Question

A cyclotron operates with a given magnetic field and at a given frequency. If R denotes the radius of the final orbit, the final particle energy is proportional to:

A) 1/R
B) R
C) R^2
D) R^3
E) R^4

Answer 1

In a cyclotron, the final particle energy is proportional to the square of the radius (R²) of the particle's orbit.

Step-by-step explanation:

The final energy of a particle in a cyclotron is directly related to the radius of its orbit. Because the cyclic motion is dominated by the Lorentz force which provides the centripetal force, the particle's kinetic energy (KE) for a given magnetic field strength and charge is proportional to the square of the velocity. Since the radius (R) of the particle's orbit is proportional to the velocity (v) for a constant magnetic field (KE = ½ mv²), this implies that KE ≈ R². Therefore, the final particle energy is proportional to the square of the radius (R²).