Betatron( Cyclic Accelerator)

Betatron ( Cyclic Accelerator)

It's the accelerator to accelerate \(\beta\) -particles (electron) to very high speed and energy. A large evaluated 'Doughnet shapped chamber is place inside changing magnetic field. The charge particle is accelerated due to induced emf and its path is circular due to magnetic field. It works on principle of transformer.

Let \(\phi\) be the amount of magnetic flux and emf induced is \(E=\frac{-d\phi}{dt}\)

The work done on electron due to induced emf is W= e.E

\(=-e\frac{d\\phi}{dt}\) where e= charge of \(\beta\) -particles.

For one complete circular orbit of radius 'r'

\(W=F_e\times 2\pi r\)

\(\therefore\) \(F_e=-\frac{e}{2\pi r}\cdot\frac{d\phi}{dt}\dotsm(1)\)

This force tries to increase the radius of orbit. Due to applied Magnetic field the charge particle experience centripetal force.

i.e \(\frac{mv^2}{r}=Bev\)

or, mv =B e r----(2)

According to Newton 2nd law

\(F_m=\frac{d}{dt}mv\)

\(=\frac{d}{dt}(Ber)\)

\(=er(\frac {dB}{dt})\dotsm(3)\)

For stable circular orbit

|(F_m+F_e=0\)

\( er\frac{dB}{dt}+\frac{-e}{2\pi r}\cdot\frac{d\phi}{dt}=0\)

\(d\phi=2\pi r^2 dB\)

Integrating on both side

\(\int_0^\phi d\phi=2\pi r^2\int_0^B dB\)

\(\phi= 2\pi r^2\cdot B\dotsm(4)\)

where, \(\phi\)= Total magnetic field through stable circular orbit.

Let Bav be the average magnetic field through orbti

\(\therefore\) \(\phi= Bav. \pi r^2\dotsm(5)\)

from (4) and (5)

\(Bav.\pi r^2=B.2\pi r^2\)

\(B=\frac{Bav}{2}\dotsm(6)\)

The strength of magnetic field should decreases progressively as we moved from center toward orbit. The strength of magnetic field at the orbit is half to that of average magnetic field.This condition is known as 'condition of Betatron'. The charge particle only gain energy in duration in which B is increasing (\(\phi\) is increasing )

Let \(\phi=\phi_\circ sin\omega t\dotsm(7)\) be the form of magnetic induced emf(E)=\(-e\frac{d\phi}{dt}\)

\(=-e\frac{d\phi}{dt}[\phi_\circ sin\omega t]\)

\(=-e\phi_\circ \frac{d(sin\omega t)}{dt}\)

The amount of average energy gain by charge particle (electron) in one cycle is ,

\(=\frac{1}{\frac T4}e\phi_\circ\int_0^{\frac T4}\frac{d}{dt}(sin\omega t)dt\)

\(=e\phi_\circ\times\frac{4}{\frac{2\pi}{\omega}}\int_0^{\frac{\pi}{2}}d(sin\omega t)\)

\(=e\phi_\circ\cdot\frac{2\omega}{\pi}\times (sing\frac{\pi}{2}-sin0\))

\(\therefore\) Average energy per=\(\frac{e\phi_\circ 2\omega}{\pi}\)

\(=\frac{e.2\pi r^2 B\cdot2\cdot\omega}{\pi}\) \([\phi_circ=2\pi r^2 B]\)

\(=4 r^2 eB\omega\dotsm(8)\)

the speed of \(\beta\) -particle is approximately equal to the speed of light.

\(\therefore\) distance covered in time (\(\frac T4)\)=\(C\times\frac T4\)

\(= C\times\frac{2\pi}{\omega}\times\frac14\) [\(T=\frac{2\pi}{\omega}]\)

\(=\frac{\pi c}{2\omega}\)

distance covered in one revolution = \( 2\pi r\)

\(\therefore\) Number of revolution made=\(\frac{ Total\; distance}{distance \; covered\;per \; revolution}\)

\(N=\frac{\pi c}{2\omega}\times\frac{1}{2\pi r}\)

\( N=\frac{c}{4\omega r}\dotsm(10)\)

\(\therefore\) Total energy gain in N-revolution =N\(\times\) average energy gain per turn

Total energy=\(4r^2eB\omega\times\frac{c}{4\omega r}\)

\(\therefore\) Total energy=\(CeBr\dotsm(11)\)

Reference:

Reviews of Modern Physics. Lancaster, P.A.: Published for the American Physical Society by the American Institute of Physics, 1952. Print.

Wehr, M. Russell, and James A. Richards. Physics of the Atom. Reading, MA: Addison-Wesley Pub., 1984. Print.

Young, Hugh D., and Roger A. Freedman. University Physics. Boston, MA: Pearson Custom, 2008. Print.

Adhikari, P.B. A Textbook of Physics. 2070 ed. Vol. II. Kathmandu: Sukunda Publication, 2070. Print.