Download Chapter I Electromagnetic field theory

Chapter VI
Applications of static fields
1.
2.
3.
4.
5.
6.
7.
8.
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
9.
10.
11.
12.
13.
14.
15.
16.
Magnetic deflection
Cyclotron
The velocity selector and the mass spectrometer
The Hall effect
Magnetohydrodynamic generator
An electromagnetic pump
A direct-current motor
Summary
2/46
1. Introduction
1.
2.
3.
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
3/46
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

V0 
E   az
L

Neglecting the effect of gravitational force on the charged
particle, the acceleration in the z direction is
az  
u z  az t

z
1 2
azt
2
qV0
mL
x  u xt
T
d
ux
The trajectory of the charged particle
2
within the parallel plates is


qV0 x
z
Electromagnetic - Field Theory Fundamentals
 
2mL  u x 
4/46
Example 6.1
Electromagnetic - Field Theory Fundamentals
5/46
Electromagnetic - Field Theory Fundamentals
6/46
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
1 2
mux  eV1
2

 2e 
u x   V1 
m 
1/ 2
The vertical displacement as the electron exits the vertical
deflection region x=d
eV0  d 
z1 
 
2mL  u x 
2
uz 
edV0
mLux
Electromagnetic - Field Theory Fundamentals
7/46


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 t2  D
edD  1 
z2  u z t2
V0  
mL  u x 

tan  
2
The total vertical displacement of the
electron as it strikes the screen is
z  z1  z2 
V 
d
[0.5d  D]  0 
2L
 V1 
Electromagnetic - Field Theory Fundamentals
ux
uz
ux
8/46
Example 6.2
Electromagnetic - Field Theory Fundamentals
9/46
Electromagnetic - Field Theory Fundamentals
10/46
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
11/46
Example 6.3
Electromagnetic - Field Theory Fundamentals
12/46
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
13/46

The velocity and the distance traveled in the x direction is
ux 

dx
 gt
dt
1 2
gt
2
The motion of the charged particle in the z direction can be
1
x
described as a  q V
u z  az t
z  azt 2
za
z

x
mL
0
The time taken by the charged particle
to exit the parallel-plate region, x=d
 2d 
T  
 g 
1/ 2
Electromagnetic - Field Theory Fundamentals
2
z
g
14/46

The velocity of the charged particle in the z direction is
constant ( t  T )
u z  a zT 
qV0  2d 
mL  g 
1/ 2
for t  T
z  uzt for t  T
z2 

2 2
u z x for t  T
g
A charged particle follows a straight-line path within the
parallel plates and a parabolic path there-after
Electromagnetic - Field Theory Fundamentals
15/46
Example 6.4
Electromagnetic - Field Theory Fundamentals
16/46
Electromagnetic - Field Theory Fundamentals
17/46
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
18/46

The potential at any point on the dome
VR 

The potential of the small sphere is
Vr 

1 Q q 

40  R R 
1  q Q
 

40  R R 
The potential difference between the spheres
V  Vr  VR 
q 1 1 
 

40  r R 
Electromagnetic - Field Theory Fundamentals
r
R
19/46
Example 6.5
Electromagnetic - Field Theory Fundamentals
20/46
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
1
Q
dQ  2 dC
C
C
 Q2  Q
Q2
dWe  d    dQ  2 dC
2C
 2C  C
Q2

dC
2C 2
2
1  Q  dC 1 2 dC
T  
 V
2  C  d 2
d
T  

1 2 dC
V
2
d
Electromagnetic - Field Theory Fundamentals
dW  Td
21/46
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
22/46
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,
m 2
u n  qBu n
R

R
m
un
qB
Time period

The time required for the charged particle
2 R 2m
to complete one cycle
T

un
Electromagnetic - Field Theory Fundamentals
qB
23/46

Cyclotron frequency

Angular frequency

The distance traveled in one period

f 
1
qB

T 2m
  2 f 
qB
m
The spacing between two adjacent turns of the helical path
2m
d  u pT 
up
qB
Electromagnetic - Field Theory Fundamentals
24/46
Example 6.6
Electromagnetic - Field Theory Fundamentals
25/46
Electromagnetic - Field Theory Fundamentals
26/46
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 Dshaped cavity, where it is ejected out
Electromagnetic - Field Theory Fundamentals
27/46

Exit velocity
u

qBR
m
The kinetic energy of the charged particle
1
q2 B2 R2
2
Wk  mu 
2
2m

The frequency of the oscillator
f 
u
2 Rc
Electromagnetic - Field Theory Fundamentals
28/46
Example 6.7
Electromagnetic - Field Theory Fundamentals
29/46
11. The velocity selector and the mass spectrometer

The operation of a velocity selector (velocity filter) is
based upon the Lorentz force
F  q E  qu  B

The net force experienced by a charged particle is zero at
one particular velocity
E  u 0  B

u0 
E
B
A positive ion having a speed of u0 will
pass through the region without
experiencing any force
Electromagnetic - Field Theory Fundamentals
30/46
Velocity selector (Courtesy of National Electrostatic Corp.)
Electromagnetic - Field Theory Fundamentals
31/46

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
qRBB '
m
E
Electromagnetic - Field Theory Fundamentals
32/46
Example 6.8
Electromagnetic - Field Theory Fundamentals
33/46
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
34/46


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
Vba 
Vba  uBw
qnA
t

t 
u
I
u
I
qnA
Electromagnetic - Field Theory Fundamentals
BIw
qAn
35/46
Example 6.9
Electromagnetic - Field Theory Fundamentals
36/46
Electromagnetic - Field Theory Fundamentals
37/46
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
38/46
Example 6.10
Electromagnetic - Field Theory Fundamentals
39/46
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
40/46
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
1
T
Electromagnetic - Field Theory Fundamentals

NIABa z
41/46
Example 6.11
Electromagnetic - Field Theory Fundamentals
42/46
16. Summary
Electromagnetic - Field Theory Fundamentals
43/46
Electromagnetic - Field Theory Fundamentals
44/46
Electromagnetic - Field Theory Fundamentals
45/46
Electromagnetic - Field Theory Fundamentals
46/46
Electromagnetic - Field Theory Fundamentals