The velocity selector

The formula $r = mv/(qB)$ contains no fewer than four quantities ($m$, $v$, $q$, $B$): a single radius cannot yield just one of them. The laboratory strategy is to cascade two devices. The first, the velocity selector, fixes the speed of the beam by filtering it.

In a region where an electric field $\vec{E}$ and a magnetic field $\vec{B}$ coexist perpendicular to each other, a particle feels the total Lorentz force:

$\vec{F} = q(\vec{E} + \vec{v}\times\vec{B})$

If the velocity $\vec{v}$ is orthogonal to both fields, the electric force $qE$ and the magnetic force $qvB$ act along the same line in opposite directions. They balance exactly when

$qE = qvB \quad\Longrightarrow\quad \ev{v_\text{sel} = \frac{E}{B}}$

Velocity selector: crossed $\vec{E}$ and $\vec{B}$. Only particles with $v = E/B$ carry straight on; the others deflect and end up against the walls.

The particle with velocity exactly $E/B$ carries straight on and passes through. Faster or slower particles are deflected and end up against the walls. The selector therefore lets through a beam monochromatic in velocity, independent of $m$ and $q$: whichever particle emerges, it has the same velocity $E/B$.

Try it — interactive simulation

In the animation below a fan of particles with different speeds enters from the left: tune $E$ and $B$ and discover that only the one with $v = E/B$ leaves straight through the slit.

Velocity selector: tune the electric and magnetic fields and watch which particles travel straight through the slit.

Links

Topics: Magnetismo Concepts: Forza di Lorentz · Campo elettrico

Related exercises: Esercizio svolto — carica in campo uscente · Frequenza di ciclotrone di un protone · Ranking della forza di Lorentz (quattro angoli)