Part 1:Electrostatics
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Transcript of Part 1:Electrostatics
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PART 1
ELECTROSTATICS
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ELECTROSTATICS
2.1 COULUMB’S LAW
2.2 ELECTRIC FIELD INTENSITY
2.3 LINE, SURFACE & VOLUME CHARGES
2.4 ELECTRIC FLUX DENSITY
2.5 GAUSS’S LAW
2.6 ELECTRIC POTENTIAL
2.7 BOUNDARY CONDITIONS
2.8 CAPACITANCE
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INTRODUCTION
Electromagnetics is the study of the effect of charges at rest and charges in motion.
Some special cases of electromagnetics:
Electrostatics: charges at rest
Magnetostatics: charges in steady motion (DC)
Electromagnetic waves: waves excited by charges in time-varying motion
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Maxwell’sequations
Fundamental laws of classical electromagnetics
Special cases
Electro-statics
Magneto-statics
Electro-magnetic
waves
Kirchoff’s Laws
Statics: 0t
d
Geometric Optics
TransmissionLine
TheoryCircuitTheory
Input from other
disciplines
INTRODUCTION (Cont’d)
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• Electrical phenomena caused by friction
are part of our everyday lives, and can be
understood in terms of electrical charge.
• The effects of electrical charge can be
observed in the attraction/repulsion of
various objects when “charged.”
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INTRODUCTION (Cont’d)
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• Charge comes in two varieties called “positive” and “negative.”
• Objects carrying a net positive charge attract those carrying a net negative charge and repel those carrying a net positive charge.
• Objects carrying a net negative charge attract those carrying a net positive charge and repel those carrying a net negative charge.
• On an atomic scale, electrons are negatively charged and nuclei are positively charged.
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INTRODUCTION (Cont’d)
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• Electric charge is inherently quantized such that the charge on any object is an integer multiple of the smallest unit of charge which is the magnitude of the electron charge e = 1.602 10-19 C.
• On the macroscopic level, we can assume that charge is “continuous.”
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INTRODUCTION (Cont’d)
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COULUMB’S LAW
In the late 18th century, Colonel Charles
Augustus Coulomb invented a sensitive
torsion balance that he used to
experimentally determine the force
exerted in one charge by another.
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COULUMB’S LAW (Cont’d)
He found that the force is proportional to the
product of two charges, inversely proportional
to the square of the distance between the
charges and acts in a line containing the two
charges.
221
RQQkF
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The proportional constant, k is:
mF
mF
36101085.8
912
0
Where the free space permittivity with a value given by:
041 rk ,
COULUMB’S LAW (Cont’d)
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Charge Q1 exerts a vector force F12 in Newton's (N) on charge Q2,
122120
2112
4a
RF
COULUMB’S LAW (Cont’d)
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COULUMB’S LAW (Cont’d)
If more than two charges, use the principle ofsuperposition to determine the force on aparticular charge.
If there are N charges, Q1,Q2...QN locatedrespectively at point with position vectorsr1,r2...rN the resultant force F on a charge Qlocated at point r is the vector sum of the forcesexerted on Q by each charges Q1,Q2...QN
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COULUMB’S LAW (Cont’d)
N
N
N
NQQ
QQQQ
rrrr
rr...
rrrr
rrrrrr
rrF
20
2
22
20
2
1
12
10
1
4
44
N
k k
kkQQ1
304 rr
rrF
Or generally,
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322320
2332
122120
2112
3212
4
4
,
aR
F
aR
F
FFF
QQTOTAL
For example,
COULUMB’S LAW (Cont’d)
2112 FF
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EXAMPLE 1
Suppose 10nC charge Q1 located at (0.0, 0.0, 4.0m)and a 10nC charge Q2 located at (0.0, 4.0m, 0.0). Findthe force acting on Q2 from Q1.
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To employ Coulomb’s Law, first find vector 12R
zy
zy
aa
aa rrR
44
441212
Magnitude of 12R
2432
44 2212
R
SOLUTION TO EXAMPLE 1
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And zyzy aa
aaRRa
21
21
2444
12
1212
Then
nN
zy
zy
aa
aa
aR
F
0.40.4
21
21
2410854.84
10101010
4
212
99
122120
2112
SOLUTION TO EXAMPLE 1 (Cont’d)
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2.2 ELECTRIC FIELD INTENSITY
If Q1 is fixed to be at origin, a second charge
Q2 will have force acting on Q1 and can be
calculated using Coulomb’s Law. We also
could calculate the force vector that would
act on Q2 at every point in space to generate
a field of such predicted force values.
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It becomes convenient to define electric field intensity E1 or force per unit charge as:
2
121 Q
FE
This field from charge Q1 fixed at origin results
from the force vector F12 for any arbitrarily
chosen value of Q2
ELECTRIC FIELD INTENSITY (Cont’d)
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Coulomb’s law can be rewritten as
to find the electric field intensity at any point in space resulting from a fixed charge Q.
RQ a
RE 2
04
ELECTRIC FIELD INTENSITY (Cont’d)
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Let a point charge Q1 = 25nC be located
at P1 (4,-2,7). If ε = ε0, find electric field
intensity at P2 (1,2,3).
EXAMPLE 2
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By using the electric field intensity,
RQ a
RE 2
04
This field will be:
12120
9
41025 aR
E
SOLUTION TO EXAMPLE 2
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zyx aaarrR 4431212
4112 R
Where,
and
aaa
RR
aR
E
zyx
4434110854.84
1025
44
2312
9
123120
122120
SOLUTION TO EXAMPLE 2 (Cont’d)
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ELECTRIC FIELD INTENSITY (Cont’d)
If there are N charges, Q1,Q2...QN locatedrespectively at point with position vectorsr1,r2...rN the electric field intensity at point r is:
N
N
N
NQQrrrr
rrrrrr
rrE
2
01
12
10
1
4..
4
N
k k
kkQ1
304
1rr
rrE
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FIELD LINES
The behavior of the fields can be visualized usingfield lines:
Field vectors plotted within a regular grid in 2D space surrounding a point charge.
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Some of these fieldvectors can easily bejoined by field lines thatemanate from thepositive point charge.
The direction of the arrow indicates the direction of electric fields
The magnitude is given by density of the lines
FIELD LINES (Cont’d)
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The field lines terminated
at a negative point charge
The field lines for a pair
of opposite charges
FIELD LINES (Cont’d)
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2.3 LINE,SURFACE & VOLUME CHARGES
Electric fields due to continuous charge distributions:
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To determine the charge for each distributions:
Line charge:
LL
L
dlQ
dldQ
Surface charge:
SS
S
dSQ
dSdQ
Volume charge:
VV
V
dVQ
dVdQ
LINE,SURFACE & VOLUME CHARGES (Cont’d)
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Infinite Length of Line Charge:
To derive the electric field intensity at any
point in space resulting from an infinite
length line of charge placed conveniently
along the z-axis
LINE CHARGE
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Place an amount of charge in coulombs along the z axis.
The linear charge density is coulombs of charge per meter length,
Choose an arbitrary point P where we want to find the electric field intensity.
mC
L
zP ,,
LINE CHARGE (Cont’d)
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LINE CHARGE (Cont’d)
The electric field intensity is:
zzEEE aaaE
But, the field is only vary with the radial distance from the line.
There is no segment of charge dQ anywhere on the z-axis that will give us . So,E
zzEE aaE
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LINE CHARGE (Cont’d)
Consider a dQ segment a distance z above radial axis, which will add the field components for the second charge element dQ.
The components cancel each other (by symmetry) , and the adds, will give:
zE
E
aE E
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LINE CHARGE (Cont’d)
Recall for point charge,
RQ a
RE 2
04
For continuous charge distribution, thesummation of vector field for each chargesbecomes an integral,
RdQ a
RE 2
04
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LINE CHARGE (Cont’d)
The differential charge,
dzdldQ
L
L
The vector from source to test point P,
z
R
zR
aa aR
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LINE CHARGE (Cont’d)
Which has magnitude, and a
unit vector,
22 z R
22 z
z zR
aaa
So, the equation for integral of continuous charge distribution becomes:
2222204 z
z
z
dz zL
aaE
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LINE CHARGE (Cont’d)
Since there is no component,
a
aE
23220
2322
0
4
4
z
dz
z
dz
L
L
za
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Hence, the electric field intensity at any point ρaway from an infinite length is:
aE
02L
For any finite length, use the limits on the integral.
LINE CHARGE (Cont’d)
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EXAMPLE 3
Use Coulomb’s Law to
find electric field
intensity at (0,0,h) for
the ring of charge, of
charge density,
centered at the origin in
the x-y plane.
L
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SOLUTION TO EXAMPLE 3
By inspection, the ring
charges delivers only
and contribution
to the field.
component will be
cancelled by symmetry.
zdEEd
Ed
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RdQ a
RE 2
04
Each term need to be determined:
SOLUTION TO EXAMPLE 3 (Cont’d)
The differential charge,
addldQ
L
L
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The vector from source to test point,
z
R
haR
aa aR
Which has magnitude, and a
unit vector,
22 ha R
22 ha
ha zR
aaa
SOLUTION TO EXAMPLE 3 (Cont’d)
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SOLUTION TO EXAMPLE 3 (Cont’d)
The integral of continuous charge distribution becomes:
2222204 ha
ha
ha
ad zL aaE
zL h
ha
ad aE2
322
04
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SOLUTION TO EXAMPLE 3 (Cont’d)
Rearranging,
2
02322
04z
L dha
ah aE
Easily solved,
zL
ha
ah aE2
32202
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EXAMPLE 4
An infinite length line of charge
exists at x = 2m and z = 4m. Find the
electric field intensity at the origin.
mnC
L 0.4
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SOLUTION TO EXAMPLE 4
Sketch in three dimensions and the cross section:
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SOLUTION TO EXAMPLE 4 (Cont’d)
The vector from line charge to the origin:
zx aaaR 42
Which has magnitude, and a unit
vector,
20R
zxR aaaa204
202
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Inserting into the infinite line charge equation:
mV
zx
zx
L
aa
aa
aE
4.142.7
2042
2010854.82104
2
12
90
SOLUTION TO EXAMPLE 4 (Cont’d)
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SURFACE CHARGE
Infinite Sheet of Surface Charge:
To derive the electric field intensity at point
P at a height h above a charge sheet of
infinite area (x-y plane).
The charge distribution, is inS 2mC
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SURFACE CHARGE (Cont’d)
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SURFACE CHARGE (Cont’d)
Consider a differential charge,
dddSdQ
S
S
The vector from surface charge to the origin:
zhaaR
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Which has magnitude, and a
unit vector,
22 h
h zR
aaa
22 h R
Where, for continuous charge distribution:
RdQ a
RE 2
04
SURFACE CHARGE (Cont’d)
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The equation becomes:
2222204 ha
h
h
dd zS aaE
zS h
h
dd aE2
322
04
Since only z components exists,
SURFACE CHARGE (Cont’d)
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SURFACE CHARGE (Cont’d)
zS
zS
zS
zS
hh
dhh
h
ddh
aE
a
a
aE
0
0
2122
0
0
2322
0
2
0 0 23
220
2
2
24
4
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SURFACE CHARGE (Cont’d)
A general expression for the field from a sheet
charge is:
NS aE02
Where is the unit vector normal from the
sheet to the test point.Na
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EXAMPLE 5
An infinite extent sheet of charge
exists at the plane y = -2m. Find the electric
field intensity at point P (0, 2m, 1m).
210mnC
S
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SOLUTION TO EXAMPLE 5
Sketch the figure:
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SOLUTION TO EXAMPLE 5 (Cont’d)
The unit vector directed away from the sheet and toward the point P is ya
mV
y
y
NS
a
a
aE
565
10854.821010
2
12
90
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VOLUME CHARGE
A volume charge is distributed over a
volume and is characterized by its volume
charge density, inV 3mC
The total charge in a volume containing a
charge distribution, is found by
integrating over the volume:V
V
V dVQ
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EXAMPLE 6
Find the total charge
over the volume with
volume charge density,
3105
5m
Ce zV
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SOLUTION TO EXAMPLE 6
V
V dVQ The total charge, with volume:
dzdddV Thus,
C
dzdde
dVQ
z
z
VV
14
01.0
0
2
0
04.0
02.0
10
10854.7
55
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VOLUME CHARGE (Cont’d)
To find the electric field intensity resulting from a volume charge, we use:
RV
RdVdQ a
Ra
RE 2
02
0 44
Since the vector R and will vary over the volume, this triple integral can be difficult. It can be much simpler to determine E using Gauss’s Law.
V
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2.4 ELECTRIC FLUX DENSITY
Consider an amount of charge
+Q is applied to a metallic
sphere of radius a.
Enclosed this charged sphere
using a pair of connecting
hemispheres with bigger radius.
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ELECTRIC FLUX DENSITY (Cont’d)
The outer shell is grounded. Remove the ground
then we could find that –Q of charge has
accumulated on the outer sphere, meaning the
+Q charge of the inner sphere has induced the –Qcharge on the outer sphere.
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Electric flux, extends from the positivecharge and casts about for a negative charge. Itbegins at the +Q charge and terminates at the–Q charge.
psi
The electric flux density, D in is:2mC
RaR
D 204
where ED
ELECTRIC FLUX DENSITY (Cont’d)
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This is the relation between D and E, where is the material permittivity. The advantage of using electric flux density rather than using electric field intensity is that the number of flux lines emanating from one set of charge and terminating on the other, independent from the media.
We can find the total flux over a surface as:
dS D
ELECTRIC FLUX DENSITY (Cont’d)
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We could also find the electric flux density, Dfor:
Infinite line of charge:
Where
aE
02L So,
aD2
L
ELECTRIC FLUX DENSITY (Cont’d)
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Where
So, NS aD
2
NS aE02
Infinite sheet of charge:
Volume charge distribution:
RV dV a
RE 2
04 So, R
V dV aR
D 24
ELECTRIC FLUX DENSITY (Cont’d)
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EXAMPLE 7
Find the amount of electric flux through the surface at z = 0 with
and
mymx 3050 ,
243 mCxxy zx aaD
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SOLUTION TO EXAMPLE 7
The differential surface vector is
zdxdyd aS
We could have chosen but the
positive differential surface vector is pointing in
the same direction as the flux, which give us a
positive answer.
zdxdyd aS
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SOLUTION TO EXAMPLE 7 (Cont’d)
Therefore,
C
xdxdy
dxdyxxy
dS
x y
zzx
aaa
D
150
4
435
0
3
0
Why?!
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EXAMPLE 8
Determine D at (4,0,3) if there is a point
charge at (4,0,0) and a line
charge along the y axis.
mC5
mCm 3
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SOLUTION TO EXAMPLE 8
How to visualize ?!
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Let total flux,
LQTOTAL DDDWhere DQ is flux densities due to point charge
and DL is flux densities due to line charge.
Thus,
R
RQ
Q
Q
aR
aR
ED
2
20
00
4
4
SOLUTION TO EXAMPLE 8 (Cont’d)
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SOLUTION TO EXAMPLE 8 (Cont’d)
Where,
za
R3
3,0,00,0,43,0,4
Which has magnitude, and a unit vector,
zz
R aaa 3
33R
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So,
2
3
2
138.0
94105
4
mCm
Q
z
z
RQ
a
a
aR
D
SOLUTION TO EXAMPLE 8 (Cont’d)
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And aD
2LL
Where, 5
340,0,03,0,40,0,03,0,4 zx aaa
So,
218.024.0
534
523
mCmzx
zxL
aa
aaD
SOLUTION TO EXAMPLE 8 (Cont’d)
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Therefore, total flux:
242240
18.024.0318.0
mC
zx
zxz
LQTOTAL
aa
aaa DDD
SOLUTION TO EXAMPLE 8 (Cont’d)
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2.5 GAUSS’S LAW
If a charge is enclosed, the net flux passing through the enclosing surface must be equal to the charge enclosed, Qenc.
Gauss’s Law states that:
The net electric flux through any closed surface isequal to the total charge enclosed by that surface
encQd SD
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It can be rearranged so that we have relation
between the Gauss’s Law and the electric flux.
VVenc
enc
dVQ
Qd
SD
V
VS
dVdQ SD
GAUSS’S LAW (Cont’d)
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GAUSS’S LAW (Cont’d)
Gauss’s Law is useful in finding the fields for
problems that have high degree of symmetry.
• Determine variables influence D and what
components D present
• Select an enclosing surface, called Gaussian
Surface, whose differential surface is directed
outward from the enclosed volume and is
everywhere (either tangent or normal to D)
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GAUSS’S LAW APPLICATION (Cont’d)
Use Gauss’s Law to determine electricfield intensity for each cases below:
Point Charge
Infinite length of Line Charge
Infinite extent Sheet of Charge
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POINT CHARGE
• Point Charge:
It has spherical coordinate
symmetry, where the field
is everywhere directed
radially away from the
origin. Thus,
rrD aD
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For a gaussian surface, we could find the differential surface vector is:
rddrd aS sin2So,
ddrD
ddrDd
r
rrr
sin
sin2
2
aaSD
POINT CHARGE (Cont’d)
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POINT CHARGE (Cont’d)
Since the gaussian surface has a fixed radius,
Dr will be constant and can be taken from
integration to yield
r
r
Dr
ddrDd
2
0
2
0
2
4
sin
SD
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POINT CHARGE (Cont’d)
By using Gauss’s Law, where:
QrD
Qd
r
enc
24
SD
So, which leads to expected result:24 r
QDr
rrQ aE 2
04 See page 16!
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INFINITE LENGTH LINE OF CHARGE
• Infinite length line of charge:
Find D and then E at any point zP ,,
A Gaussian surface containing the point P is placed around a section of an infinite length line of charge density L
occupying the z-axis.
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LINE CHARGE (Cont’d)
An element of charge dQ along the line will give Dρ and Dz. But second element of dQ will result in cancellation of Dz. Thus,
aD DThe flux through the closed surface is:
side
bottomtop
d
ddd
SD
SDSDSD
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Where,
aS , aS
, aS
dzddddd
ddd
sidezbottom
ztop
Then, we know that Dρ is constant on the side of gaussian surface
DhdzdD
dzdDdh
z
side
22
0 0
aaSD
LINE CHARGE (Cont’d)
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The charge enclosed by the gaussian surface:
hdzQ L
h
Lenc 0
encL QhDhd 2SD
We know that,
So,
2LD Thus, as
expected: aE
02L
LINE CHARGE (Cont’d)
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INFINITE EXTENT SHEET OF CHARGE
• Infinite extent sheet of charge:
Determine the field everywhere resulting from an infinite extent sheet of charge ρS
placed on the x-y plane at z = 0.
Locate a point at which we want to find the field along the z axis at height h.
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Gaussian surface must contain this point and surround some portion of the charged sheet.
A rectangular box is employed as the Gaussian surface surrounding a section of sheet charge with sides 2x, 2y and 2z
SHEET OF CHARGE (Cont’d)
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Only a DZ component will be present, and the
charge enclosed is simply:
xy
dydxdSQ
S
y
y
x
xSS
4
No flux through the side of the box, so find
the flux through the top and bottom surface
SHEET OF CHARGE (Cont’d)
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z
bottomzzz
topzzz
bottomtop
Dxy
dxdyD
dxdyD
ddd
42
aa
aa
SDSDSD
SHEET OF CHARGE (Cont’d)
Notice that the answer is independent of the height of the box.
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Then we have:
2
442
Sz
Sz
D
xyDxy
Qd
SD
zS aD
2
or
And electric field intensity, as expected:
NS aE02
SHEET OF CHARGE (Cont’d)
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GAUSS’S LAW (Cont’d)
Related to Gauss’s Law, where net flux isevaluated exiting a closed surface, is theconcept of divergence.
Expression for divergence by applying Gauss’sLaw might be too lengthy to derive, but it can bedescribed as:
V D
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The expression is also called the point form of Gauss’s Law, since it occurs at some particular point in space. For instance,
Plunger stationary – no net movement of molecules
Plunger moves up – net movement where air molecules diverging air is expanding
Plunger pushes in – net flux is negative and molecules diverging air is compressing
GAUSS’S LAW (Cont’d)
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EXAMPLE 9
Suppose:
Find the flux through the surface of a cylinder
with and by evaluating the
left side and the right side of the divergence
theorem.
aD 2
hz 0 a
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SOLUTION TO EXAMPLE 9
Remember the divergence theorem?
V
dVdS DD
We can first evaluate the left side of the divergence theorem by considering:
side
bottomtop
d
ddd
SD
SDSDSD
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A sketch of this cylinder is shown withdifferential vectors.
The integrals over the top and bottom surfaces are each zero, since:
0 zaa
SOLUTION TO EXAMPLE 9 (Cont’d)
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Thus,
3
2
0 0
2
2 ha
dzd
ddh
z
side
a
SDSD
SOLUTION TO EXAMPLE 9 (Cont’d)
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For evaluation of the right side of the divergence
theorem, first find the divergence in cylindrical
coordinate:
31
1
3
D D
SOLUTION TO EXAMPLE 9 (Cont’d)
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Performing a volume integration on this divergence,
3
2
0 0 0
2
2
3
3
ha
dzdd
dzdd
dV
a h
z
V
D
This is the same!
SOLUTION TO EXAMPLE 9 (Cont’d)
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2.6 ELECTRIC POTENTIAL
To develop the concept of electric potential and show its relationship to electric field intensity.
In moving the object from point a to b, the work can be expressed by:
b
adW LF
dL is differential length vector along some portion of the path between a and b
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ELECTRIC POTENTIAL (Cont’d)
The work done by the field in moving the charge from a to b is
b
afieldE dQW LE
If an external force moves the charge against the field, the work done is negative:
b
adQW LE
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We can defined the electric potential difference, Vba
as the work done by an external source to move a charge from point a to point b as:
b
aba d
QWV LE
Where,abba VVV
ELECTRIC POTENTIAL (Cont’d)
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ELECTRIC POTENTIAL (Cont’d)
Consider the potential difference between twopoints in space resulting from the field of apoint charge located at origin, where theelectric field intensity is radially directed, thenmove from point a to b to have:
b
arr
b
aba dr
rQdV aaLE 2
04
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Thus,
ab
br
arba
VVab
Q
rQV
114
4
0
0
The absolute potential at some finite radius from a point charge fixed at the origin:
rQV
04
ELECTRIC POTENTIAL (Cont’d)
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If the collection of charges becomes a continuous distribution, we could find:
rdQV
04
Where,
rdVV
rdSV
rdLV
V
S
L
0
0
0
4
4
4
Line charge
Surface charge
Volume charge
ELECTRIC POTENTIAL (Cont’d)
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N
NQ
QQV
rr...
rrrr
0
20
2
10
1
4
44
N
k k
kQV104
1rr
Or generally,
The principle of superposition, where applied to electric field also applies to potential difference.
ELECTRIC POTENTIAL (Cont’d)
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ELECTRIC POTENTIAL (Cont’d)
Based on figure, if a closed path is chosen, the integral will return zero potential:
Three different paths to calculate work moving from the origin to point P against an electric field.
0 LE d
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EXAMPLE 10
Two point charges -4 μC and 5 μC are
located at (2,1-,3) and (0,4,-2)
respectively. Find the potential at
(1,0,1).
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SOLUTION TO EXAMPLE 10
Let and CQ 41 CQ 52
So,
20
2
10
144 rrrr
QQV
Where,
262,4,12,4,01,0,1
62,1,13,1,21,0,1
2
1
rr
rr
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SOLUTION TO EXAMPLE 10 (Cont’d)
264105
64104
441,0,1
0
6
0
620
2
10
1
rrrrQQV
Therefore,
kVV 872.51,0,1
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The electrostatic potentialcontours from a point chargeform equipotential surfacessurrounding the point charge.The surfaces are alwaysorthogonal to the field lines.The electric field can bedetermined by finding the max.rate and direction of spatialchange of the potential field.
ELECTRIC POTENTIAL (Cont’d)
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Therefore,
VEThe negative sign indicates that the field is pointing in the direction of decreasing potential.
By applying to the potential field:
rr rQ
rQ
rV aaE 2
00 44
ELECTRIC POTENTIAL (Cont’d)
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IMPORTANT!!Three ways to calculate E:
If sufficient symmetry, employ Gauss’s Law.
Use the Coulomb’s Law approach.
Use the gradient equation.
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EXAMPLE 11
Consider a disk of charge ρS, find the
potential at point (0,0,h) on the z-axis and
then find E at that point.
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SOLUTION TO EXAMPLE 11
Find that,
dddSdQ
S
S
and 22 hr
With rdQV
04then,
a
SrddV
0
2
004
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Let and leads to
integral then,
22 hu
How to calculate the integral?
ddu 2
duu 21
hah
hV
S
aS
22
0
0
22
0
2
2
SOLUTION TO EXAMPLE 11 (Cont’d)
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To find E, need to know how V is changing with position. In this case E varies along the z-axis, so simply replace h with z in the answer for V, then proceed with the gradient equation.
zS
zS
z
az
z
az
zzVV
aa
aE
2202201
212
21
2
SOLUTION TO EXAMPLE 11 (Cont’d)
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BOUNDARY CONDITIONS
So far we have considered the existence of electric
field in a region consisting of two different media,
the condition that the field must satisfy at the
interfacing separating the media called “boundary
condition”. Thus, we could see how the fields
behave at the boundary between a pair of
dielectrics or between a dielectric and a conductor.
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First boundary condition can be determined by performing a line integral of E around a closed rectangular path,
BOUNDARY CONDITIONS
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Fields are shown in each medium along with normal and tangential components. For static fields,
0 LE d
Integrate in the loop clockwise starting from a,
0 a
d
d
c
c
b
b
adddd LELELELE
BOUNDARY CONDITIONS
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Evaluate each segment,
221
2
02
0
21
10
1
hEE
dLEdLEd
wEdLEd
NN
h
NNNh
NNN
c
b
T
w
TTT
b
a
aaaaLE
aaLE
BOUNDARY CONDITIONS
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221
2
01
0
22
2
0
2
hEE
dLEdLEd
wEdLEd
NN
h
NNNh
NNN
a
d
Tw
TTT
d
c
aaaaLE
aaLE
Summing for each segment, then we have the first boundary condition:
21 TT EE
BOUNDARY CONDITIONS
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Second boundary condition can be determined by applying Gauss’s Law over a small pillbox shaped Gaussian surface,
BOUNDARY CONDITIONS
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encQd SD
BOUNDARY CONDITIONS
The Gauss’s Law,
Where,
sidebottomtop
dddd SDSDSDSD
The pillbox is short enough, so the flux passes through the side is negligible.
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So, only top and bottom where:
SDdSDd
SDdSDd
NNNNbottom
NNNNtop
22
11
aaSD
aaSD
Which sums to:
encNN QSDD 21
BOUNDARY CONDITIONS
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And the right side of Gauss’s Law,
SdSQ SSenc
Thus, it leads to the second boundary condition:
SNN DD 21
This is when the normal direction from medium 2 to medium 1.
BOUNDARY CONDITIONS
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If the normal direction is from medium 1 to medium 2,
SNN DD 12
BOUNDARY CONDITIONS
Generally,
S 2121 DDa
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For a boundary conditions between a dielectric and a good conductor,
0TEBecause in a good conductor, E = 0. And since the electric flux density is zero inside the conductor,
SN D
BOUNDARY CONDITIONS
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EXAMPLE 12
Consider that the field E1 is known as:
Find the field E2 in the other dielectrics.
mVzyx aaaE 5431
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SOLUTION TO EXAMPLE 12
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BOUNDARY CONDITIONS
We can employ Poisson’s and Laplace’sequations to help find the potential functionwhen conditions at the boundaries are specified.
From divergence theorem expression,
V DBy considering ,ED
V E
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From the gradient expression,
VEWhich gives us Poisson’s equation,
VV 2
In charge free medium in which , it becomes Laplace’s equation:
0V
02 V
BOUNDARY CONDITIONS
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EXAMPLE 13
Determine the electric potential in the dielectric region between a pair of concentric spheres that have a potential difference Vab.
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SOLUTION TO EXAMPLE 13
The charge distribution is: 30
mC
rV
This employs Poisson’s equation and the potential is only a function of r,
0
022
02
)(1
r
V
rrrVr
rr
rV
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Multiply with r2 and integrate to obtain,
ArrrVr
r
0
202
2)(
Dividing both sides with r2 and integrate again,
BrArrV
r
0
02
)(
SOLUTION TO EXAMPLE 13
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Assume that the Vab consists of a voltage Va on the inner conductor and the outer conductor is grounded. So,
22
0)(
12
)(
0
0
0
0
BbAbVbrV
BaAaVarV
rb
ra
SOLUTION TO EXAMPLE 13 (Cont’d)
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ababAabV
BbAbB
aAaVVV
ra
rrbaab
0
0
0
0
0
0
2
22
From (3) you can get:0
02
r
a abba
abVA
SOLUTION TO EXAMPLE 13
(3)
ba
abVbaB a
r
0
02
Thus,
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Then, the potential between the spheres is
given by:
raVr
rarV a
r
0
02
)(
SOLUTION TO EXAMPLE 13
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CAPACITANCE
The amount of charge that accumulates as a function of potential difference is called the capacitance.
VQC
The unit is the farad (F) or coulomb per volt.
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CAPACITANCE (Cont’d)
Two methods for determining capacitance:
Q Method
• Assume a charge +Q on plate a and a charge –Q on plate b.
• Solve for E using the appropriate method (Coulomb’s Law, Gauss’s Law, boundary conditions)
• Solve for the potential difference Vab between the plates (The assumed Q will divide out)
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V Method
• Assume Vab between the plates.
• Find E , then D using Laplace’s equation.
• Find ρS, and then Q at each plate using conductor dielectric boundary condition (DN = ρS )
• C = Q/Vab (the assumed Vab will divide out)
CAPACITANCE (Cont’d)
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EXAMPLE 14
Use Q method to find the capacitance for
the parallel plate capacitor as shown.
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SOLUTION TO EXAMPLE 14
Place charge +Q on the inner surface of the top plate, and –Q charge on the upper surface of the bottom plate, where the charge density,
Use conductor dielectric boundary, to obtain:
dSQ SSQ
S from
zSQ aD from SN D
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We could find the electric field intensity, E
zrSQ aE0
The potential difference across the plates is:
SQddz
SQ
dV
rz
d
zr
a
bab
00 0
aa
LE
SOLUTION TO EXAMPLE 14
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Finally, to get the capacitance:
SQd
QVQC
rab
0
SOLUTION TO EXAMPLE 14
dSC r0
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EXAMPLE 15
Use V method to find the capacitance for a length Lof coaxial line of inner conductor radius a and
outer radius b, filled with dielectric permittivity as
shown.
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SOLUTION TO EXAMPLE 15
Employ Laplace’s equation to find the potential
field everywhere in the dielectric. Assume that
fringing fields are neglected and the field is only
in function of ρ.
Laplace’s equation becomes:
0
V
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SOLUTION TO EXAMPLE 15
Integrating twice to obtain:
BAV ln
Apply boundary condition to determine A and B. Let V(b) = 0 and V(a) = Vab to get:
ab
VAbAB abln
,ln So,
abbVV ab
lnln
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Apply the gradient to obtain E:
aE
E
abV
VV
abln
and also..
aD
abVabr
ln0
SOLUTION TO EXAMPLE 15
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We can find Q on the inner conductor with:
abVLaL
abaVSQ abrabr
S ln22
ln00
SOLUTION TO EXAMPLE 15 (Cont’d)
At inner conductor, the flux is directed outward indicating a positive surface charge density,
abaVabr
S ln0
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Thus, we can find the capacitance:
abL
VQC r
ln2 0
SOLUTION TO EXAMPLE 15 (Cont’d)
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EXAMPLE 16
Conducting spherical shells with radius a =10cmand b = 30cm are maintained at a potentialdifference of 100V, such that V(r = a) = 100V andV(r = b) = 0V. Determine:
• The V and E in the region between the shells.
• If εr = 2.5 in the shells, determine the totalcharges induced on the shells and the capacitanceof the capacitor.
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With:
5.23.01.0
r
mbma
EXAMPLE 16 (Cont’d)
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SOLUTION TO EXAMPLE 16
Employ Laplace’s equation to find the potential
field and the field is only in function of r.
Laplace’s equation becomes:
01 22
2
rVr
rrV
Multiply by r2,02
rVr
r
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Integrating once gives
22
rA
rVA
rVr
Integrating again gives
BrAV
SOLUTION TO EXAMPLE 16 (Cont’d)
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To obtain the value of constant A and B, use boundary conditions, where:
When,
bAB
BbAVbr
00,
Thus,
rbAV 11
SOLUTION TO EXAMPLE 16 (Cont’d)
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SOLUTION TO EXAMPLE 16 (Cont’d)
When,
ab
VA
abAV
rbAVV
VVar
11
11
11,
0
0
0
0
:
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Therefore, the potential difference V
ab
rbVV11
11
0
SOLUTION TO EXAMPLE 16 (Cont’d)
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So, the electric field intensity E
r
ab
Vr
aE
11
1 02
SOLUTION TO EXAMPLE 16 (Cont’d)
Apply the gradient to obtain E:
rr rA
rVV aaE 2
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The total charges Q is:
ddr
abr
V
ddQ
r sin111 2
2
0 02
00
SESD
SOLUTION TO EXAMPLE 16 (Cont’d)
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This yields to:
ab
V
dd
ab
VQ
r
r
114
sin11
00
2
0 0
00
SOLUTION TO EXAMPLE 16 (Cont’d)
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Thus, substituting the values of a, b and V0
to get the total charges, Q
nC
Q
1723.41.0
13.0
11005.210854.84 12
SOLUTION TO EXAMPLE 16 (Cont’d)
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And for the potential difference V = V0.
So, the capacitance, C
pFVnC
VQC
723.41
1001723.4
SOLUTION TO EXAMPLE 16 (Cont’d)
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PART 1
END
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PRACTICAL APPLICATION
Laser Printer
Electret Microphone
Electrolytic Capacitors
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LASER PRINTER
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• OPC drum : Organic Photoconductive Cartridge –has a special coating will hold electrostatic charge.
• The surface is photoconductive – will discharge if surface hit by light.
i. A portion of drum passes under a negativecharged wire large negative charges induces apositive charge on the drum.
ii. Image to be printed is delivered to this chargedregion by laser and spinning mirror combination.
iii.Wherever the laser light strikes the drum, thephotoconductive material is discharged.
LASER PRINTER (Cont’d)
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iv. The drum rolls past a toner The toner is blackpowder and positive charge drawn to thoseportions of the charge that have been dischargedby the laser.
v. Paper is fed through same speed as drum itpasses over positively charged wire that gives thepaper a strong negative charge.
vi. The positively charged toner drum is transferred tothe stronger negative charged on the paper thenit passes near a negatively charged wire thatremoves the negative charge from the paper,prevents it from statically clinging to the drum
LASER PRINTER (Cont’d)
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vii.The paper and loose toner powder passed throughheated fuser rollers powder melts into the paperfiber the warm paper exits the printer.
• The drum continues rolling, passing through highintensity light discharges all the photoconductorsto erase the image from the drum, and ready forapplication of positive charge again from coronawire.
LASER PRINTER (Cont’d)
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SUMMARY (1)
•The force exerted on a charge Q1 on charge Q2 in a medium of permittivity ε is given by Coulomb’s Law:
12212
2112
4a
RF
Where is a vector from charge Q1 to Q2121212 aR R
2
121 Q
FE
•Electric field intensity E1 is related to force F12 by:
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The Coulomb’s Law can be rewritten as:
RQ a
RE 2
04
For a continuous charge distribution:
RRdQ aE 2
04•For a point charge at origin:
rrQ aE 2
04
SUMMARY (2)
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•For an infinite length line charge ρL on the z axis
aE
2L
•For an infinite extent sheet of charge ρS
NS aE
2
•Electric flux density related to field intensity by:
ED 0 r
SUMMARY (3)
Where εr is the relative permittivity in a linear, isotropic and homogeneous material.
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SUMMARY (4)
• Electric flux passing through a surface is given by:
dS D
• Gauss’s Law states that the net electric flux through any closed surface is equal to the total charge enclosed by that surface:
encQd SD
Point form of Gauss’s Law is
V D
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SUMMARY (5)
• The electric potential difference Vab between a pair of points a and b in an electric field is given by:
ab
b
aba VVdV LE
Where Va and Vb are the electrostatics potentials at aand b respectively.
For a distribution of charge in the vicinity of the origin, where a zero reference voltage is taken at infinite radius:
rdQV4
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SUMMARY (6)
• E is related to V by the gradient equation:
VEWhich for Cartesian coordinates is:
zyx zV
yV
xVV aaa
• The conditions for the fields at the boundary between a pair of dielectrics is given by:
S 2121 DDa21 TT EE and
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Where ET1 and ET2 are the electric field components tangential to the boundary, a21 is a unit vector from medium 2 to 1 and ρS is the surface charge at the boundary. If no surface charge is present, the components of D normal to the boundary are equal:
21 NN DD
SUMMARY (7)
At the boundary between a conductor and a dielectric, the conditions are:
0TE and SN D
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• Poisson’s equation is:
VV 2
Where the Laplacian of V in Cartesian coordinates is given by:
2
2
2
2
2
22
zV
yV
xVV
In a charge free medium, Poisson’s equation reduces to Laplace’s equation
02 V
SUMMARY (8)
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• Capacitance is a measure of charge storage capability and is given by:
VQC
SUMMARY (9)
For coaxial cable:
For two concentric spheres:
abLC
ln2
abV Lab ln
2
So,
baC
114
baQVab
114
So,