electrostatics chapter 3 - جامعة نزوى€¦ · electrostatics chapter 3 •the...
TRANSCRIPT
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Electrostatics
Chapter 3
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• The electromagnetic constitutive parameters of a material
medium are its electrical permittivity 𝜀,magnetic
permeability 𝜇 and conductivity 𝜍.
** A material is said to be homogeneous if 𝜀, 𝜇 and 𝜍
do not vary from point to point.
** A material is said to be Isotropic if 𝜀, 𝜇 and 𝜍 do not
vary with direction.
3-6 Electric Properties of Materials
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• The conductivity of material is a measure of how
easily electron can travel through the material
under the influence of an external electric field.
• Materials are classified as:
conductors(metals).
dielectrics(insulators).
semiconductors.
based on the magnitudes of their conductivity 𝜎.
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• Upon applying an external electric field, the
electrons in the conductor migrate from one atom
to the next along a direction opposite that of the
external field. The average velocity of the
electrons are called "electron drift velocity ue "
which gives rise to a “conductor current”.
A perfect dielectric : 𝜍 = 0
A perfect conductor : 𝜍 = ∞
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Note :
The unit of conductivity 𝝈 is :
S/m (Siemens per meter)
OR (𝟏 𝛀.𝒎)
(𝑺 = 𝑨/𝑽 (ampere per volt) or 𝟏
𝛀 )
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Typical metals : 𝝈 = 𝟏𝟎𝟔 𝐭𝐨 𝟏𝟎𝟕 𝑺
𝒎
Typical dielectrics : 𝝈 = 𝟏𝟎−𝟏𝟎 𝐭𝐨 𝟏𝟎−𝟏𝟕 𝑺
𝒎
Semiconductors : 𝝈 is in between metals and
dielectrics such as Germanium(Ge) has 𝝈 = 𝟐. 𝟐 𝑺/𝒎
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• The conductivity of a material depends on
temperature and the presence of impurities.
• At very low temperatures in the region of
absolute zero, some conductors become
“superconductors“.
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3-7 Conductors
• The drift velocity of electrons in a conducting
material is related to the externally applied electric
field 𝐸 through :
𝑢𝑒 = −𝜇𝑒𝐸 (𝑚 𝑠 )
Electric field Electron
mobility
with units
of 𝒎𝟐 𝑽. 𝒔
Drift
velocity
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𝑢ℎ = 𝜇ℎ𝐸 ( 𝑚 𝑠 )
hole
mobility
• In a semiconductor, current flow is due to the
movements of both electrons and holes and since the
holes are positive charge carriers, the “hole drift
velocity” 𝑢ℎ is in the same direction of 𝐸 ,
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• The current density 𝐽 in a medium containing a volume
density 𝜌𝑣 moving with velocity 𝑢 is :
𝐽 = 𝜌𝑣𝑢
• The current density consists of components from both
electrons ( 𝐽 𝑒 ) and holes ( 𝐽 ℎ ).
• Thus the total conduction current density 𝐽 is :
𝐽 = 𝐽 𝑒 + 𝐽 ℎ = 𝜌𝑣𝑒𝑢𝑒 + 𝜌𝑣ℎ𝑢ℎ (𝐴 𝑚2 )
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• Using 𝑢 = 𝜇𝐸 :
𝐽 = (−𝜌𝑣𝑒𝜇𝑒 + 𝜌𝑣ℎ𝜇ℎ)𝐸
,Where 𝜌𝑣𝑒 = −𝑁𝑒𝑒 and 𝜌𝑣ℎ = 𝑁ℎ𝑒
𝑒 = 1.6 × 10−19 𝐶
# of electrons
per unit
volume
# of holes per
unit volume
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• The conductivity of the material, 𝜍, is defined as :
𝜍 = 𝑁𝑒𝜇𝑒 + 𝑁ℎ𝜇ℎ 𝑒 (𝑆 𝑚 ) (semiconductor)
= −𝜌𝑣𝑒𝜇𝑒 + 𝜌𝑣ℎ𝜇ℎ
• For a good conductor usually :
𝑁ℎ𝜇ℎ ≪ 𝑁𝑒𝜇𝑒
𝜍 = −𝜌𝑣𝑒𝜇𝑒 = 𝑁𝑒𝜇𝑒𝑒 (𝑆 𝑚 ) (conductor)
• In either case :
𝐽 = 𝜍𝐸 ( 𝐴 𝑚2) (ohm′s law − point form)
𝑨
𝒎𝟐 =𝟏
𝜴𝒎×𝑽
𝒎
→ 𝜴 =𝑽
𝑨 (𝒐𝒉𝒎′𝒔 𝒍𝒂𝒘)
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Perfect dielectric : 𝐽 = 0
(𝜍 = 0,regardless of 𝐸)
Perfect conductor : 𝐸 = 0
(𝜍 = ∞,regardless of 𝐽 )
• A perfect conductor is an “equipotential medium” .it means that
the electric potential is the same at every point in the conductor.
• Since 𝐸 = 0 everywhere in the perfect conductor, the voltage
difference 𝑉21 = 0
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Example
The conductor shown in the figure is applied to an
electric field of (20 mV m ) . Find :
(a) Volume charge density 𝜌𝑣of free electrons.
(b) Current density J.
(c) The current flowing in the wire.
(d) The electron drift velocity.
(e) Volume density of free electrons 𝑁𝑒.
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solution
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3-7.1 Resistance
*** What are 𝐽 and 𝐸 directions?!
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𝐼 , 𝐽 , and 𝐸 have the same direction from high potential to low potential as shown in the figure.
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• Using the point form of Ohm’s law, we can derive an expression
for the resistance R of a conductor of length 𝑙 and uniform
cross section A .
𝑉 = 𝑉1 − 𝑉2
= − 𝐸. 𝑑𝑙 = − 𝑥 𝐸𝑥 . 𝑥 𝑑𝑙 = −𝐸𝑥 𝑥1 − 𝑥2 = 𝐸𝑥𝑙𝑥1
𝑥2
𝑥1
𝑥2
𝑙 = 𝑥2 − 𝑥1
𝐸 = 𝒙 𝐸𝑥
𝑉 = 𝐸𝑥 𝑥2 − 𝑥1 = 𝐸𝑥 𝑙
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• The current flowing through the cross section A at 𝑥2 is
𝐼 = 𝐽 . 𝑑𝑠 𝐴
= 𝜍𝐸. 𝑑𝑠 𝐴
= 𝜍𝐸𝑥𝐴
∴ R =𝑉
𝐼=
𝐸𝑥𝑙
𝜍𝐸𝑥𝐴=
𝑙
𝜍𝐴 (Ω)
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• For any arbitrary shape, the resistance R can be
expressed as :
𝑅 = 𝑉
𝐼 =
− 𝐸 .𝑑𝑙 𝑙
𝐽 .𝑑𝑠 𝑠
= − 𝐸 .𝑑𝑙 𝑙
𝜎𝐸. 𝑑𝑠 𝑠
** The reciprocal of R (1
𝑅 ) is called the conductance G
and the unit of G is (Ω−1) or siemens (S) .
** For linear resistor:
𝐺 =1
𝑅=
𝜎𝐴
𝑙 (S)
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Example 1
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solution
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Example 2 Conductance of Coaxial Cable
Obtain an expression for G′, the conductance per unit length of
the insulation layer .
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• Since the current is radial, the area through which the
current flows is 𝐴 = 2𝜋𝑟𝑙. Hence,
𝐽 = 𝒓 𝐼
𝐴= 𝒓
𝐼
2𝜋𝑟𝑙
𝐽 = 𝜍𝐸 → 𝐸 =𝐽
𝜎= 𝒓
𝐼
2𝜋𝜎𝑟𝑙
• Current 𝐼 flows from higher potential to lower potential .
𝑉𝑎𝑏 = − 𝐸. 𝑑𝑙 = − 𝐼
2𝜋𝜎𝑙
𝑟 .𝑟 𝑑𝑟
𝑟=
𝐼
2𝜋𝜎𝑙
𝑑𝑟
𝑟=
𝐼
2𝜋𝜎𝑙ln(
𝑏
𝑎
𝑏
𝑎
𝑎
𝑏
𝑎
𝑏)
solution
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• The conductance per unit length is then:
𝐺′ =𝐺
𝑙=
1
𝑅𝑙=
𝐼
𝑉𝑎𝑏𝑙=
𝐼
𝐼2𝜋𝜍𝑙
ln(𝑏𝑎) 𝑙
=2𝜋𝜍
ln(𝑏𝑎) (𝑆 𝑚 )
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3-7.2 Joule's Law
Let’s now consider the power dissipated in a conducting medium
in the presence of an electrostatic field 𝐸. The electric force acting
on charge 𝑞𝑒 and 𝑞ℎ are:
𝐹 𝑒 = 𝑞𝑒𝐸 = 𝜌𝑣𝑒∆𝑣𝐸 , ∆𝑣 is the element of volume.
𝐹 ℎ = 𝑞ℎ𝐸 = 𝜌𝑣ℎ∆𝑣𝐸 The energy(work) expended by electric field in moving 𝑞𝑒 by
distance ∆𝑙𝑒 is :
∆𝑤 = 𝐹𝑒. ∆𝑙𝑒 + 𝐹ℎ. ∆𝑙ℎ
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• The power measured in watts(W) is:
∆𝑃 =∆𝑤
∆𝑡= 𝐹 𝑒 .
∆𝑙 𝑒
∆𝑡+ 𝐹 ℎ .
∆𝑙 ℎ
∆𝑡
=𝐹 𝑒 . 𝑢𝑒 + 𝐹 ℎ. 𝑢ℎ
= (𝜌𝑣𝑒𝐸. 𝑢𝑒 + 𝜌𝑣ℎ𝐸𝑢ℎ)∆𝑣
=𝐸 . 𝐽 ∆𝑣
𝑃 = 𝐸 . 𝐽 𝑑𝑣 𝑣
(𝑊) (Joule’s law)
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• Since 𝑗 = 𝜍𝐸 → 𝑃 = 𝜍 𝐸 2𝑑𝑣𝑣
𝑣 = 𝑙𝐴 → separating the integral:
𝑃 = 𝜍 𝐸 2𝑑𝑣
𝑣 = 𝜍𝐸𝑥𝑑𝑠𝐴 𝐸𝑥𝑑𝑙 𝑙
→ 𝑃 = 𝜍𝐸𝑥𝐴 𝐸𝑥𝑙 = 𝐼𝑉 (𝑊)
• Using 𝑉 = 𝐼𝑅 → 𝑷 = 𝑹𝑰𝟐
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• In dielectric (insulator),an externally applied electric
field 𝐸𝑒𝑥𝑡 cannot cause mass migration of charges since
they are not able to move freely, but it can “polarize” the
atoms or molecules in the material by distorting the
center of the cloud and the location of the nucleus.
3-8 Dielectrics
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• The “induced” electric field ,called a polarization field,
is weaker than and opposite in direction to 𝐸𝑒𝑥𝑡 .
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• Each dipole exhibits a dipole moment. The
materials described here are called a nonpolar
materials. Nonpolar molecules become
polarized only when an external electric field
is applied ,and when the field is terminated, the
molecules return to their original unpolarized
state.
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• Since 𝐷 and 𝐸 are related by 𝜀0 in free space, the presence
of microscopic dipoles in dielectric material alters that
relationship in that material to :
𝐃 = 𝛆𝟎𝐄 + 𝐏
,where P is called the “electric polarization field”, accounts
for the polarization properties of material.
• The polarization field is produced by the electric field 𝐸
and depends on the material properties.
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• In linear, isotropic and homogeneous media, the polarization field is
directly proportional to 𝐸 and is expressed as :
𝑷 = 𝜺𝟎𝒙𝒆𝑬 ,where 𝑥𝑒 is called electric susceptibility of the material.
*** Note 𝑥𝑒 is a dimensionless quantity.
𝑫 = 𝜺𝟎𝑬 + 𝜺𝟎𝒙𝒆𝑬
= 𝜺𝟎 𝟏 + 𝒙𝒆 𝑬
𝐷 = 𝜺𝑬
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• Which defines the permittivity 𝜀 of the material
as
𝜀 = 𝜀0𝜀𝑟 = 𝜀0(1 + 𝑥𝑒)
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** For air 𝜀𝑟≅ 1.0006 at see level.
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• The polarized atom or molecule may be
represented by an electric dipole consisting of
charge +𝑞 at the center of the nucleus and
charge −𝑞 at the center of electric cloud. Each
such dipole sets up a small electric ,pointing
from the positively charged nucleus to the
center of the equally but negatively charged
electron cloud.
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Dielectric Breakdown
• If a dielectric material is placed in a very
strong electric field (exceeds a certain critical
value ,known as the dielectric strength Eds
,electrons can be torn from their corresponding
nuclei causing large currents to flow and
damaging the material. This phenomenon is
called dielectric breakdown.
*** dielectric strength Eds is the highest magnitude of E that the material can
sustain without breakdown.
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Dielectric Breakdown (Cont’d)
• The dielectric strength of a material may vary by several orders of magnitude depending on various factors including the exact composition of the material as well as other factors such as temperature and humidity. Some typical values of dielectric strength for some common insulators are:
• Usually dielectric breakdown does not permanently
damage gaseous or liquid dielectrics, but does ruin solid
dielectrics.
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Dielectric Breakdown (Cont’d)
• Capacitors typically carry a maximum voltage rating.
Keeping the terminal voltage below this value insures
that the field within the capacitor never exceeds Eds
for the dielectric.
• If 𝑉 is sufficiently large so that 𝐸 exceeds the
dielectric strength of air, ionization occurs and
discharge (lightning) follows.