In a cyclotron, resonance occurs when the frequency of the alternating electric field ($v$) matches the cyclotron frequency ($v_c$). According to the sources, the cyclotron frequency is $v_c = \frac{qB}{2\pi m}$ . For a proton with charge $e$, we have $v = \frac{eB}{2\pi m}$. Rearranging this for the operating magnetic field gives $B = \frac{2\pi m v}{e}$ . The maximum velocity ($v_{max}$) of the proton is attained at the radius of the dees ($R$), where $v_{max} = \frac{eBR}{m} = \omega R$ . Since $\omega = 2\pi v$, we get $v_{max} = 2\pi v R$. The kinetic energy is $K = \frac{1}{2} m v_{max}^2 = \frac{1}{2} m (2\pi v R)^2 = 2m\pi^2 v^2 R^2$ .