} = 0$ . Substitution into the position formulas gives the displayed coordinates and $x^{3} = x_{0}^{3}$ .

2.1step 1.1F1algebra∎

The radius clause of [F1] gives $∣ p_{⊥} ∣ / ∣ q B_{0} ∣ = p_{*} / (q B_{0})$ , and the energy is $c S_{0}$ . At $Ω t = π / 2$ , sine is $1$ and cosine is $0$ , so $p = (0, - p_{*}, 0)$ . The orbit is planar because $p_{03} = 0$ makes $x^{3}$ constant.

Depends on

Exact relativistic helix in a uniform magnetic field

A circular magnetic orbit with zero axial momentum

Example

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources