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Motion of charged particles in electric and magnetics fields

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  • Lorentz force:$$F=qE + q[vB].$$
  • Motion equation of a relativistic particle:$$\frac{d}{dt} \frac{m_0 v}{\sqrt{1-(\frac{v}{c})'}}=F.$$
  • 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,$$m=m_0 l\sqrt{1-(\frac{v}{c})^2}.$$
  • Betatron condition, that is the condition for an electron to move along a circular orbit in a betatron:$$B_0=\frac{1}{2}(B),$$ where Bois the magnetic induction at an orbit's point, (B) is the mean value of the induction inside the orbit.

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