4.4 MOTION IN COMBINED ELECTRIC AND MAGNETIC FIELDS

4.4.1 Velocity selector

You know that a charge q moving with velocity \( \mathbf{v} \) in presence of both electric and magnetic fields experiences a force given by Eq. (4.3), that is, \[ \mathbf{F}=q(\mathbf{E}+\boldsymbol{v} \times \mathbf{B})=\mathbf{F}_{\mathrm{E}}+\mathbf{F}_{\mathrm{B}} \]

4.7

We shall consider the simple case in which electric and magnetic fields are perpendicular to each other and also perpendicular to the velocity of the particle, as shown in Fig. 4.7. We have, \[ \begin{array}{l} \mathbf{E}=E \hat{\mathbf{j}}, \mathbf{B}=B \hat{\mathbf{k}}, \mathbf{v}=v \hat{\mathbf{i}} \\ \mathbf{F}_{E}=q \mathbf{E}=q E \hat{\mathbf{j}}, \mathbf{F}_{B}=q \mathbf{v} \times \mathbf{B},=q(v \hat{\mathbf{i}} \times B \hat{\mathbf{k}})=-q B \hat{\mathbf{j}} \end{array} \] Therefore, \( \mathbf{F}=q(E-v B) \hat{\mathbf{j}} \)

Thus, electric and magnetic forces are in opposite directions as shown in the figure. Suppose, we adjust the value of \( \mathbf{E} \) and \( \mathbf{B} \) such that magnitudes of the two forces are equal. Then, total force on the charge is zero and the charge will move in the fields undeflected. This happens when,

\[ q E=q v B \text { or } v=\frac{E}{B} \](4.7)

This condition can be used to select charged particles of a particular velocity out of a beam containing charges moving with different speeds (irrespective of their charge and mass). The crossed \( E \) and \( B \) fields, therefore, serve as a velocity selector. Only particles with speed \( E/B \) pass undeflected through the region of crossed fields. This method was employed by J. J. Thomson in 1897 to measure the charge to mass ratio (\( e/m \)) of an electron. The principle is also employed in Mass Spectrometer – a device that separates charged particles, usually ions, according to their charge to mass ratio.

4.4.2 Cyclotron

The cyclotron is a machine to accelerate charged particles or ions to high energies. It was invented by E.O. Lawrence and M.S. Livingston in 1934 to investigate nuclear structure. The cyclotron uses both electric and magnetic fields in combination to increase the energy of charged particles. As the fields are perpendicular to each other they are called crossed fields. Cyclotron uses the fact that the frequency of revolution of the charged particle in a magnetic field is independent of its energy. The particles move most of the time inside two semicircular disc-like metal containers, \( D_1 \) and \( D_2 \), which are called dees as they look like the letter \( D \). Figure 4.8 shows a schematic view of the cyclotron. Inside the metal boxes the particle is shielded and is not acted on by the electric field. The magnetic field, however, acts on the particle and makes it go round in a circular path inside a dee. Every time the particle moves from one dee to another it is acted upon by the electric field. The sign of the electric field is changed alternately in tune with the circular motion of the particle. This ensures that the particle is always accelerated by the electric field. Each time the acceleration increases the energy of the particle. As energy increases, the radius of the circular path increases. So the path is a spiral one.

4.8

A schematic sketch of the cyclotron. There is a source of charged particles or ions at \( P \) which move in a circular fashion in the dees, \( D_1 \) and \( D_2 \), on account of a uniform perpendicular magnetic field \( B \). An alternating voltage source accelerates these ions to high speeds. The ions are eventually ‘extracted’ at the exit port.

The whole assembly is evacuated to minimise collisions between the ions and the air molecules. A high frequency alternating voltage is applied to the dees. In the sketch shown in Fig. 4.8, positive ions or positively charged particles (e.g., protons) are released at the centre \( P \). They move in a semi-circular path in one of the dees and arrive in the gap between the dees in a time interval \( T/2 \); where \( T \), the period of revolution, is given by Eq. (4.6),

\[ \begin{array}{l} T=\frac{1}{v_{c}}=\frac{2 \pi m}{q B} \\ \text { or } v_{c}=\frac{q B}{2 \pi m} \end{array} \](4.8)

This frequency is called the cyclotron frequency for obvious reasons and is denoted by \( v_c \) .

The frequency \( v_a \) of the applied voltage is adjusted so that the polarity of the dees is reversed in the same time that it takes the ions to complete one half of the revolution. The requirement \( v_a=v_c \) is called the resonance condition. The phase of the supply is adjusted so that when the positive ions arrive at the edge of \( D_1 \), \( D_2 \) is at a lower potential and the ions are accelerated across the gap. Inside the dees the particles travel in a region free of the electric field. The increase in their kinetic energy is \( qV \) each time they cross from one dee to another (\( V \) refers to the voltage across the dees at that time). From Eq. (4.5), it is clear that the radius of their path goes on increasing each time their kinetic energy increases. The ions are repeatedly accelerated across the dees until they have the required energy to have a radius approximately that of the dees. They are then deflected by a magnetic field and leave the system via an exit slit. From Eq. (4.5) we have,

\[ v=\frac{qBR}{m} \](4.9)

where \( R \) is the radius of the trajectory at exit, and equals the radius of a dee.

Hence, the kinetic energy of the ions is,

\[ \frac{1}{2} m v^{2}=\frac{q^{2} B^{2} R^{2}}{2 m} \](4.10)

The operation of the cyclotron is based on the fact that the time for one revolution of an ion is independent of its speed or radius of its orbit. The cyclotron is used to bombard nuclei with energetic particles, so accelerated by it, and study the resulting nuclear reactions. It is also used to implant ions into solids and modify their properties or even synthesise new materials. It is used in hospitals to produce radioactive substances which can be used in diagnosis and treatment.

Accelerators in India

India has been an early entrant in the area of accelerator-based research. The vision of Dr. Meghnath Saha created a 37" Cyclotron in the Saha Institute of Nuclear Physics in Kolkata in 1953. This was soon followed by a series of Cockroft-Walton type of accelerators established in Tata Institute of Fundamental Research (TIFR), Mumbai, Aligarh Muslim University (AMU), Aligarh, Bose Institute, Kolkata and Andhra University, Waltair.

The sixties saw the commissioning of a number of Van de Graaff accelerators: a 5.5 MV terminal machine in Bhabha Atomic Research Centre (BARC), Mumbai (1963); a 2 MV terminal machine in Indian Institute of Technology (IIT), Kanpur; a 400 kV terminal machine in Banaras Hindu University (BHU), Varanasi; and Punjabi University, Patiala. One 66 cm Cyclotron donated by the Rochester University of USA was commissioned in Panjab University, Chandigarh. A small electron accelerator was also established in University of Pune, Pune.

In a major initiative taken in the seventies and eighties, a Variable Energy Cyclotron was built indigenously in Variable Energy Cyclotron Centre (VECC), Kolkata; 2 MV Tandem Van de Graaff accelerator was developed and built in BARC and a 14 MV Tandem Pelletron accelerator was installed in TIFR.

This was soon followed by a 15 MV Tandem Pelletron established by University Grants Commission (UGC), as an inter-university facility in Inter-University Accelerator Centre (IUAC), New Delhi; a 3 MV Tandem Pelletron in Institute of Physics, Bhubaneswar; and two 1.7 MV Tandetrons in Atomic Minerals Directorate for Exploration and Research, Hyderabad and Indira Gandhi Centre for Atomic Research, Kalpakkam. Both TIFR and IUAC are augmenting their facilities with the addition of superconducting LINAC modules to accelerate the ions to higher energies.

Besides these ion accelerators, the Department of Atomic Energy (DAE) has developed many electron accelerators. A 2 GeV Synchrotron Radiation Source is being built in Raja Ramanna Centre for Advanced Technologies, Indore.

The Department of Atomic Energy is considering Accelerator Driven Systems (ADS) for power production and fissile material breeding as future options.