A particle of charge q and mass m is accelerated from rest...

A particle of charge $q$ and mass $m$ is accelerated from rest through a potential difference $V$, after which it encounters a uniform magnetic field $B$. If the particle moves in a plane perpendicular to $B$, what is the radius of its circular orbit?

Key Concepts

Energy Conversion in Charged Particles

This concept explains how a charged particle gains kinetic energy when it is accelerated by an electric potential difference. The electrical work done on the particle is converted into kinetic energy, providing the particle with a velocity that can be calculated by equating the potential energy to the kinetic energy.

Lorentz Force

The Lorentz force is the force experienced by a moving charged particle in a magnetic field. It acts perpendicular to both the velocity of the particle and the magnetic field, and its magnitude is given by the product of the charge, the speed of the particle, and the magnetic field strength. This force is responsible for changing the direction of the particle’s motion.

Centripetal Force and Circular Motion

In a uniform magnetic field, when the velocity of the particle is perpendicular to the magnetic field, the Lorentz force acts as a centripetal force. This causes the particle to follow a circular trajectory. Understanding the relationship between the centripetal force, the mass and velocity of the particle, and the radius of the circular path is crucial for analyzing such motion.

Relationship Between Kinetic Energy and Velocity

This concept involves using the energy conversion principle to derive the velocity of a particle after acceleration. By equating the gained kinetic energy to the work done by the electric field, one can derive an expression for the particle’s speed, which is then used to determine parameters such as the radius of curvature in the magnetic field.

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