The document summarizes the working principles of a betatron, which is a device that uses changing magnetic fields to accelerate electrons in a circular path. It describes how Donald Kerst constructed the first betatron in 1941 at the University of Illinois and successfully accelerated electrons, producing x-rays. The betatron operates by injecting electrons into a doughnut-shaped vacuum chamber located between the poles of an electromagnet. As the alternating magnetic field increases over time, it induces an electric field that accelerates the electrons in their circular orbit to high energies without changing their radius, following the betatron condition of ΦB = 2πr2B.

3. 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.”

4. 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.

5. K.W. Kerst with BETATRON

6. Betatron Differs
from cyclotron
The electrons are accelerated
by expending magnetic field.
 The circular orbit has a
constant radius.

7. Different Betatrons
According to their
Generations

10. 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 injected
into 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

11. Constant Radius of Betatron

13. 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.

15. OPERATION
Electrons 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

17. BETATRON CONDITION
Consider 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)

18. 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)

19. 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