A long solenoid has 100 turns/cm and carries current i. An...

A long solenoid has 100 turns/cm and carries current $i$. An electron moves within the solenoid in a circle of radius $2.30 \mathrm{~cm}$ perpendicular to the solenoid axis. The speed of the electron is $0.0460 c(c=$ speed of light $) .$ Find the current $i$ in the solenoid.

Key Concepts

Circular Motion in a Magnetic Field

When a charged particle moves perpendicular to a magnetic field, the Lorentz force acts as a centripetal force, causing the particle to move in a circular path. By equating qvB to the centripetal force expression (mv²/r), one can relate the particle's speed, the magnetic field strength, and the radius of its circular motion. This relationship is fundamental in analyzing the dynamics of charged particles in magnetic environments.

Charged Particle Dynamics

The study of how charged particles interact with electromagnetic fields, including their trajectories under the influence of forces like the Lorentz force, is a cornerstone of classical electromagnetism. This involves understanding how variables such as particle mass, charge, velocity, and the magnetic field combine to determine motion, especially in contexts like cyclotron motion.

Magnetic Field of a Solenoid

Inside an ideal long solenoid, the magnetic field is uniform and described by the equation B = ?? n i, where ?? is the permeability of free space, n is the number of turns per unit length, and i is the current. This concept is crucial for understanding how the solenoid generates a magnetic environment that can influence the motion of charged particles within it.

Lorentz Force on a Charged Particle

A charged particle moving through a magnetic field experiences the Lorentz force, which is given by F = q(v × B). When the motion is perpendicular to the magnetic field, the magnitude simplifies to F = qvB. This force is essential because it governs how charged particles, like electrons, respond to magnetic fields by altering their trajectories.

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