Physics · Ch 3 — Magnetism and Magnetic Effects of Electric Current

Motion of a Charged Particle Under Crossed Electric and Magnetic Field (Velocity Selector)

3.10.3

Motion of a Charged Particle Under Crossed Electric and Magnetic Field (Velocity Selector)

Set up a region where a uniform electric field $\vec E$ (from parallel capacitor plates) and a uniform magnetic field $\vec B$ act simultaneously and perpendicular to each other -- a velocity selector. A charge $q$ entering with velocity $\vec v$ feels a total force $\vec F = q\vec E + q(\vec v\times\vec B)$ ; for a positive charge, the electric force acts one way while the magnetic (Lorentz) force acts the opposite way. When these two exactly balance,

$qE = qvB \qquad\Longrightarrow\qquad \boxed{v = \frac{E}{B}}$ …

Figure 3.48Velocity selector

What this figure shows. A charged particle q moves rightward with velocity v through the gap between two parallel capacitor plates that maintain a uniform electric field E pointing from the positive (top) plate to the negative (bottom) plate, while a uniform magnetic field B (drawn as crosses, into the page) fills the same region. Two force arrows are drawn on the particle: qE pointing downward (the electric force) and q(v x B) pointing upward (the magnetic force); the particle travels straight thr …

Misc Example 3.22Finding the null-deflection potential

Worked out. With E=6.0x10^6 N/C and B=0.83 T, the velocity selector passes undeflected only particles with v=E/B = 6.0x10^6/0.83 approx 7.23x10^6 m/s. An electron accelerated through 200 V acquires speed v200 = sqrt(2eV/m) approx 8.39x10^6 m/s, which is faster than the selector's pass-speed, so at 200 V the electron is deflected, not undeflected. To achieve zero deflection the accelerating potential must instead be chosen so that (1/2)m v^2 = eV gives exactly v=7.23x10^6 m/s, i.e. …

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