Particle accelerators are used to create well$-$controlled beams of high$-$energy particles. Such beams have many uses, both in research and industry.

One common type of accelerator is the cyclotron, as shown in the figure below. In a cyclotron, a magnetic field confines charged particles to circular paths while an oscillating electric field accelerates them.

Consider a cyclotron in which a beam of particles of positive charge $q$ and mass $m$ is moving along a circular path restricted by the magnetic field vec$(B)($which is perpendicular to the velocity of the particles$).$

Page $6$ of $10($a$)$ Before entering the cyclotron, the particles are accelerated by a potential difference $V.$ Find the speed v with which the particles enter the cyclotron.

$($b$)$ Find the radius $r$ of the circular path followed by the particles. The magnitude of the magnetic field is B$.$ Express your answer in terms of $v,m,B,$ and q $.$

$($c$)$ Find the period of revolution $T$ and angular frequency $$ for the particles. Express your answer in terms of $m,B,$ and $q.$