An electron moves in a circle of radius r around a very long...

An electron moves in a circle of radius $r$ around a very long uniformly charged wire in a vacuum chamber, as shown in Fig. $76 .$ The charge density on the wire is $\lambda = 0.14 \mu \mathrm { C } / \mathrm { m } .$ (a) What is the electric field at the electron (magnitude and direction in terms of $r$ and $\lambda ) ? ( b )$ What is the speed of the electron?

Key Concepts

Electric Field Due to an Infinite Line Charge

This concept involves calculating the electric field generated by a uniformly charged, infinitely long wire. The field at a distance r from the wire is derived using Gauss’s law, resulting in the formula E = ?/(2???r), where ? is the linear charge density and ?? is the permittivity of free space. The field is directed radially outward (or inward, if the charge is negative) and decreases inversely with distance, reflecting the cylindrical symmetry of the charge distribution.

Force on a Charged Particle in an Electric Field

A charged particle placed in an electric field experiences a force given by F = qE, where q is the charge on the particle and E is the electric field. This fundamental concept of electromagnetism explains how the electric field applies a force that influences the motion of charged particles, and it is central to analyzing the dynamics of particles in electrostatic environments.

Circular Motion and Centripetal Force

When a particle moves in a circular path, a net inward force, known as the centripetal force, is required to maintain its curved trajectory. This force is described by the equation F_c = mv²/r, where m is the mass of the particle, v is its speed, and r is the radius of the circle. In problems where a charged particle moves in a circle under the influence of an electric field, the electric force provides the necessary centripetal force, allowing one to equate the two and solve for the particle's speed.

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