Velocity selector
= Velocity selector
{title2=$v_0=E/B_s$}
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> $\mathbf E=E\widehat{\mathbf y}$ and <magnetic field> $\mathbf B_s=B_s\widehat{\mathbf z}$, with $E,B_s>0$, and an incident <velocity> $\mathbf v=v\widehat{\mathbf x}$. The <cross product> gives $\mathbf v\times\mathbf B_s=-vB_s\widehat{\mathbf y}$, so the transverse <Lorentz force> is
$$
F_y=q(E-vB_s),\qquad F_y=0\ \Longrightarrow\ v_0=\frac{E}{B_s}.
$$
For 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.
\Image[/electron-velocity-selector.png]
{title=An ideal <velocity selector> passes <Electrons> with <speed> $E/B_s$}
{description=Original teaching geometry with upper negative and lower positive plates, upward <electric field>, and <magnetic field> out of the page. The three trajectories solve the ideal nonrelativistic <Lorentz force> equations in rescaled units. The right panel shows the opposite <forces> on a selected <Electron>. This is not a drawing of the 1897 apparatus.}
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.