# Betatron

## The Betatron

Betatron is a device used for particle acceleration. We call the Betatron a type of particle accelerator that helps to accelerate the electrons with high speed in a circular orbit. To do so, it uses the electric field induced due to the alternating magnetic field.

The term ‘Betatron’ is derived from the beta-particles of highly energized electrons. This concept got originated by a scientist named Rolf Widerøe. He developed the ‘induction accelerator’ which got failed later.

Max Steenbeck was a scientist from Germany who had developed a method to accelerate electrons in 1935. However, the concepts are originally adapted from Rolf Widerøe. Since his experiment on the induction accelerator was unsuccessful, he was unable to develop the project.

In 1940, the cyclotron was the first particle accelerator discovered by Ernest Lawrence. In this article, we will learn about the betatron oscillation, betatron particle accelerator, etc.

### Betatron Oscillation

The oscillation of particles in all circular accelerators about their equilibrium orbits is known as Betatron oscillation. In the horizontal and vertical planes, these oscillations are stable around the equilibrium orbit.

We can obtain Betatron oscillation from Hill’s equation:

$\frac{d^{2}x}{ds^{2}}$ + K(s)x = 0

### Betatron Accelerator

A Betatron consists of a doughnut-shaped vacuum chamber surrounded by coils. The two ends of the coils are attached to an alternating voltage source. As a result, the coil produces an alternating magnetic field in a direction perpendicular to the doughnut-shaped vacuum chamber.

The working principle of this device is dependent on two phenomena, such as:

1. Lorentz Magnetic Force

2. Electromagnetic Induction

### Betatron Particle Accelerator

The Lorentz magnetic force acting on a charged particle starts to move in some external magnetic field. Electromagnetic induction is a phenomenon where an induced EMF is developed in a circle. Also, there is a variation of magnetic flux linked with that circle.

During the first quarter of the magnetic field cycle, the electrons are injected from the filament into the chamber. At the same time, the magnetic field rapidly starts rising from zero value.

The accelerated electrons are created to strike the target during the time of completion of the 1st quarter of the cycle. The velocity of the injection of electrons is kept in a direction perpendicular to the external magnetic field. After that, electrons follow a circular orbit.

Also, due to continuous variation in the magnetic field, an EMF is induced in the chamber due to electromagnetic induction. Here, the induced EMF accelerates the electrons. The arrangements are made in a way that the electrons do not follow a spiral-like path as in the case of cyclotrons.

But the electrons follow a circular path in a fixed radius. All this process takes place during the first quarter of the magnetic field cycle. During the second quarter of the magnetic field cycle, there is a decline in the magnetic field from peak value to zero.

So, the induced EMF decelerates the electrons. What should we do to avoid the loss of energy during the second quarter of the magnetic field cycle? The accelerated electrons are needed to strike the target just after the completion of the first quarter of the cycle.

The working of Betatron is mainly controlled by a formula called Betatron condition or 2: 1 rule.

The relation is:  $(\frac{dϕ}{dt})$ = $2[πR^{2}(\frac{dB}{dt})]$

The energy of an electron at the end of the 1st quarter cycle of the alternating magnetic field is given as:

E= $\frac{ecφ_{0}}{2πR}$

Here,

c = Maximum velocity of the electron

e = Charge on the electron

R = Radius of orbit of an electron revolving in the chamber

B = Magnetic field at any time

ϕ = Magnetic flux at time ‘t’

ф = фsin⁡(ωt) is magnetic flux at any time ‘t’

ω = Angular velocity

### Uses of Betatron

Some of the applications of Betatron are mentioned below:

1. Betatron delivers about 300 MeV of highly energized beam electrons.

2. When the electron beam is required to strike on a metal plate, Betatron is used as X-rays and gamma rays source.

3. X-rays developed from Betatron have huge usage among industries and medical fields.

4. To study the applications of particle-physics, high energy of electrons is needed.

5. It can be a possible mechanism to learn about solar flare.

### Limitations of Betatron

Here are some lists that explain the limitations of betatron:

1. The maximum energy of the particle has an impact on the strength of the magnetic field.

2. The reason for declining in the magnetic field is the physical size of the magnet core and saturation of iron.

3. A betatron acts as a secondary coil of the transformer.

4. It helps to accelerate the electrons only in a vacuum.

5. The process of acceleration can only be conducted within a circular vacuum tube.

6. Betatron is functional under the conditions of the variable magnetic field and constant electric field.

Q1. Assume that a particle of mass m and charge q goes in a circular path of radius r with a uniform magnetic field B. The particle mass is then replaced by a new one. Find the radius of the circular path made by the particle of a new mass m/2.

Ans: We know that the radius of the circular path, r = mv/Bq

Assume, mo is the mass of the new particle, mo = m/2

The radius of the new circular path is:

r0 = (m/2)v/Bq = r/2

Q2. What do you mean by Magnetic Poles?

Ans: Magnetic poles are two ends of a magnet. In these regions, the strongest magnetic field can be experienced. On our planet, there are two magnetic poles viz: the north magnetic pole and the south magnetic pole.

Q3. Explain a Cyclotron in three lines.

Ans:

• A cyclotron was first invented by Ernest O. Lawrence at the University of California, Berkeley, in the middle of 1929–30.

• It helps to accelerate charged particles away from the center of a flat cylindrical vacuum chamber.

• The chamber is a spiral tube.

Q4. Write a short note on Magnetic Flux.

Ans: We can define magnetic flux as the number of magnetic field lines that flows via a surrounded surface. We can manipulate the net magnetic field for a given surface area with the help of magnetic flux.