Motion of charged particles in electric and magnetics fields

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Lorentz force:$$\vec{F}=q\vec{E} + q\vec{v}\times\vec{B}.$$
Motion equation of a relativistic particle:$$\frac{d}{dt} \frac{m_0 \vec{v}}{\sqrt{1-(\frac{v}{c})^2}}=\vec{F}.$$
Period of revolution of a charged particle in a uniform magnetic field:$$T=\frac{2\pi m}{qB},$$ where $m$ is the relativistic mass of the particle,$$m=m_0/\sqrt{1-\left (\frac{v}{c}\right )^2}.$$

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Lorentz force:$$\vec{F}=q\vec{E} + q\vec{v}\times\vec{B}.$$
Motion equation of a relativistic particle:$$\frac{d}{dt} \frac{m_0 \vec{v}}{\sqrt{1-(\frac{v}{c})^2}}=\vec{F}.$$
Period of revolution of a charged particle in a uniform magnetic field:$$T=\frac{2\pi m}{qB},$$ where $m$ is the relativistic mass of the particle,$$m=m_0/\sqrt{1-\left (\frac{v}{c} \right )^2}.$$
Betatron condition, that is the condition for an electron to move along a circular orbit in a betatron:$$B_0=\frac{1}{2}\langle B \rangle,$$ where $B_0$ is the magnetic induction at an orbit's point, $\langle B\rangle$ is the mean value of the induction inside the orbit.