A beam of αparticles (helium nuclei) is used to treat a...

A beam of $\alpha$ particles (helium nuclei) is used to treat a tumor located $10.0 \mathrm{cm}$ inside a patient. To penetrate to the tumor, the $\alpha$ particles must be accelerated to a speed of $0.458 c,$ where $c$ is the speed of light. (Ignore relativistic effects.) The mass of an $\alpha$ particle is $4.003 \mathrm{u}$ and its charge is $+2 e .$ The cyclotron used to accelerate the beam has radius $1.00 \mathrm{m}$. What is the magnitude of the magnetic field?

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Key Concepts

Charged Particle Dynamics in Magnetic Fields

When a charged particle moves in a magnetic field, it experiences the Lorentz force, which acts perpendicular to its velocity. This perpendicular force causes the particle to move in a circular path, a principle that underlies many applications in physics such as mass spectrometry and cyclotron operation.

Centripetal Force and Lorentz Force Balance

In uniform circular motion of a charged particle within a magnetic field, the necessary centripetal force (given by mv²/r) is provided entirely by the magnetic Lorentz force (qvB). Equating these forces allows the derivation of relationships between the particle's mass, charge, speed, the radius of its path, and the magnetic field strength.

Non-Relativistic Approximation

When the speeds involved are a significant fraction of the speed of light but relativistic effects are neglected, classical formulas such as p = mv are used without the relativistic correction factors. This assumption simplifies calculations by allowing the use of Newtonian mechanics even when dealing with high-speed particles.

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