In Fig. 28-36 , a particle moves along a circle in a region...

In Fig. $28-36$ , a particle moves along a circle in a region of uniform magnetic ficld of magnitude $B=4.00 \mathrm{mT}$ . The particle is either a proton or an electron (you must decide which). It experiences a magnetic force of magnitude $3.20 \times 10^{-15} \mathrm{N}$ . What are (a) the particle's speed, (b) the radius of the circle, and (c) the period of the motion?

In Fig. $28-36$ , a particle moves along a circle in a region of uniform magnetic ficld of
magnitude $B=4.00 \mathrm{mT}$ . The particle is either a proton or an electron (you must decide which). It experiences a magnetic force of magnitude $3.20 \times 10^{-15} \mathrm{N}$ . What are (a) the particle's speed,
(b) the radius of the circle, and (c) the period of the motion?

Key Concepts

Magnetic Lorentz Force

This is the force experienced by a moving charged particle in a magnetic field. It is given by F = |q|vB when the velocity is perpendicular to the magnetic field. This concept is central to understanding how particles are deflected and forced into circular trajectories when moving through a magnetic field.

Circular Motion in Magnetic Fields

When a charged particle moves perpendicular to a uniform magnetic field, the Lorentz force acts as a centripetal force, causing the particle to follow a circular path. By equating the magnetic force (|q|vB) to the centripetal force (mv²/r), one can derive expressions for the radius of the path and relate the particle’s speed, mass, and charge.

Cyclotron Frequency and Period

The cyclotron frequency is defined as ? = |q|B/m, representing the angular frequency of the particle's circular motion. The period of the motion, T, is obtained by T = 2?/? = 2?m/(|q|B). Notably, the cyclotron period is independent of the particle's speed, an important insight in accelerator physics and related applications.

Particle Identification in Magnetic Fields

Determining whether a charged particle is a proton or an electron in a given magnetic field scenario involves comparing the computed quantities (like speed, radius, and period) using known values of mass and charge for each particle. Since electrons and protons have vastly different masses and opposite charges, the values calculated from the magnetic force condition can help in identifying which particle is responsible for the observed behavior.

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