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

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

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

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

Example 6.1 Electromagnetic - Field Theory Fundamentals

Electromagnetic - Field Theory Fundamentals

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

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 Electromagnetic - Field Theory Fundamentals

Example 6.2 Electromagnetic - Field Theory Fundamentals

Electromagnetic - Field Theory Fundamentals

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 Electromagnetic - Field Theory Fundamentals

Example 6.3 Electromagnetic - Field Theory Fundamentals

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

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 Electromagnetic - Field Theory Fundamentals

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 Electromagnetic - Field Theory Fundamentals

Example 6.4 Electromagnetic - Field Theory Fundamentals

Electromagnetic - Field Theory Fundamentals

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

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 Electromagnetic - Field Theory Fundamentals

Example 6.5 Electromagnetic - Field Theory Fundamentals

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

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

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 Electromagnetic - Field Theory Fundamentals

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 Electromagnetic - Field Theory Fundamentals

Example 6.6 Electromagnetic - Field Theory Fundamentals

Electromagnetic - Field Theory Fundamentals

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 Electromagnetic - Field Theory Fundamentals

The kinetic energy of the charged particle Exit velocity The kinetic energy of the charged particle The frequency of the oscillator Electromagnetic - Field Theory Fundamentals

Example 6.7 Electromagnetic - Field Theory Fundamentals

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 Electromagnetic - Field Theory Fundamentals

Velocity selector (Courtesy of National Electrostatic Corp.) Electromagnetic - Field Theory Fundamentals

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 Electromagnetic - Field Theory Fundamentals

Example 6.8 Electromagnetic - Field Theory Fundamentals

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 Electromagnetic - Field Theory Fundamentals

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 Electromagnetic - Field Theory Fundamentals

Example 6.9 Electromagnetic - Field Theory Fundamentals

Electromagnetic - Field Theory Fundamentals

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 Electromagnetic - Field Theory Fundamentals

Example 6.10 Electromagnetic - Field Theory Fundamentals

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

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 Electromagnetic - Field Theory Fundamentals

Example 6.11 Electromagnetic - Field Theory Fundamentals

16. Summary Electromagnetic - Field Theory Fundamentals

Electromagnetic - Field Theory Fundamentals

Electromagnetic - Field Theory Fundamentals

Electromagnetic - Field Theory Fundamentals

Electromagnetic - Field Theory Fundamentals