1. Chapter VI Applications of static fields
Introduction
Deflection of a charged particle
Cathode-ray oscilloscope
Ink-jet printer
Sorting of minerals
Electrostatic generator
Electrostatic voltmeter
Magnetic separator

2. Chapter VI Applications of static fields
Magnetic deflection
Cyclotron
The velocity selector and the mass spectrometer
The Hall effect
Magnetohydrodynamic generator
An electromagnetic pump
A direct-current motor
Summary

3. 1. Introduction
To discuss some of the applications in their entirety requires knowledge of both electrostatic and magnetostatic fields. For instance, the acceleration of a charged particle in a cyclotron is accomplished by an electric field, whereas the rotation is imparted by a magnetic field.
By presenting the major applications of static fields in one chapter we hope to convince the reader of their importance. We have seen some recently published textbooks that tend to skip over the subject of static fields as if they are of no significance.
If there is not enough time to discuss the applications of static fields in the classroom, we presume that this chapter epitomizes a very good reading assignment for the student
Electromagnetic - Field Theory Fundamentals

4. 2. Deflection of a charged particle
Ignoring the effects of fringing of the electric field lines, the electric field intensity within the parallel plates is
Neglecting the effect of gravitational force on the charged particle, the acceleration in the z direction is
The trajectory of the charged particle
within the parallel plates is
Electromagnetic - Field Theory Fundamentals

5. Example 6.1
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7. 3. Cathode-ray oscilloscope
The velocity of the electron as it exits the anode can be obtained from the gain in its kinetic energy as
The vertical displacement as the electron exits the vertical deflection region x=d
Electromagnetic - Field Theory Fundamentals

8. The velocity u now makes an angle θ with the x axis
The time required by the electron to travel a distance D on emerging from the deflection plates to the screen is
The total vertical displacement of the
electron as it strikes the screen is
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9. Example 6.2
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11. 4. Ink-jet printer
A nozzle vibrating at ultrasonic frequency sprays ink in the form of very fine
Uniformly sized droplets separated by a certain spacing
These droplets acquire charge proportional to the character to be printed while passing through a set of charged plates
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12. Example 6.3
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13. 5. Sorting of minerals Ore separator
Phosphate ore containing granules of phosphate rock and quartz is dropped onto a vibrating feeder
During the rubbing process each quartz granule acquires a positive charge and each phosphate particle acquires a negative charge
The sorting of the oppositely charged particles is accomplished by passing them through an electric field set up by a parallel-plate capacitor
Electromagnetic - Field Theory Fundamentals

14. The velocity and the distance traveled in the x direction is
The motion of the charged particle in the z direction can be described as
The time taken by the charged particle
to exit the parallel-plate region, x=d
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15. The velocity of the charged particle in the z direction is constant
A charged particle follows a straight-line path within the parallel plates and a parabolic path there-after
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16. Example 6.4
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18. 6. Electrostatic generator
Let us now introduce a positively charged small sphere with charge q through the opening into the cavity
If the small sphere is now made to touch the inner surface of the dome, the positive charge of the small sphere will be completely neutralized by the negative charge on the inner surface of the dome
However, the outer surface will still maintain the positive charge q
Electromagnetic - Field Theory Fundamentals

19. r R The potential at any point on the dome
The potential of the small sphere is
The potential difference between the spheres r R
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20. Example 6.5
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21. 7. Electrostatic voltmeter
When the applied voltage is held constant and the pointer takes up its final position at an angel θ
The increase in electrostatic energy is equal to the amount of mechanical work done
Electromagnetic - Field Theory Fundamentals

22. 8. Magnetic separator
Which is designed to separate magnetic form nonmagnetic material
The magnetic pulley consists of an iron shell containing an exciting coil that produces the magnetic field
The nonmagnetic material immediately drops off into a bin while the magnetic material is held by the pulley until the belt leaves the pulley
Electromagnetic - Field Theory Fundamentals

23. 9. Magnetic deflection
The motion of a charged particle in a uniform B field exhibits similar traits, so we expect the charged particle to move in a circular path
By equating the magnetic and centripetal forces acting on the particle with charge q and mass m,
Time period
The time required for the charged particle
to complete one cycle
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24. The distance traveled in one period
Cyclotron frequency
Angular frequency
The distance traveled in one period
The spacing between two adjacent turns of the helical path
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25. Example 6.6
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27. 10. Cyclotron
Requires one electron gun through which the charged particle is made to pass again and again
The electric field will exist only within the gap between the cavities, and the charged particle will gain energy only while passing through the gap
This process continues until the charted particle reaches the outer edge of the D-shaped cavity, where it is ejected out
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28. The kinetic energy of the charged particle
Exit velocity
The kinetic energy of the charged particle
The frequency of the oscillator
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29. Example 6.7
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30. 11. The velocity selector and the mass spectrometer
The operation of a velocity selector (velocity filter) is based upon the Lorentz force
The net force experienced by a charged particle is zero at one particular velocity
A positive ion having a speed of u0 will
pass through the region without
experiencing any force
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31. Velocity selector (Courtesy of National Electrostatic Corp.)
Electromagnetic - Field Theory Fundamentals

32. Mass spectrometer
The ion source produces the positively charged particles, and the velocity selector produces a beam of these charged particles moving with the same speed
Each charged particle will follow a semicircular trajectory before being detected by the ion detector
The radius of the orbit depends upon the mass of
each charged particle
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33. Example 6.8
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34. 12. The Hall effect Hall effect
To determine the density of free electrons in a metal
To measure the magnetic flux density in the air gap of an electric machine
A positive charge moving with a velocity u at right angles to a magnetic field B will experience a force that will tend to move it toward side b of the strip
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35. There will be an excess of positive charges on side b, whereas side a will experience a deficiency of these charges
Hall-effect voltage
This results in a potential difference
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36. Example 6.9
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38. 13. Magnetohydrodynamic generator
Hot ionized gas or plasma is made to flow through a rectangular channel in a plane perpendicular to the uniform magnetic field
HMDs can play a major role in the development of electrical energy from the burning of fossil fuel
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39. Example 6.10
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40. 14. An electromagnetic pump
The magnetic force exerted by the magnetic field on a moving charge has also led to the development of a pumping device without any moving parts
Electromagnetic - Field Theory Fundamentals

41. 15. A direct-current motor
The field winding wound on the two poles of the stationary member of the motor carries a constant current in order to establish the required magnetic flux in the machine
The total torque exerted on the conductors in the armature is
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42. Example 6.11
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43. 16. Summary
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