In a cyclotron (one type of particle accelerator), a...

In a cyclotron (one type of particle accelerator), a deuteron (of mass 2.00 u) reaches a final speed of 10.0$\%$ of the speed of light while moving in a circular path of radius 0.480 $\mathrm{m}$ . The deuteron is maintained in the circular path by a magnetic force. What magnitude of force is required?

Key Concepts

Centripetal Force in Circular Motion

This concept refers to the net force required to keep an object moving in a circular path. For uniform circular motion, the magnitude of the centripetal force is given by the formula F = m v² / r, where m is the mass of the object, v is its speed, and r is the radius of the circular path. Understanding this principle is essential for analyzing any system where a continuous inward force is necessary to maintain circular motion, such as in particle accelerators.

Magnetic Force on a Charged Particle

When a charged particle moves through a magnetic field, it experiences a magnetic (Lorentz) force perpendicular to both the velocity of the particle and the magnetic field. This force is responsible for causing circular motion when the velocity and magnetic field are perpendicular. The magnetic force can be equated to the centripetal force needed for circular motion, linking electromagnetic theory with dynamics in accelerators.

Cyclotron Operation and Design

A cyclotron is a type of particle accelerator that uses a constant magnetic field to bend charged particles into a semicircular path while an electric field accelerates them each time they cross the gap between the dees. The interplay between the magnetic force (providing the centripetal force) and the accelerating electric field allows particles to gain energy incrementally. This device illustrates fundamental principles of electromagnetism and circular motion as applied in modern physics and engineering.

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