Introduction
A charged particle that goes through a region where both electric and magnetic fields are present cannot be attributed to be moving as a result of either one of the elements of force. Instead, the forces due to the electric and magnetic fields are present simultaneously and superpose to give a complex and thought-provoking dynamical response. It is the combination of these effects that forms the basis of a host of modern scientific equipment, including the velocity selector and the cyclotron.
Here we shall consider systematically the forces acting on a charged particle under the simultaneous influence of electric and magnetic fields, derive the main conditions of equilibrium and acceleration, and finally consider the working examples.
Forces on a Charged Particle in Combined Fields
1. Lorentz Force Law
The force acting on a charged particle of charge q, moving with velocity v, in the presence of both electric field E and magnetic field B is given by the Lorentz force law:
F = q(E + v × B)
This equation has two components:
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Electric force:
FE = qE -
Magnetic force:
FB = q(v × B)
Thus, the total force is simply the vector sum:
F = FE + FB
2. Special Case: Perpendicular Field Configuration (Fig.)
Let us consider the case where:
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The electric field is along the y-axis (E = Eĵ),
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The magnetic field is along the z-axis (B = Bk̂),
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The velocity of the particle is along the x-axis (v = vî).
Now,
FE = qEĵ
and
FB = q(v × B) = q(vî × Bk̂) = -qvBĵ
Thus,
F = q(E - vB)ĵ
From this, it is clear that the electric force and magnetic force act in opposite directions.
Velocity Selector
1. Working Principle
If the magnitudes of electric and magnetic forces are equal, they cancel out, and the particle moves undeflected. This condition is:
qE = qvB ⇒ v = E/B
Hence, only particles with velocity v = E/B pass through undeflected.
This arrangement is known as a velocity selector.
2. Applications
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Mass Spectrometer: Used to select charged particles of specific velocities for measuring charge-to-mass ratio (e/m).
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Thomson’s Experiment (1897): Used by J. J. Thomson to determine the e/m ratio of electrons.
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Particle Filters: Applied in particle accelerators to ensure beams have uniform velocities.
Thus, the velocity selector acts as a filter for charged particles, irrespective of their charge or mass.
Cyclotron
1. Introduction and Principle
The cyclotron is a particular type of particle accelerator, originally proposed by E.O. Lawrence and M.S. Livingston in 1934. It enables the acceleration of charged particles, such as protons, deuterons, and alpha particles to very high kinetic energies.
The principle behind the operation of the cyclotron is based on the principle that a charged particle that passes through a magnetic field perpendicularly will move in a circular motion with a frequency that is independent of its speed or orbital radius.
2. Structure of a Cyclotron (Fig.)
The cyclotron consists of two hollow semicircular metallic enclosures called dees D1 and D2, which are placed in a uniform magnetic field of B.
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A time-dependent potential difference between the dees is produced by an oscillator.
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The central point P, where the particles are introduced, is accelerated, and the particles go through the interstitial gap.
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In the dees, it is under the magnetic field that the particles can move in semicircular motions.
3. Motion of Particles in a Cyclotron
The particle moves in a circular orbit under the influence of the magnetic field. The time period of one revolution is:
T = 2πm / qB
Thus, the frequency of revolution (called cyclotron frequency) is:
νc = qB / 2πm
Notice that this frequency is independent of the speed and radius of the orbit.
4. Resonance Condition
The oscillator is tuned to supply an alternating electric field of frequency equal to the cyclotron frequency:
νa = νc
This ensures that the particle always receives acceleration in the correct direction each time it crosses the gap.
5. Energy of Accelerated Particles
If R is the radius of the particle’s final orbit, then the velocity is:
v = qBR / m
The corresponding kinetic energy is:
E = 1/2 * mv2 = q2B2R2 / 2m
Thus, the cyclotron can accelerate particles to very high energies.
Applications of Cyclotron
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Nuclear Physics Research – Used to bombard nuclei with energetic particles.
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Medical Applications – Production of isotopes for diagnosis and treatment (e.g., PET scans).
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Material Science – Implanting ions into solids, modifying materials, and synthesizing new compounds.
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Radioactive Source Production – Used in hospitals to generate radioactive substances for cancer treatment.
Worked Example: Cyclotron Problem (Example 4.4)
Problem
A cyclotron’s oscillator frequency is 10 MHz. What should be the operating magnetic field for accelerating protons? If the radius of its dees is 60 cm, what is the kinetic energy (in MeV) of the proton beam produced?
(Given: e = 1.6×10−19 C, mp = 1.67×10−27 kg, 1 MeV = 1.6×10−13 J)
Solution:
Final Answer:
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Operating Magnetic Field: 0.66 T
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Kinetic Energy of Protons: 7 MeV
Table 4.1: Summary of Key Results
| Quantity | Expression | Remark |
|---|---|---|
| Lorentz Force | F = q(E + v × B) | Total force in combined fields |
| Velocity Selector Condition | v = E/B | Undeflected motion |
| Time Period of Revolution | T = 2πm / qB | Independent of speed |
| Cyclotron Frequency | νc = qB / 2πm | Resonance condition |
| Kinetic Energy in Cyclotron | E = q2B2R2 / 2m | Depends on the maximum radius |
Conclusion
Studies of the dynamics of charged particles in combined magnetic and electric fields not only enhance our theoretical understanding of electromagnetism but also have far-ranging practical consequences. A good example of such use is the velocity selector, which shows how very selective fields can be made to isolate particles in terms of their velocity; the cyclotron shows the synergistic behaviour of opposing electric fields when used in combination with stationary magnetic fields, thus enabling particles to be accelerated to energies many orders of magnitude higher than their native rest energies.
These concepts form a foundation on which a continuum of modern scientific and technological activity is based, including nuclear physics exploration on one end and medical diagnostic techniques on the other end.