An electron moves in a circular path perpendicular to a...

An electron moves in a circular path perpendicular to a uniform magnetic field with a magnitude of 2.00 $\mathrm{mT}$ . If the speed of the electron is $1.50 \times 10^{7} \mathrm{m} / \mathrm{s}$ , determine (a) the radius of the circular path and (b) the time interval required to complete one revolution.

Key Concepts

Period of Revolution

The period of revolution is the time taken by a particle to complete one full circle along its path. It can be determined by relating the circumference of the circular path to the speed of the particle or by using the angular frequency, which is derived from the ratio of the force to the mass of the particle. This concept is essential for understanding the timing of periodic motion in systems involving magnetic forces.

Lorentz Force

The Lorentz force is the force experienced by a charged particle moving in a magnetic field. It is given by the cross product of the particle's velocity and the magnetic field, scaled by the charge. This fundamental concept explains how magnetic fields influence the motion of charged particles, leading to forces that are perpendicular to both the velocity and the magnetic field vectors.

Uniform Magnetic Field

A uniform magnetic field is one in which the magnetic field strength and direction are constant in space. In such a field, the force acting on a moving charge will have a consistent magnitude and direction (relative to the charge's velocity), making it a predictable environment for analyzing particle trajectories, such as circular motion.

Centripetal Force

Centripetal force is the inward force required to keep an object moving in a circular path. For a charged particle moving in a magnetic field, the magnetic (Lorentz) force can act as the centripetal force, enabling the particle to follow a circular trajectory. The balance between this inward force and the tendency of the particle to move in a straight line is critical in circular motion analysis.

Circular Motion

Circular motion refers to the movement of an object along the circumference of a circle. In the context of charged particles in magnetic fields, circular motion results from the constant perpendicular Lorentz force providing the necessary centripetal acceleration. The analysis of circular motion involves understanding the relationships between radius, speed, and the forces involved.

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