The betatron accelerator is specifically used for accelerating light charged particles at or near the speed of light, which means electrons. There are three basic physics principles which are important in the basic betatron accelerator operation.
1) Faraday's Law of induction
The voltage around a loop containing a changing magnetic field is
[tex]\oint E \space d \ell = V = - \int n \cdot \frac{dB}{dt}dA[/tex]
2) Lorentz force law
The vector force on a charged particle with velocity v moving in a perpendicular magnetic field B is a force perpendicular to the velocity.
[tex]\overrightarrow{F}=q\overrightarrow{v}\times \overrightarrow{B}[/tex]
which leads to
3) The rigidity of a charged particle of mass m in a magnetic field B is
[tex]B\rho=mc^2\frac{\beta\gamma}{c} \space \space Tesla-meters[/tex]
where ρ ≡ radius of curvature R. Combining these three equations will show that if the average magnetic field inside a loop (orbit) of radius R is twice the magnetic field on the orbit, then the electron will gain enough voltage every turn maintain a constant radius R orbit as it gains energy.
Thus the magnetic field dB/dt creating the acceleration also creates the magnetic field B keeping the electron in a constant radius orbit.