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 $2.0 \mathrm{mT}$. If the speed of the electron is $1.5 \times 10^{7} \mathrm{~m} / \mathrm{s}$, determine (a) the radius of the circular path and (b) the time it takes to complete one revolution.

Key Concepts

Lorentz Force

The Lorentz force describes the force experienced by a charged particle moving in a magnetic field. This force, given by F = qvB sin?, acts perpendicular to both the particle's velocity and the magnetic field. In cases where the particle's velocity is perpendicular to the magnetic field (? = 90°), the expression simplifies to F = qvB, which is essential for understanding how charged particles are deflected and can follow curved, circular paths in magnetic fields.

Centripetal Force in Circular Motion

When a charged particle moves in a circular path, a centripetal force is required to keep it moving along that curved trajectory. This force is defined by F = mv²/r, where m is the mass of the particle, v is its speed, and r is the radius of the circle. In the context of charged particles in magnetic fields, the magnetic Lorentz force provides this necessary centripetal force.

Radius of the Trajectory

By equating the magnetic Lorentz force (qvB) with the centripetal force (mv²/r), one can derive the radius of the circular path: r = mv/(qB). This relationship is fundamental in analyzing the motion of charged particles in magnetic fields, as it relates the physical properties of the particle and the properties of the magnetic field to the curvature of the particle's path.

Period of Circular Motion

The period of revolution, or the time taken for one complete rotation, is a key concept in circular motion. It can be calculated from the circumference of the circular path divided by the speed of the particle (T = 2?r/v). When the radius is expressed in terms of other known quantities, the period can also be directly related to the particle's mass, charge, and the magnetic field strength. This concept is crucial for determining the timing and dynamics of circular motion in magnetic fields.

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