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Betatron is nothing but a cyclic particle accelerator which is developed in 1940 to accelerate the negatively charged particles (electrons). It was considered as the first machine to accelerate electrons. Ac current through the circuit produces the magnetic field and this induced an electric field. These two fields cause the electron to accelerate so fast through this circular loop. The working principle of the betatron is explained in detail in the following section. 

Definition

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Betatron is a circular particle accelerator, which is mainly used to accelerate the electrons to high speed. The combined effect of varying magnetic filed and induced electric field causes the acceleration of the electrons through this circular loop.

Working Principle

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The electron accelerator called the betatron makes use of the phenomenon that a time varying magnetic field produces an electric field. The betatron consists of a ring-shaped or 'doughnut' accelerating chamber located between the poles of an electromagnet. Electrons emitted by a source move in a circular path. The magnetic field of the electromagnets performs a dual function in the betatron. First, it causes the electrons to bend and keeps them in a circular orbit and second, it includes a voltage that gives the electrons an acceleration tangential to the orbit. During one revolution along the orbit an electron gains a relatively low energy, of the order of a few volts to at most several tens of kiloelectron volts. But the number of revolutions can be high, eg.,one million, the final energy reaches a value of the order of millions or tens of millions of electron volts.

A characteristic feature of the magnetic circuit is that in addition to the main poles, there are auxiliary magnetic poles, called controlling poles. The accelerating particles are then acted upon by both the magnetic field produced by the main poles and the field generated by the controlling poles. A variable magnetic flux $\Phi$ similar to that in a transformer arises in the magnetic core, which is excited by an AC coil. The time dependence of this flux corresponds to that of the excitation current. In the betatron, the beam of electrons circulating inside the chamber plays the role of the secondary winding of the transformer.The accelerating voltage is obtained as the result of the azimuthal electric field arising from the changing magnetic field by Faraday's law. Along the orbit in which the electrons circulate this electric field produces a small accelerating voltage that causes a steady increase in the energy of the accelerated electron beam.

Accelerator

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Betatron accelerator is a kind of circular particle accelerator, to accelerate electrons to very high speed. In this particle accelerator induced electric field by varying magnetic field is caused to the acceleration of the particle. The modern design of betatron is used to produce high energy X-rays for different applications. It consists of an evacuated circular loop placed in between a strong electromagnet. The windings of the electromagnet are parallel to the circular loop. If an AC current passing through this windings results the varying magnetic field. This varying magnetic fields produces the induced electric fields. This effect of electric field causes the electron to accelerate through this circular loop with a high velocity.

Oscillation

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The magnetic field at the position of the orbit of the electrons in a betatron can be resolved into two components, one along the axis of symmetry of the machine defined as z, the other in the radial direction, r. If the particles are to be in stable motion in the orbit, any displacements in either the z or the r directions must bring restoring forces into play so that the equilibrium orbit is attained again. Detailed analysis show that, if the z component of the magnetic field at the equilibrium orbit position. Then the beam executes betatron oscillations in the z and the r directions with frequencies given by

$\omega_{0}\sqrt{n}$ and $\omega_{0}\sqrt{1-n}$, respectively, where $\omega_{0}$ = $\frac{eB_{0}}{m}$.
The condition for a stable beam implies that the magnetic field must decrease from the center of the machine to the outer edge.