Velocity Selectors

Send a positive ion to the right through a region where B points into the page and E points down, from a positive top plate to a negative bottom one. The electric force qE is downward and is the same size whatever the ion is doing — E does not know the speed. The magnetic force qv × B points up and has magnitude qvB, zero for a particle at rest and growing without limit as v rises. Plot both against v and you have a horizontal line and a straight line through the origin: they cross once. Below the crossing the electric force wins and the ion is pushed down; above it the magnetic force wins and the ion is pushed up; at the crossing the net transverse force is exactly zero and the ion goes straight. Setting qE = qvB locates it. With E = 1.50 × 10⁵ V m⁻¹ and B = 0.250 T the crossing sits at 6.00 × 10⁵ m s⁻¹.

Divide qE = qvB by q and the charge is gone: vₛₑₗ = E/B. Nothing on either side mentions mass, so a proton, a ²⁰Ne⁺ ion and a ²³⁸U³⁺ ion all cross the fields above undeflected provided each is doing 6.00 × 10⁵ m s⁻¹. A negative particle is no exception: reverse the sign of q and both forces reverse together, so they still cancel at the same speed, and an electron at 6.00 × 10⁵ m s⁻¹ passes that setting exactly as the proton does. The cancellation even holds at relativistic speeds, because relativity changes how a force produces acceleration, not the force laws themselves — qE and qv × B keep their form, and the balance never invokes m or γ. This blindness to the species is the whole point of the device. A magnetic analyser on its own measures mv/q and cannot separate a slow heavy ion from a fast light one; fix v first and the analyser is left measuring mass.

Write the balance as vectors and the direction falls out: the forces cancel when v = (E × B)/B², a vector of magnitude E/B pointing along E × B. Only that one velocity passes. Send the beam in the opposite direction and the magnetic force reverses while the electric force does not, so the two add instead of cancelling: a particle at the selected speed is swept aside by a net qE + qvₛₑₗ B = 2qE, twice what the plates would give on their own. Enter at a small angle θ instead, and only the component of v along E × B enters the transverse balance, which now reads v cos θ = E/B: a particle arriving at θ = 3.0° must be 0.14% fast to balance. Worse, the tilt gives the magnetic force a component qvB sin θ along the axis, which no electric field perpendicular to the axis can cancel. The selector is therefore only as good as the collimator in front of it.

Nothing is stopped or slowed. Put v = vₛₑₗ + Δv into the transverse force: q(E − vB) = qB(vₛₑₗ − v) = −qB Δv, proportional to the speed error and pointing toward qv × B when the particle is too fast. Treat it as constant over the plate length L, take the transit time as t = L/v, and the particle leaves on a parabola of miss distance y = ½(qB|Δv|/m)(L/v)². Give the 600 V, 0.250 T selector above plates 50.0 mm long: a proton (m = 1.673 × 10⁻²⁷ kg) that is 1.0% fast at 6.00 × 10⁵ m s⁻¹ sees a = qB|Δv|/m = 1.44 × 10¹¹ m s⁻² for a transit of t = 8.25 × 10⁻⁸ s, and leaves 0.49 mm off axis. There is an exact picture behind that parabola: in the frame moving at vₛₑₗ the electric field vanishes to first order in v/c, every particle simply circles at radius r = m|Δv|/(qB), and expanding 1 − cos(ωt) with ω = qB/m to second order returns the parabola.

A selector with no aperture selects nothing, because the fields only bend. Put a slit of half-width h at the exit: a particle passes when its miss distance is smaller than h. Writing the fractional error as δ = Δv/vₛₑₗ and setting y = h in y = qEδL²/(2 m vₛₑₗ²) gives the pass band δ = 2 h m vₛₑₗ²/(q E L²). Everything the designer controls sits in that expression. Halve the slit and the band halves; double the plate length and it falls by four; raise E and B together so that vₛₑₗ is unchanged and the band tightens as 1/B, since δ = 2 h m vₛₑₗ/(q B L²). It also carries m, so the same hardware filters electrons far more sharply than ions at the same speed. Give the proton selector above a slit of half-width h = 0.400 mm and δ = 8.0 × 10⁻³, a window of only ±4.8 × 10³ m s⁻¹ — which is why the proton that was 1.0% fast, missing by 0.49 mm, lands on the jaw instead.

Since vₛₑₗ = E/B is a quotient, the fractional uncertainties add in quadrature: u(vₛₑₗ)/vₛₑₗ = √[(u_E/E)² + (u_B/B)²]. How much that costs depends on how narrow the slit already is. With the 0.400 mm slit above the pass band is ±8.0 × 10⁻³, so a magnet drifting by 0.10% moves the selected speed by 6.0 × 10² m s⁻¹, an eighth of a half-width, and almost nothing happens. Close the slit to 50.0 μm to sharpen the selection and the band collapses to ±1.0 × 10⁻³: the same 0.10% drift now walks the band a full half-width along the speed axis, and the transmitted current falls away although nothing about the beam has changed. Three further limits bite in real hardware. Fringing at the plate ends means the effective length is not the length you measure with a rule, so L is calibrated. The magnet gap must be uniform over the same region, or different parts of the beam see different B. And the particles are assumed non-interacting and collisionless, which fails at high current, where space charge spreads the beam, and at poor vacuum, where a scattered ion can reach the slit with a speed the fields had rejected. One limit is built into the formula: if E/B exceeds c no particle can balance, and every one is swept toward qE.