Download Electrostatic fields • Why study electrostatics? • Many applications in

Electrostatic fields
• Why study electrostatics?
• Many applications in biology and medicine:
-Diagnosis as incorporated in electrocardiograms,
electroencephalograms and other recordings of
organs with electrical activity)
-Therapy as incorporated in diathermy, electroporation, ……
• Electrostatics is the branch of science that deals
with the phenomena arising from what seem to
be stationary electric charges.
• Two fundamental laws govern electrostatic
fields:
•
1. Coulomb’s law
•
2. Gauss’s law
• Both laws are based on experimental studies.
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Coulomb’s Law
• The force F between two point charges Q1 and Q2 is:
1. Along the line joining them
2. Directly proportional to the product of Q1 and Q2
3. Inversely proportional to the square of the distance R
between them
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â
Qa
Fa
4per2
Fa =-QaâQb
Fb =+QbâQa
4per2
4per2
Fb =+QbEa
Where Eb =-Qbâ
Fb
In air, e= 8.85 x 10-12 Fm-1
|â| = 1, Fa = -Fb
Fb =+QbâQa
4per2
Fa =+QaEb
Qb
r
Fa =-QaâQb
Where Ea = +Qaâ
4per2
4per2
Eb(r)
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is the electric field
Ea(r)
is the electric field
set up by charge b at
set up by charge a at
distance r (point a)
distance r (point b)
Charge Qa sets up a field Ea at the location of charge Qb and the Force on Qb is
given by F = QbEa or Ea =F/Qb (force per unit charge)
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Several Charges?
• If there are N charges
+Qc
Ea
+Qd
Qa,Qb,….located
-Qe Eb
at points with position
+Qa
Ec
vectors r1,r2,…
-Qb
Ed
the resultant force F on
Ee
a charge Q located at point r
is the vector sum of the forces exerted on Q by each of the charges
Q1,Q2,…..
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Many charges …
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Many charges …
• Q1, Q2, Q3 …QN
-For a small numbers of charges:
Q1(r1), Q2(r2) … QN(rN) is OK to describe a charge Q1 at
position r1 etc.
-For a large N: Instead use r(r) as the density
(in Cm-3) of charge at a point r
aˆQN
EN =
4pe 0 r 2
• E = E1 + E2 + E3 … EN
aˆQN
• E = SNEN = SN 4pe r 2
0
SNQN becomes ∫r(r)dxdydz = ∫∫∫volr(r)dv
r = charge density
• OK for a handful of charges
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Electric Flux Density :
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Gauss’s Law: Integral form
• Flux : number of field lines that cross a normal area
(measured in Coulombs per square meter)
D = e0E
 = Qenc
 = D.ds
• One line of electric flux emanates from +1 C and
terminates on – 1 C.
• ε0 = 8.85 x 10-12 in a vacuum
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Gauss’s Law - Integral form
Gauss’s Law - Differential form
 
D
s  ds = v rv dv
 = Qen c
 =
Qen c

  D = rv
 d =  D.ds
=  r dv
This equation says that the electric flux density D diverges
from free charge rv
v
v
 D.ds =  r
v
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dv
v
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Gauss’s Law : Cartoon Version
Flux of rain (rainfall) through an area ds
• The number of electric field lines leaving a closed surface is
equal to the charge enclosed by that surface
S(E-field-lines) a Charge Enclosed
Rainfall
Rainfall
ds
This area gets
wetter!
N Coulombs  aN lines
The surface is chosen such that D is normal or tangential to the Gaussian
surface. When D is normal to the surface, D • dS = D dS. When D is tangential
to the surface, D • dS = 0.
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Flux rain = D.ds
|D||ds|  cos()
Dds cos()
Flux rain = 0 for 90° … cos() = 0
Flux rain = -Dds for 180° … cos() = -1
Generally, Flux rain = Dds cos()
-1 < cos() < +1
E, D
Gauss’s law - Example
Long straight “rod” of charge
Evaluate D.ds
Construct a “Gaussian Surface” that reflects the symmetry
of the charge - cylindrical in this case, then evaluate D.ds
E, D
E, D
ds
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D.ds =  D.ds curved surface
+ D.ds flat end faces
ds
r
• End faces, D & ds are perpendicular
ds
– D.ds on end faces = 0
–  D.ds flat end faces = 0
rl Coulombs/m
• Flat end faces do not contribute!
r
L
ds
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Evaluate D.ds
Evaluate D.ds
D.ds =  D.ds curved surface only
E, D
ds
 D.ds curved surface only =  Dds
D & ds parallel,
D.ds = |D|´|ds| = Dds
rl Coulombs/m
E, D
D has the same strength
D(r) everywhere on this
surface.
rl Coulombs/m
ds
L
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L
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Evaluate D.ds
Evaluate D.ds
 D.ds curved surface only =  ds D
• = D ds = D  area of curved surface
• =D2prL
• So 2Dp r L = charge enclosed
• Charge enclosed?
• Charge/length  length L = rl  L
 D.ds = charge enclosed
2pDr L = rl  L
• D(r) = rl
2pr
• D(r) = rl âr
2pr
2pr
rl Coulombs/m
L
r
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Discussion
• |D| is proportional to 1/r
– Gets weaker with distance
 D.ds =  ρ
• D points radially outwards (âr)
• |D| is proportional to rl
v
dv
v
– More charge density = more field
– Intuitively correct
 D.ds = r dv
v
v
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Postulates of Electrostatics in Free Space
Differential Form
Integral Form
r
e0
 E  ds = e
 E = 0
 E  dl = 0
E =
Q
s
0
c
Thus, the E-Field is conservative or irrotational. If a
vector field is curl-free, then it can be expressed as the
gradient of a scalar field: E = V
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Material Classification
• Materials may be classified in terms of their conductivity,
in mhos per meter or Siemens per meter, as conductors and
nonconductors, or technically as metals and insulators (or
dielectrics).
• A material with high conductivity ( >> 1) is referred to as
a metal whereas one with low conductivity ( << 1) is referred to as
an insulator.
• A material whose conductivity lies somewhere between
those of metals and insulators is called a semiconductor.
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• The conductivity of a material usually depends on temperature
and frequency.
• The conductivity of metals generally increases with decrease in
temperature.
• At temperatures near absolute zero (T = 0°K), some conductors
exhibit infinite conductivity and are called superconductors. Lead
and aluminum are typical examples of such metals.
• Microscopically, the major difference between a metal and an
insulator lies in the amount of electrons available for conduction
of current.
• Dielectric materials have few electrons available for conduction
of current in contrast to metals, which have an abundance of free
electrons.
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• Based on the values of conductivity, materials such as
copper and aluminum are metals, silicon and germanium
are semiconductors, and glass and rubber are insulators.
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Electrical Conductivity
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Conductors and Dielectrics
Conductors in Electric Fields
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Dielectrics in Electric Fields: Polarization
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Polarization vector
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Conductors and Dielectrics
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Conduction & Displacement Currents
• Electric current is generally caused by the motion of electric charges.
• The current (in amperes) through a given area is the electric charge
passing through the area per unit time.
• Current I = dQ / dt
• If current I flows through a surface S, the current density
Jn= I / S
or
I = Jn S
assuming that the current density is perpendicular to the surface. If the
current density is not normal to the surface, I = J • S
• Thus, the total current flowing through a surface S is
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I = ∫ J . dS
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Electrical properties of body tissues
Examples: (Bone Cancellous)
• The human body is made of a large number of materials (tissues),
each of them having specific properties.
• Since biological tissues mainly consist of water, they behave
neither as a conductor nor a dielectric, but as a lossy dielectric.
•  represents the ability of the material’s charge to be transported
throughout its volume by an applied E-field.
• e represents the ability of the material’s molecular dipoles to
rotate or its charge to be stored by an applied external applied Efield.
• The permittivity and conductivity of biological tissues are
functions of frequency.
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Grey Matter
Cerebellum
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Magnetostatic Fields
• The other half of electromagnetics is the magnetic field.
• Whereas electric fields emanate directly from individual charges,
magnetic fields arise in a subtle manner because there are no
magnetic charges.
•
Moreover, because there are no magnetic charges, magnetic field
lines can never have a beginning or an end (Magnetic field lines
always form closed loops).
•
Some physicists have been searching for magnetic charges (or
“magnetic monopoles,” as the particle physicists call them) in high
energy experiments.
• Without magnetic charges, magnetic fields can only arise indirectly.
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• Magnetic fields are generated indirectly by moving electric
charges.
• It is a fundamental fact of nature that moving electrons, as well
as any other charges, produce a magnetic field when in motion.
•
Electrical currents in wires also produce magnetic fields because
a current is basically the collective movement of a large number
of electrons.
• A steady (DC) current through a wire produces a magnetic field
that encircles the wire.
B
I
right hand rule
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• Magnetic Dipole
•
If a current travels in a loop, the magnetic field is donut-shaped.
•
•
The magnetic field flows out of one side and back in the other side.
Although the field lines still form closed loops, they now have a
sense of direction. The side where the field lines emanate is called
the north pole, and the side they enter is called the south pole.
• Such a structure is called a magnetic dipole.
• Now if a wire is wound in many spiraling loops, a solenoid is
formed. A solenoid concentrates the field into even more of a dipole
structure.
• A magnetostatic field is produced by a constant current flow (or
direct current).
• This current flow may be due to magnetization currents as in
permanent magnets, electron-beam currents as in vacuum tubes,
or conduction currents as in current-carrying wires.
• There are two major laws governing magnetostatic fields:
•
(1) Biot-Savart's law, and
(2) Ampere's circuit law.
Similar to Coulomb's law, and Gauss's law in electrostatics
S
N
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The Biot-Savart's Law
•
The Biot-Savart Law
The Biot-Savart's law states that the magnetic field intensity dB produced at a point
X by the differential current element dl is proportional to the product dl and the sine
of the angle between the element and the line joining X to the element and is
inversely proportional to the square of the distance r between X and the element.
dB =
0I dl  ar
4p r 2
âr
x
dB =
0I dl  ar
4p r 2
dB
dB
âr
x
r
dl
r
I
I
d = b  c = |b||c|sin(F) â
dl
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Example of Biot-Savart Law : Infinite Line
of Current
dB =
0I dl  ar
, B=
4p r 2
l =




l =
dB =
0I dl  ar
4p r 2
âr
.

0I dl  ar
4p r 2
df
âr
dB
dB
.
R = r sin (f 
r
I
r
f
dl
dl
df
f

I
f
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dl
sin f =
rdf
rdf
, dl =
dl
sin f
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âr
 I dl  ar
dB = 0
4p r 2
F
dB =
dl

B =  dB =

dl  ar = dl 1 sin (f  , out of the diagram
 Idl sin (f 
dB = dB = 0
=
4p r 2
0I sin (f  df
4p R
 rd
f 
rdf
 sin (f 
in (ff 
 ssin
0I 
4p r 2
=
0Idf 0I sin (f  df
=
4p r
4p R
0I sin (f  df 0I f =180
=
 sin (f df
4p R
4p R f = 0
B=
0I
I
I
f =180
  cos (f  f = 0 = 0 1  1 = 0
4p R 
4p R
2p R
B=
0I
I
I
f =180
  cos (f  f = 0 = 0 1  1 = 0
4p R 
4p R
2p R
Β=
0I
a, where a points out of the page
2p R
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Gauss’s Law for the Magnetic Field-Integral
Form
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Gauss’s Law-Differential Form
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Ampere’s Law –Integral Form
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Ampere’s Law –Differential Form
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Outside … r>a, H.dl = Ienclosed
Example
• Calculate the magnetic field H both
Outside (r>a)
and
Inside (r<a)
A wire with uniformly-distributed current I,
current density

c2
H  dl = H f rdfaF = I enclosed
I
H = H f af =
2pr
|J| = I/A
B?
B?
af
B
I
a
I
H, B
r
a
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Inside … r<a, H.dl = Ienclosed
2p
r
 Haf  rdfaf = I
0
 Hr (2p  =
I
pa 2
H inside = H f af =
B?
(pr 
2
H =
I
raf
2pa 2
r
I
2pr
I, |J|=I/pa2
a
I
H =
Ir
2pa 2
B,H
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Magnetization in materials
•
Assuming that our atomic model is that of an electron orbiting about a positive
nucleus.
A given material is composed of atoms. Each atom may be regarded as
consisting of electrons orbiting about a central positive nucleus; the electrons also
rotate (or spin) about their own axes.
• Thus an internal magnetic field is produced by electrons orbiting around the
• nucleus (a) or electrons spinning (b).
• Both of these electronic motions produce internal magnetic fields B, that are
similar to the magnetic field produced by a current loop.
•
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Permeability &Susceptibility
Classification of Magnetic Materials
• In general, we may use the magnetic susceptibility m or the
relative permeability µr to classify materials in terms of their
magnetic property or behavior.
• A material is said to be nonmagnetic if m = 0 (or µr = 1); it is
magnetic otherwise.
• Roughly speaking, magnetic materials may be grouped into three
major classes: diamagnetic, paramagnetic, and ferromagnetic.
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Diamagnetism
•
•
•
•
•
•
•
•
Paramagnetism
Diamagnetism occurs in materials where the magnetic fields due to electronic
motions of orbiting and spinning completely cancel each other.
Thus, the permanent (or intrinsic) magnetic moment of each atom is zero and
the materials are weakly affected by a magnetic field.
For most diamagnetic materials (e.g., bismuth, lead, copper, silicon, diamond,
sodium icon, chloride), m is of the order of 10 -5.
In certain types of materials called superconductors at temperatures near
absolute zero, "perfect diamagnetism" occurs: m = -1 or 1 or µr = 0 and B = 0.
Thus superconductors cannot contain magnetic fields.
Except for superconductors, diamagnetic materials are seldom used in
practice.
Diamagnetism is an extremely weak effect. Even though all materials exhibit
diamagnetism, the effect is so weak that you can usually ignore it.
This explains the commonly known fact that most materials are not affected by
magnets.
Materials whose atoms have nonzero permanent magnetic moment may be
paramagnetic be paramagnetic or ferromagnetic.
•
•
•
•
•
•
•
•
•
•
•
Paramagnetism occurs in materials where the magnetic fields produced by orbital and
spinning electrons do not cancel completely.
Unlike diamagnetism, paramagnetism is temperature dependent..
For most paramagnetic materials (e.g., air, platinum, tungsten, potassium), m is of the
order +10-5 to +10 -3 and is temperature dependent.
Paramagnetism, while stronger than diamagnetism, is another very weak effect and can
usually be ignored.
The reason for its weakness is that the electrons in each atom are always grouped in
pairs that spin opposite to one another.
Hence, paramagnetism can only occur in atoms that have an odd number of electrons.
For example, aluminum has an atomic number of 13 and thus has an odd number of
electrons. It therefore exhibits paramagnetic properties.
The random thermal motions of the atoms tend to prevent the dipole moments from
lining up well, even when exposed to an external field.
Such materials find application in masers (a device or object that emits coherent
microwave radiation produced by the natural oscillations of atom or molecules between
energy levels).
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Ferromagnetism
• A ferromagnetic material is like a paramagnetic material with the added
feature of “domains.”
• Each domain is a microscopic patch of billions of atoms that have all
lined up their dipole moments in the same direction.
• It so happens that the quantum mechanical properties of certain
materials, notably iron, cause these domains to form spontaneously.
This is due to the electron spin and to the collective behavior of the
outermost electrons of large groups of atoms.
• Normally, the domains are randomly oriented so that the material still
has no overall magnetic dipole.
• However, when a magnetic field is applied the domains that align to
the field grow, while domains of other orientations shrink.
• In addition, the domains have a tendency to freeze in place after
aligning. In other words, ferromagnetic materials have memory. For
example, if a bar of iron is placed in a strong field and then removed,
the bar retains a net magnetic dipole moment. It has become a magnet.
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•
Some other examples of ferromagnetic materials are nickel and cobalt.
• Several metal alloys are also ferromagnetic.
• Ferromagnetic materials have the following properties: They lose their
ferromagnetic properties and become linear paramagnetic materials
when the temperature is raised above a certain temperature known as
the curie temperature.
• Thus if a permanent magnet is heated above its curie temperature
(770°C for iron), it loses its magnetization completely.
• They are nonlinear; that is, the constitutive relation the B = µoµrH does
not hold for ferromagnetic materials because µr depends on B and
cannot be represented by a single value.
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•Permeability quantifies how a material responds to magnetic fields in a manner
analogous to how permittivity quantifies the material response to an electric field.
•Permeability is a measure of the magnetic energy storage capabilities of a material. A
material with a relative permeability of 1 is magnetically identical to a vacuum, and
therefore stores no magnetic energy.
Paramagnetic and ferromagnetic materials have relative permeability greater than 1,
which implies that the material aligns its dipole moments to an induced field and
therefore stores energy.
•Higher permeability translates to a larger reaction and higher energy storage.
• Diamagnetic materials are characterized by relative permeabilities less than 1.This
fact implies that the material aligns its dipole moments opposite to an induced73field.
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