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Lecture 3
Engineering Electromagnetics
Dr.-Ing. Erwin SitompulPresident University
http://zitompul.wordpress.com
2 0 1 4
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Chapter 3Electric Flux Density, Gauss’s Law,
and Divergence
Engineering Electromagnetics
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About 1837, the Director of the Royal Society in London, Michael Faraday, was interested in static electric fields and the effect of various insulating materials on these fields.
This is the lead to his famous invention, the electric motor. He found that if he moved a magnet through a loop of wire, an
electric current flowed in the wire. The current also flowed if the loop was moved over a stationary magnet.
►Changing magnetic field produces an electric field.
Electric Flux DensityChapter 3 Electric Flux Density, Gauss’s Law, and Divergence
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Electric Flux Density In one of his experiments, Faraday had a pair of concentric
metallic spheres constructed, the outer one consisting of two hemispheres that could be firmly clamped together.
He also prepared shells of insulating material (or dielectric material), which would occupy the entire volume between the concentric spheres.
Chapter 3 Electric Flux Density, Gauss’s Law, and Divergence
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Electric Flux DensityChapter 3 Electric Flux Density, Gauss’s Law, and Divergence
Faraday found out, that there was a sort of “charge displacement” from the inner sphere to the outer sphere, which was independent of the medium.
We refer to this flow as displacement, displacement flux, or simply electric flux.
ψ Q
Where ψ is the electric flux, measured in coulombs, and Q is the total charge on the inner sphere, also in coulombs.
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Electric Flux DensityChapter 3 Electric Flux Density, Gauss’s Law, and Divergence
At the surface of the inner sphere, ψ coulombs of electric flux are produced by the given charge Q coulombs, and distributed uniformly over a surface having an area of 4πa2 m2.
The density of the flux at this surface is ψ/4πa2 or Q/4πa2 C/m2.
The new quantity, electric flux density, is measured in C/m2 and denoted with D.
The direction of D at a point is the direction of the flux lines at that point.
The magnitude of D is given by the number of flux lines crossing a surface normal to the lines divided by the surface area.
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Electric Flux DensityChapter 3 Electric Flux Density, Gauss’s Law, and Divergence
Referring again to the concentric spheres, the electric flux density is in the radial direction :
At a distance r, where a ≤ r ≤ b,
2 (inner sphere)4 rr a
Qa
D a
2 (outer sphere)4 rr b
Qb
D a
24 rQr
D a
If we make the inner sphere smaller and smaller, it becomes a point charge while still retaining a charge of Q. The electrix flux density at a point r meters away is still given by:
24 rQr
D a
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204 rQr
E a
Electric Flux DensityChapter 3 Electric Flux Density, Gauss’s Law, and Divergence
0D E
2vol04v
RdvR
E a
2vol 4v
RdvR
D a
Comparing with the previous chapter, the radial electric field intensity of a point charge in free space is:
Therefore, in free space, the following relation applies:
For a general volume charge distribution in free space:
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Electric Flux DensityChapter 3 Electric Flux Density, Gauss’s Law, and Divergence
ExampleFind the electric flux density at a point having a distance 3 m from a uniform line charge of 8 nC/m lying along the z axis in free space.
02L
E a2
L
D a98 10
2
a
921.273 10 C m
a
For the value ρ = 3 m, 91.273 10
3
D 10 24.244 10 C m
a 20.424 nC m a
• Can you determine the electric flux density at (1,7,7) and (3,4,5)?
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Electric Flux DensityExample
Calculate D at point P(6,8,–10) produced by a uniform surface charge density with ρs = 57.2 μC/m2 on the plane x = 9.
Chapter 3 Electric Flux Density, Gauss’s Law, and Divergence
02s
N
E a2s
N
D a657.2 10
2 N
a 228.6 C mN a
At P(6,8,–10),
N xa = a 228.6 C mx D a
• Can you determine D at (1,8,2) and (12,–2,7)?
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Gauss’s LawThe results of Faraday’s experiments with the concentric
spheres could be summed up as an experimental law by stating that the electric flux passing through any imaginary spherical surface lying between the two conducting spheres is equal to the charge enclosed within that imaginary surface.
Chapter 3 Electric Flux Density, Gauss’s Law, and Divergence
Faraday’s experiment can be generalized to the following statement, which is known as Gauss’s Law:
“The electric flux passing through any closed surface is equal to the total charge enclosed by that surface.”
ψ Q
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Gauss’s Law Imagine a distribution of charge, shown as a cloud of point
charges, surrounded by a closed surface of any shape.
Chapter 3 Electric Flux Density, Gauss’s Law, and Divergence
If the total charge is Q, the Q coulombs of electric flux will pass through the enclosing surface.
At every point on the surface the electric-flux-density vector D will have some value DS (subscript S means that D must be evaluated at the surface).
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Gauss’s LawΔS defines an incremental element of area with magnitude of
ΔS and the direction normal to the plane, or tangent to the surface at the point in question.
At any point P, where DS makes an angle θ with ΔS, then the flux crossing ΔS is the product of the normal components of DS and ΔS.
Chapter 3 Electric Flux Density, Gauss’s Law, and Divergence
ψ flux crossing S cosSD S S D S
closedsurface
ψ ψ Sd d D S
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Gauss’s LawThe resultant integral is a closed surface integral, with dS
always involves the differentials of two coordinates ► The integral is a double integral.
We can formulate the Gauss’s law mathematically as:
Chapter 3 Electric Flux Density, Gauss’s Law, and Divergence
nQ Q LQ dL SSQ dS vvol
Q dv
The charge enclosed meant by the formula above might be several point charges, a line charge, a surface charge, or a volume charge distribution.
ψ charge enclosedS
S d Q D S
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Gauss’s LawChapter 3 Electric Flux Density, Gauss’s Law, and Divergence
We now take the last form, written in terms of the charge distribution, to represent the other forms:
Illustration. Let a point charge Q be placed at the origin of a spherical coordinate system, and choose a closed surface as a sphere of radius a.
The electric field intensity due to the point charge has been found to be:
204 rQr
E a
0D E 24 rQr
D a
volSS vd dv D S
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Gauss’s LawChapter 3 Electric Flux Density, Gauss’s Law, and Divergence
24S rQa
D a
2 sin rd a d d S a
22 sin sin
4 4S r rQ Qd a d d d da
D S a a
At the surface, r = a,
2
00
cos4Q
ψS
S d D S2
0 0sin
4 r a
Q d d
Q
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Application of Gauss’s Law: Some Symmetrical Charge DistributionsChapter 3 Electric Flux Density, Gauss’s Law, and Divergence
Let us now consider how to use the Gauss’s law to calculate the electric field intensity DS:
The solution will be easy if we are able to choose a closed surface which satisfies two conditions:
1. DS is everywhere either normal or tangential to the closed surface, so that DSdS becomes either DSdS or zero, respectively.
2. On that portion of the closed surface for which DSdS is not zero, DS is constant.
For point charge ► The surface of a sphere.For line charge ► The surface of a cylinder.
SSQ d D S
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Application of Gauss’s Law: Some Symmetrical Charge DistributionsChapter 3 Electric Flux Density, Gauss’s Law, and Divergence
D D a
From the previous discussion of the uniform line charge, only the radial component of D is present:
The choice of a surface that fulfill the requirement is simple: a cylindrical surface.
Dρ is every normal to the surface of a cylinder. It may then be closed by two plane surfaces normal to the z axis.
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Application of Gauss’s Law: Some Symmetrical Charge DistributionsChapter 3 Electric Flux Density, Gauss’s Law, and Divergence
SSQ d D Ssides top bottom 0
z z z zzz L
D dS D dS D dS
2
0 0
L
zD d dz
2D L
2QDL
We know that the charge enclosed is ρLL,
2LD
02LE
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Chapter 3 Electric Flux Density, Gauss’s Law, and Divergence
Application of Gauss’s Law: Some Symmetrical Charge Distributions
The problem of a coaxial cable is almost identical with that of the line charge.
Suppose that we have two coaxial cylindrical conductors, the inner of radius a and the outer of radius b, both with infinite length.
We shall assume a charge distribution of ρS on the outer surface of the inner conductor.
Choosing a circular cylinder of length L and radius ρ, a < ρ < b, as the gaussian surface, we find:
2SQ D L
The total charge on a length L of the inner conductor is:2
0 02
L
S SzQ ad dz aL
S
SaD
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Chapter 3 Electric Flux Density, Gauss’s Law, and Divergence
Application of Gauss’s Law: Some Symmetrical Charge Distributions
For one meter length, the inner conductor has 2πaρS coulombs, hence ρL = 2πaρS,
2L
D a
Every line of electric flux starting from the inner cylinder must terminate on the inner surface of the outer cylinder:
outer cyl ,inner cyl2 SQ aL
,outer cyl ,inner cyl2 2S SbL aL
,outer cyl ,inner cylS Sab
If we use a cylinder of radius ρ > b, then the total charge enclosed will be zero. ► There is no external field,
0SD
• Due to simplicity, noise immunity and broad bandwidth, coaxial cable is still the most common means of data transmission over short distances.
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ExampleA 50-cm length of coaxial cable has an inner radius of 1 mm and an outer radius of 4 mm. The space between conductors is assumed to be filled with air. The total charge on the inner conductor is 30 nC. Find the charge density on each conductor and the expressions for E and D fields.
Chapter 3 Electric Flux Density, Gauss’s Law, and Divergence
Application of Gauss’s Law: Some Symmetrical Charge Distributions
inner cyl ,inner cyl2 SQ aL inner cyl
,inner cyl 2S
QaL
9
3
30 102 (10 )(0.5)
29.55 C m
outer cyl ,outer cyl inner cyl2 SQ bL Q inner cyl
,outer cyl 2S
QbL
9
3
30 102 (4 10 )(0.5)
22.39 C m
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Chapter 3 Electric Flux Density, Gauss’s Law, and Divergence
Application of Gauss’s Law: Some Symmetrical Charge Distributions
,inner cylSD a
63 (9.55 10 )10
29.55 nC m
0
DE
9
12
9.55 108.854 10
1079 V m
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Chapter 3 Electric Flux Density, Gauss’s Law, and Divergence
Homework 3D3.3. D3.4. D3.5. All homework problems from Hayt and Buck, 7th Edition.
Due: Monday, 28 April 2014.Quiz 1: Tuesday, 29 April 2014.
Make Up Class: Tue, 27 May 2014 (Ascension of the Prophet Muhammad)on Monday, 26 May 2014 at 07:30