sBetatron

On July 17, 1940, F. Wheeler Loomis, Head of the Department of Physics at the University of Illinois, received a letter from Donald W. Kerst.

“Monday afternoon the electron accelerator started to work. It was its first trial with the new glass doughnut and the new pole pieces. By evening the intensity of the X-rays produced when the electrons strike the target was up to about the effect of 10 mill curies of radium gamma rays (radium at target distance) according to the calibration on the electron-scope.”

Betatron is a device for speeding up electron to

extremely high energies with the help of expending

magnetic field.

It was constructed in 1941 by D.W.Kerst.

K.W. Kerst with BETATRON

Betatron Differs

from cyclotronThe electrons are accelerated

by expending magnetic field.

The circular orbit has a

constant radius.

Different Betatrons According to their Generations

Construction

Betatron consists of highly evacuated angular tube D known

as doughnut chamber.

The chamber is placed between the poles of an

electromagnet excited by an alternating current (frequency of

60 or 180 Hz)

Electrons are produced by electron gun and are injectedinto doughnut at the beginning of each cycle of alternate

current.

The increasing magnetic flux gives rise to a voltage

gradient(electric field) round the doughnut which accelerates

the orbiting electrons

Constant Radius of Betatron

PRINCIPLE

The principle of the betatron is the same as that of a transformer in which an

Alternating current applied to the primary coil induces an alternating current

In the secondary.

In betatron secondary coil is replaced by a doughnut shaped vaccum

chamber.

When the electron is injected in doughnut, the alternating magnetic field has

two effects :

An electromotive force is produced in the electron orbit by changing

magnetic flux that gives an additional energy to the electrons.

A radial force is produced by the reaction of magnetic field whose direction

is perpendicular to the electron velocity which keeps the electrons moving

in the circular part.

OPERATIONElectrons from the electron gun are injected into doughnut shaped

vacuum chamber when the magnetic field is just rising from its zero

value in the first quarter cycle.

The electrons now make several thousand revolution and gain energy.

When the magnetic field has reached its maximum value, the electrons

are pulled out from their orbit.

Either they strike a target and produce X-rays or emerge from the

apparatus through a window

BETATRON CONDITIONConsider an electron is moving in a circular orbit of radius ’r’ in the magnetic field.

Let at any instant, B be the magnetic field at this orbit and the total magnetic flux through

the orbit is ΦB. The flux ΦB increases at the rate of d/dt (ΦB) and the induced e.m.f. In

the orbit is given by

Induced e.m.f. = d /dt (ΦB) .....(i)

work done on the electron in one revolution

= induced e.m.f. X Charge

= - d/dt (ΦB) x e

Thus work done must be equal to the tangential force F acting on the electron

multiplied by the length of the orbit path i.e.,

work done = Force x Distance

= F x 2 π r

Therefore, F x 2 π r = - d/dt (ΦB) x e

F =- e/ 2 π r x {d/dt (ΦB)} ....(ii)

The force F will increase the electron energy and which in turn would tend to

increase the orbit of large radius. In order to maintain the radius of the orbit,

The force experienced by the electron must be counteracted. Suppose the velocity

Of the electron is v and its mass is m. When the electron moves in an orbit of

Radius r under the action of field of magnetic induction B, the inward radial

force B e v is to be equal to the upwards centrifugal force mv2/r .

Therefore, B e v = m v2 / r

m v = B e r ....(iii)

According to Newton’s law, the force is defined at the rate of change of

momentum (p=m v) i.e.,

F = d/dt (B e r)

= er dB/dt ....(iv)

To maintain the radius constant, the value of F given in equation (ii) and

Equation (iv), should numerically, hence

e/ 2 π r x {d/dt (ΦB)} = e r dB/dt

d/dt (ΦB) = 2 π r2 dB/dt

Integrating, we get

ΦB = 2 π r2 B

This is known as Betatron condition