An electron is accelerated from rest by a potential...

An electron is accelerated from rest by a potential difference of $380 \mathrm{~V}$. It then enters a uniform magnetic field of magnitude $200 \mathrm{mT}$ with its velocity perpendicular to the field. Calculate (a) the speed of the electron and (b) the radius of its path in the magnetic field.

Key Concepts

Magnetic Force and Lorentz Force

A charged particle moving through a magnetic field experiences a force due to the Lorentz force, which is given by F = qvB sin(?). When the particle's velocity is perpendicular to the magnetic field (? = 90°), the force is maximized and acts at a right angle to the motion, causing the particle to follow a curved path rather than a straight line.

Circular Motion in a Magnetic Field

For a particle moving in a circular path under the influence of a magnetic field, the magnetic Lorentz force provides the necessary centripetal force required for circular motion. Equating the magnetic force, qvB, to the centripetal force, mv²/r, allows one to derive the radius of the curvature of the path. This process is essential in understanding how the geometry of the particle’s trajectory is determined by the balance between its velocity, the magnetic field strength, and its inherent charge and mass.

Conversion of Electric Potential Energy to Kinetic Energy

When a charged particle such as an electron is accelerated from rest through a potential difference, its electric potential energy is converted into kinetic energy. This principle is represented by the equation qV = (1/2)mv², where q is the charge of the particle, V is the potential difference, m is the mass, and v is the velocity. This relationship is fundamental in calculating the speed of the electron after acceleration.

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