The document summarizes the betatron, a device that accelerates electrons to high energies. It works on the principle of transformer induction, where a changing magnetic field induces an electric field that accelerates electrons in a circular path. The betatron consists of an evacuated doughnut chamber with electromagnets that produce an increasing magnetic field. This causes an induced electric field to accelerate electrons injected from a cathode, keeping them moving in a constant radius orbit through magnetic force balance. The maximum energy electrons attain is directly proportional to the strongest magnetic field achieved.

1. R.SUYA PADHRA HARIDHA
ASSISTANT PROFESSOR
DEPARTMENT OF PHYSICS
A.P.C.MAHALAXMI COLLEGE FOR WOMEN
THOOTHUKUDI, TAMILNADU
Betatron

2. Betatron is a device used to accelerate electrons/beta
particles to very high energies.
It was developed by Donald William Kerst

3. Principle
 It is same as that of the transformer. In transformer,
if an alternating current is passed through the
primary coil an alternating magnetic field will appear
in the coil. This field produces an induced e.m.f. in
the secondary coil. Similarly the changing magnetic
flux induces an e.m.f. tangentially along a circular
path for the electron which accelerates the electrons
to high energies. The electrons is kept accelerating in
circular path of constant radius with the help of
increasing magnetic field.

4. Construction
 The Betatron consists of an evacuated doughnut
chamber in which electrons are produced by
indirectly heated cathode.
 The doughnut tube is placed between two strong
electromagnet such that, when the a.c current is
passed in the electromagnets the flux increases in the
centre of doughnut (single coil).

6. Working
 When the electron appears at K (cathode) in
doughnut tube and the electromagnets are energized
the magnetic field increases, the increasing magnetic
field has two effects
(i) Induced e.m.f. is produced in electron orbit by
changing magnetic flux that gives an additional
energy to electron. According to Faraday’s law
induced e.m.f. E= - dφ/dt

7. (ii) A radial force (magnetic force) is produced by
action of magnetic field whose direction is
perpendicular to the electron orbit, velocity which
keeps the electron moving in circular path. The
magnetic force is balanced by centripetal force,
qvB=mv2/r

8. Induced e.m.f in the coil from Faraday’s law of electromagnetic induction
E= dφ/dt  (1)
Work done on an electron in one revolution
W = e.m.f. * charge of electron
W=e * dφ/dt  (2)
Work done = tangential Force ‘F’ on electron x
distance traveled in one revolution
W=F* 2πr =e * dφ/dt
F=(e/ 2πr ) * dφ/dt  (3)
The electron moves in circular path. The magnetic
force is balanced by centripetal force,
evB=mv2/r

9. eBr=mv=p  (4)
From Newtons second law radial force
F=dp/dt = d(mv)/dt
F= d(eBr)/dt
In order to maintain path of constant radius (r is
const.)
F=er( dB/ dt)  (5)
Equating equations (3) & (5)
F=er( dB/ dt)=(e/ 2πr ) * dφ/dt

10. dφ/dt = 2πr2 (dB/ dt )  (6)
Integrating the above equation
φ = 2πr2 B  (7)
The relation is known as Betatron condition.
The electron moves in circular path of constant radius,
the magnetic flux within the orbit of radius R is always
twice what it would have been if magnetic field were
uniform throughout the orbit.

11. Energy gained by electron
The particles have maximum energy when the
magnetic field is at its strongest value but the formula
used for the cyclotron will not work for Betatron
because the electron motion is relativistic.
However, if the total energy is much greater than the
rest energy then
E=pc  (8)
p=E/c
mv=E/c
E= mvc  (9)

12. For electron in the stable orbit,
centripetal force = magnetic force,
The momentum of the electron will
eBr=mv=p  (10)
hence Energy of the electron can be given by,
E=Berc  (11)

13. Thank You