Knetic Energy An electron with kinetic energy 1.20 keV circles in a plane perpendicular to a uni

step 1 Hey so kinetic energy and electron with the kinetic energy is a 1 .2 kilo electron volt and so b

Knetic Energy An electron with kinetic energy 1.20 keV...

Knetic Energy An electron with kinetic energy $1.20 \mathrm{keV}$ circles in a plane perpendicular to a uniform magnetic field. The orbit radius is $25.0 \mathrm{~cm}$. Find (a) the speed of the electron, (b) the magnetic field, (c) the frequency, and (d) the period of the motion.

Key Concepts

Kinetic Energy

Kinetic energy is the energy possessed by an object due to its motion and is given by the formula KE = ½ m v². In problems involving charged particles, this concept is used to determine the speed when the energy and mass of the particle are known. The conversion of energy units, such as from electron volts to joules, is also an essential aspect of applying this formula.

Magnetic Force on a Charged Particle

A charged particle moving in a magnetic field experiences a magnetic (Lorentz) force given by F = qvB when the velocity is perpendicular to the magnetic field. This force acts perpendicular to the direction of motion, causing the particle to move in a circular path, and is fundamental in relating the particle's speed, charge, and the strength of the magnetic field.

Centripetal Force in Circular Motion

For a particle moving in a circular path, the centripetal force required to maintain this motion is m v² / r. In the context of charged particles in a magnetic field, the magnetic force provides the necessary centripetal force, allowing one to set qvB equal to m v² / r to solve for unknown quantities like the magnetic field strength or the particle's speed.

Cyclotron Frequency and Period

The cyclotron frequency is the frequency with which a charged particle orbits in a uniform magnetic field and is given by ? = qB/m or f = qB/(2?m), independent of the particle's speed. The period of the motion is the reciprocal of the frequency, T = 1/f. These concepts are central to understanding the dynamics of charged particles in magnetic confinement systems and in various applications such as cyclotrons.

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