Let’s assume that the reader has a basic knowledge of classical and relativistic mechanics.
Consier a particle of mass m and speed v. If c is the speed of light, it is customary to define a normalized velocity \beta as \beta =v/c, and a relativistic mass factor \gamma , defined as \gamma=1/\sqrt(1-\beta^2).

Some other important definitions include the relativistic momentum of a particle, p=\gamma mv, the kinetic energy, W=(\gamma-1)mc^2 , the rest energy, mc^2, and the total energy, U=W+mc^2=\gamma mc^2. The nonrelativistic limit applied when \beta <<1.

It is often convenient to convert between velocity, energy, and momentum, and the following relationship are helpful. The conversion from velocity \beta to kinetic energy W is
\gamma=1/\sqrt(1-\beta^2), W=(\gamma-1)mc^2

The inverse conversion is
\gamma =(W+mc^2)/mc^2, \beta=\sqrt(1-1/\gamma^2)

The following relationships between small differences are sometimes useful:
\delta\gamma=\gamma^3\beta\delta\beta, \delta\gamma=\beta\delta(\beta\gamma), \delta W=mc^2\delta\gamma,\delta p=mc\delta(\beta\gamma).

Particle dynamics is obtained from Newton’s law relating the force and the rate of change of momentum:
F=dp/dt=md(\gamma v)/dt
For a particle of charge q in an electromagnetic field, the Lorentz force on particle with charge q and velocity v in an electric field E and a magnetic field B, is given by

From book of RF linear accelerators, second edition.

News Reporter
Dr. Lu


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