(a) What is the speed of a beam of electrons when the...

(a) What is the speed of a beam of electrons when the simultaneous influence of an electric field of $1.56 \times 10^{4} \mathrm{V} / \mathrm{m}$ and a magnetic field of $4.62 \times 10^{-3} \mathrm{T}$ , with both fields normal to the beam and to each other, produces no deflection of the electrons? (b) In a diagram, show the relative orientation of the vectors $\vec{\boldsymbol{v}}$ , $\vec{\boldsymbol{E}}$ and $\vec{\boldsymbol{B}}$ . (c) When the electric field is removed, what is the radius of the electron orbit? What is the period of the orbit?

Key Concepts

Cyclotron Frequency

The cyclotron frequency is the constant frequency at which a charged particle orbits in a magnetic field, defined as ? = |q|B/m for a particle of charge q and mass m. This frequency is independent of the particle’s speed and the radius of its path, which makes it a critical parameter in devices like cyclotrons and in the analysis of charged particle dynamics in magnetic fields.

Cyclotron Motion

Cyclotron motion refers to the circular path taken by a charged particle moving perpendicular to a uniform magnetic field. Under the influence of the steady magnetic force, the particle undergoes uniform circular motion, with the radius of the orbit determined by the balance between the magnetic Lorentz force and the centripetal force required for circular motion. This concept leads directly to determining the radius and period of the particle’s orbit.

Lorentz Force

The Lorentz force is the force exerted on a charged particle moving in electric and magnetic fields. It is given by the equation F = q(E + v × B), where q is the charge of the particle, E is the electric field, v is the velocity of the particle, and B is the magnetic field. This concept is fundamental in understanding how charged particles are accelerated or deflected by electromagnetic fields.

Velocity Selector

A velocity selector is a device that uses perpendicular electric and magnetic fields to filter particles of a specific speed. In a setup where the electric force and the magnetic force on a charged particle cancel each other, only particles with a particular velocity (v = E/B) pass through undeflected. This principle is essential in experiments and applications that require precise control of particle speeds.

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