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For motion perpendicular to a uniform magnetic field, the Lorentz force gives $r=p/(|q|B)$. An ultrarelativistic massive particle has $pc\simeq E$, hence $r\simeq E/(|q|Bc)$. Total energy and kinetic energy differ by the rest energy.

For constant-density incompressible ideal magnetohydrodynamics about $\mathbf{B}=Jr{\mathbf{e}}_{\varphi }$, the cylindrical basis contributions make the linearized Lorentz force equal to $imJ\mathbf{b}+2J{\mathbf{e}}_z\times \mathbf{b}$, apart from a magnetic pressure gradient. The induction equation reduces to $i\omega \mathbf{b}=imJ\mathbf{u}$. Taking the curl after eliminating velocity gives the displayed relation, with $\tilde{J}=J/\sqrt{{\mu }_0\rho }$. A second curl gives a vector Helmholtz equation and squared frequencies ${\omega }^2={\tilde{J}}^2[m^2\pm 2|mk|/({\alpha }^2+k^2{)}^{1/2}]$. The lower $|m|=1$ branch grows when $3k^2>{\alpha }^2$.

Under the magnetic-only Lorentz force, planar velocity has constant magnitude and an accumulated phase determined by the time integral of the axial field. For $B_z(t)=\beta t$, the planar position involves Fresnel integrals. With initial planar velocity $(1,0)$ and zero initial position, its limit is $(\sqrt{\pi m/(4q\beta )},-\sqrt{\pi m/(4q\beta )})$. The model prescribes a magnetic-only force; a physical time-dependent field may also produce an induced electric field.

// Target: number-theory.bigb

A velocity selector uses perpendicular electric field and magnetic field to transmit a narrow range of speeds along a chosen beam direction. Assume uniform static electric field $\mathbf{E}=E\hat{\mathbf{y}}$ and magnetic field ${\mathbf{B}}_s=B_s\hat{\mathbf{z}}$, with $E,B_s>0$, and an incident velocity $\mathbf{v}=v\hat{\mathbf{x}}$. The cross product gives $\mathbf{v}\times {\mathbf{B}}_s=-v{B}_s\hat{\mathbf{y}}$, so the transverse Lorentz force isFor an electron, the electric force points downward and the magnetic force upward. A faster electron initially bends upward; a slower one initially bends downward. At the selected speed, both forces cancel along the whole straight trajectory. The cancellation is independent of mass and nonzero electric charge; it selects velocity, not a particular charge-to-mass ratio. A component of velocity parallel to the magnetic field is not constrained by the cancellation, so the incident beam must also have its direction restricted.

A real exit slit accepts a finite range of speeds and incident directions. A zero displacement at one screen alone does not establish zero force throughout the path: a curved trajectory could return to that screen position. The ideal velocity selector interpretation requires a straight aligned beam, known overlapping electric field and magnetic field, and sufficiently narrow entrance and exit slits. Its useful role in the Thomson charge-to-mass experiment teaching model is to supply speed independently of the subsequent magnetic bending.

With small viscosity, the azimuthal cylindrical balance permits Lorentz force torque to be cancelled by viscous stress. For ordinary no-slip rotating boundaries an Ekman layer gives stress of order $\rho \Omega LU\sqrt{E}$, larger than the order-$E$ bulk viscous correction at fixed velocity. Thus bounded small-$E$ flows approach a magnetostrophic Taylor state but need not satisfy its constraint exactly at finite viscosity. An order-one uncompensated magnetic torque would require a divergent flow as $E\to 0$, invalidating the assumed small-inertia balance. Stress-free boundaries remove the leading no-slip friction and require their own torque/regularity conditions.