chapter 2 - coulomb’s law and electric field intensity
TRANSCRIPT
Chapter 2 - Coulomb’s Law and Electric Field Intensity
The Experimental Law of Coulomb
F kQ1 Q2
R2 k
1
4 00 8.85410 12
1
36 10 9
F
mF
Q1 Q2
4 0 R2
FQ1 Q2
4 0 R2a12 a12
r2 r1r2 r1
Electric Field IntensityFt
Q1 Qt
4 0 R1t 2a1t
Ft
Qt
Q1
4 0 R1t 2a1t
EFt
Qt
E r( )Q
4 o r r1 2
r r1r r1
E r( )Q x x1( ) ax y y1( ) ay z z1( ) az
4 o x x1( )2 y y1( )2 z z1( )2
3
2
Electric Field Intensity
E r( )
1
n
m
Qm
4 0 r rm 2am
Electric Field Intensity – Example 2.2
E
6.821
6.821
32.785
EQ
4 0
P P1R1
1
R12
P P2R2
1
R22
P P3R3
1
R32
P P4R4
1
R42
P P4
0
2
1
P P3
2
2
1
P P2
2
0
1
P P1
0
0
1
R4 2.236R4 P4 P
R3 3R3 P3 P
R2 2.236R2 P2 P
V mQ
4 026.963
R1 1R1 P1 P
0 8.85410 12Q 3 10 9
P4
1
1
0
P3
1
1
0
P2
1
1
0
P1
1
1
0
P
1
1
1
Field Due To A Continuous Volume Charge Distribution
Q zyxv
d
d
d
Example 2.3
v z 5 10 6 e 105 z
Q0.02
0.04
z0
2
0
0.01
v z
d
d
d
Q 7.854 10 14
Field of a Line Charge
E
zL
4 0 2z2
3
2
d
EL
2 0
EL
2 0 a
Field of a Line Charge (neglect symmetry)
E z1y1x1vq
4 0
r r1( )
r r1 3
d
d
d
v1 L dz1
r a z az r1 z1 az
R r r1 a z z1( ) az
R 2
z z1( )2 aR
a z z1( ) az
2
z z1( )2
E
z1L dz1 a z z1( ) az
4 0 2
z z1( )2
3
2
d
EL
4 0 a
z1 dz1
2
z z1( )2
3
2
d az
z1z z1( )( )
2
z z1( )2
3
2
d
to to
EL
4 0 a 1
2
z z1( )
2
z z1( )2
az1
2
z z1( )2
EL
4 0 a2
az 0
L
2 0 a
Field of a Line Charge (neglect symmetry)
Field of a Line Charge
Field of a Sheet of Charge
R x2
y2
dEx sdy1
2 0 x2
y2
cos s
2 0 x dy1
x2
y2
to
Exs
2 0
y1x
x2
y2
ds
2 0 atany1
x
Exs
2 0E
s
2 0aN
This is a very interesting result. The field is constant in magnitude and direction. It is as strong a million miles away from the sheet as it is right of the surface.
Streamlines and Sketches of Fields
Cross-sectional view of the line charge.
Lengths proportional to the magnitudes of E and pointing in the direction of E
Streamlines and Sketches of FieldsEy
Ex
dy
dz
E1
a E
x
x2
y2
axy
x2
y2
ay
dy
dx
Ey
Ex
y
x
dy
y
dx
x
ln y( ) ln x( ) C1 ln y( ) ln x( ) ln c( )
y C x
E x y( ) 5 x3 ax 15 x
2 y ay P
4
2
1
dy
dx
Ey
Ex
15 x2 y
5 x3
3y
x
dy
y3
y
x ln y( ) 3 ln x( ) ln C( )
y e3 ln x( )
eln C( )
C
x3
At P x 4 y 2 C 1
given
y e3 ln x( )
eln C( ) C Find C( ) C 128
10 5 0 5 1040
20
0
20
4025.448
25.448
128
xx3
10
xx3
300
xx3
1010 xx
E 400y ax( ) 400x ay A
2
1
2
dy
dx
Ey
Ex
x
yx dx y dx
x2
y2
C At A C 3
x2
3
y2
3 1 y x( ) 3 x
2
4 3 2 1 0 1 2 3 40
1
2
3
43.162
3.154 107
Fy x y( )
Fy x 3( )
3.13.1 x
Field of a Line Charge
MathCAD ExampleForce field in the x = 0 plane.
f y f z
Electrostatic Precipitators for Power Plants
Many countries around the world, including our own, depend on coal and other fossil fuels to produce electricity. A natural result from the burning of fossil fuels, particularly coal, is the emission of flyash. Ash is mineral matter present in the fuel. For a pulverized coal unit, 60-80% of ash leaves with the flue gas. Historically, flyash emissions have received the greatest attention since they are easily seen leaving smokestacks. Two emission control devices for flyash are the traditional fabric filters and the more recent electrostatic precipitators. The fabric filters are large baghouse filters having a high maintenance cost (the cloth bags have a life of 18 to 36 months, but can be temporarily cleaned by shaking or backflushing with air). These fabric filters are inherently large structures resulting in a large pressure drop, which reduces the plant efficiency. Electrostatic precipitators have collection efficiency of 99%, but do not work well for flyash with a high electrical resistively (as commonly results from combustion of low-sulfur coal).
Electrostatic precipitators are not only used in utility applications but also other industries (for other exhaust gas particles) such as cement (dust), pulp & paper (salt cake & lime dust), petrochemicals (sulfuric acid mist), and steel (dust & fumes).