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 field of magnitude $B=4.00 \mathrm{~m} \mathrm{~T}$. 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 field of magnitude $B=4.00 \mathrm{~m} \mathrm{~T}$. 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

Lorentz Force on a Charged Particle

This concept explains that a charged particle moving in a magnetic field experiences a force given by F = qvB sin?. For motion perpendicular to the magnetic field (? = 90°), the force is qvB. This force is always perpendicular to the particle’s velocity, causing the particle to change direction but not speed, which is essential in understanding how magnetic fields affect moving charges.

Uniform Circular Motion in a Magnetic Field

When the magnetic force acts as the sole force on the charged particle and is always perpendicular to its velocity, it causes the particle to move in a circular path. The magnetic force in this instance acts as the centripetal force required to maintain circular motion. This framework allows us to derive expressions for the radius of the path (r = mv/(qB)) and the period of the motion (T = 2?m/(qB)).

Cyclotron Motion Parameters

Cyclotron motion refers to the circular motion of a charged particle in a magnetic field. Key parameters of this motion include the radius, speed, and period, all of which are interrelated through the particle’s mass and charge, as well as the magnetic field strength. The period, interestingly, is independent of the speed and radius, and depends only on the charge, mass, and magnetic field.

Charge-to-Mass Ratio Considerations

Determining the identity of the particle (whether it is an electron or a proton) hinges on its charge-to-mass ratio. Since the formulas for radius and period involve both the particle’s charge and mass, comparing the calculated values with known properties of common charged particles can help decide which particle is involved. This concept is crucial for making inferences about a particle's nature based on its motion in a magnetic field.

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