Question

How is energy of particles increased by action of electric field on a charged particle in a cyclotron?

Answer 1

In a cyclotron particle is injected with a very small velocity near the canter of the cyclotron. Due to the Lorentz force acting on it, it starts rotating in a circle but the radius of the locus of the particle increases monotonously. Let us assume that that the DEE's of the cyclotron has a radius $R$. So this is the maximum radius of the path followed by the particle.
Here,
$$\begin{gathered} q{v}_{max}B=m{v_{max}}^2/R \\ \Rightarrow v_{max}=\frac{qBR}{m} \end{gathered}$$
This is the maximum speed achieved by the particle (where, $q,B,R,m$ has usual meaning).
As the velocity increases its KE also increases and becomes
$K{E}_{max}=\frac{m}{2}{\left(\frac{qBR}{m}\right)}^2$
just before exiting the cyclotron. This is very large compared to the initial KE ie. almost zero.