You want to build a cyclotron to accelerate protons to a...

You want to build a cyclotron to accelerate protons to a speed of $3.0 \times 10^{7} \mathrm{~m} / \mathrm{s}$. The largest magnetic field strength you can attain is $1.5 \mathrm{~T}$. What must be the minimum radius of the dees in your cyclotron? Show how your answer comes from Newton's second law.

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Key Concepts

Newton's Second Law

This fundamental principle states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration (F = ma). It forms the basis for analyzing the forces in circular motion, such as determining the balance between the force provided by a magnetic field and the need for centripetal acceleration in a cyclotron.

Lorentz Force

The Lorentz force is the force exerted on a charged particle moving through a magnetic field, mathematically expressed as F = qvB for perpendicular motion. It is the driving force in a cyclotron that causes charged particles to follow a circular path by acting perpendicular to their velocity.

Centripetal Force

Centripetal force is the inward force required to keep an object moving in a circular trajectory, given by the equation F = mv²/r. In a cyclotron, the magnetic Lorentz force supplies this centripetal force, allowing the analysis of the particle’s circular motion to determine characteristics like the radius of the particle's path.

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