Medical cyclotrons need efficient sources of protons to...

Medical cyclotrons need efficient sources of protons to inject into their center. In one kind of ion source, hydrogen atoms (i.e., protons with one orbiting electron) are fed into a chamber where there is a strong magnetic field. Electrons in this chamber are trapped in tight orbits, which greatly increases the chance that they will collide with a hydrogen atom and ionize it. One such source uses a magnetic field of $50 \mathrm{mT}$, and the electrons' kinetic energy is $2.0 \mathrm{eV}$. If the electrons travel in a plane perpendicular to the field, what is the radius of their orbits?

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Key Concepts

Lorentz Force

The Lorentz force is the fundamental force acting on a charged particle moving in a magnetic field. When the particle's velocity is perpendicular to the field, the force is always directed toward the center of its circular path, causing the particle to undergo uniform circular motion.

Cyclotron Motion

Cyclotron motion describes the circular movement of a charged particle in a uniform magnetic field due to the Lorentz force. This motion is characterized by a radius that depends on the particle's mass, speed, charge, and the magnetic field strength, following the relation r = mv/(qB).

Kinetic Energy and Velocity Relationship

The kinetic energy of a particle is directly related to its velocity by the equation KE = ½mv². This relationship enables the calculation of the particle's speed from its known kinetic energy and mass, which is essential for subsequent motion analysis in a magnetic field.

Unit Conversion and Dimensional Consistency

Accurate physical calculations require consistent units, so converting measurements such as energy from electron volts to joules or magnetic field strength from millitesla to tesla is crucial. This ensures that all quantities are in compatible units for applying formulas like the one for the cyclotron radius.

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