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

Explanation

In a cyclotron, charged particles (such as electrons or ions) are accelerated by a magnetic field and an oscillating electric field. The key principle in a cyclotron is that themagnetic field causes the charged particles to move in a circular path, and the oscillating electric field alternates the polarity of the accelerating voltage at just the right frequency to keep the particles accelerating. The kinetic energy of a charged particle in a cyclotron can be expressed as: K.E. = (1/2)mv^2 In a cyclotron, the magnetic field (B) and the frequency of the oscillating electric field (f) are kept constant. As the particle is accelerated, its velocity increases, and its kinetic energy also increases. The radius (R) of the circular path is related to the velocity and the magnetic field strength by the equation: mv = qBR Solving for v, we get: v = (qBR) / m Now, we can substitute this expression for v into the kinetic energy equation: K.E. = (1/2)m[(qBR) / m]^2 K.E. is proportional to (BR)^2. So, in a cyclotron, the kinetic energy of the particle is proportional to the square of the product of the magnetic field strength (B) and the radius (R) of the circular path.