Cyclotron motion. The archtypical motion of a charged particle in a magnetic field is circular, with the magnetic force providing the centripetal acceleration. In Fig. 5.5, a uniform magnetic field points into the page; if the charge Q moves counterclockwise, with speed v, around a circle of radiu

Question

Cyclotron motion. The archtypical motion of a charged particle in a magnetic field is circular, with the magnetic force providing the centripetal acceleration. In Fig. 5.5, a uniform magnetic field points into the page; if the charge Q moves counterclockwise, with speed v, around a circle of radius R, the magnetic force points inward, and has a fixed magnitude QvB—just right to sustain uniform circular motion:

Accepted Answer

[latex] QvB=m\frac{v^{2}}{R}, or P=QBR, [/latex] (5.3)

where m is the particle’s mass and p = mv is its momentum. Equation 5.3 is known as the cyclotron formula because it describes the motion of a particle in a cyclotron—the first of the modern particle accelerators. It also suggests a simple experimental technique for finding the momentum of a charged particle: send it through a region of known magnetic field, and measure the radius of its trajectory. This is in fact the standard means for determining the momenta of elementary particles.
I assumed that the charge moves in a plane perpendicular to B. If it starts out with some additional speed v parallel to B, this component of the motion is unaffected by the magnetic field, and the particle moves in a helix (Fig. 5.6). The radius is still given by Eq. 5.3, but the velocity in question is now the component perpendicular to B, [latex]v_{⊥}[/latex].

Question 5.1: Cyclotron motion. The archtypical motion of a charged partic...

Related Answered Questions