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Chapter 1
Electro Magneto-statics : Highlights
2
The following presentations were designed, drawn and edited by Motti Deutsch (0544 767 567, [email protected]).
โข Mathematics will be teached in respect to the physical problem.
โข Exercises: questions and solutions will be set in the courseโs site.
โข There is no obligation to submit solved exercises.
โข Three exams: 45 minutes each, questions taken from exercises (closed material) (30%).
โข Test, 3 hours, 3 out of 4 question should be shoved (open material) (70%).
โข Bibliography:
1. Umran & Aziz S. Inan: Engineering Electromagnetics
2. D.J. Griffiths : Introduction to Electrodynamics
3. Berkley: Electricity & Magnetism
4. J. D. Jackson: Classical Electrodynamics (Third Edition)
5. Alonso โ Finn, Fundamental University Physics:
Mechanics I: Chapters 6 and 11 Fields and Waves: Chapter 17
Coulomb's lawGauss Theorem
Divergence Theorem
3
Units
4
Chargescgs: when two equal charges, 1cm apart, act upon each other by a force of 1 dyne the charge is defined as 1 electrostatic unit โ esu or statcoulomb.
MKS: Coulomb (C - International System). It is the charge carried by a constant current of one ampere in one second.
Coefficients:
5
Given that:
๐๐
6
๐
S
ิฆs๐
๐ธ
๐๐=๐ธ๐๐ = ๐ธ๐บ๐๐๐ ๐ = ๐ธ โ ๐บ
๐๐๐=๐ธ โ ๐๐
๐๐ = เถฑ๐ธ โ ๐ ๐
๐บ
Where ๐ธ the force per unit charge and/or the flow density. When ๐ธ is space-dependent then:
The total flow lines through S:
The general expression for line of flux
๐ โ ๐๐๐๐๐๐๐๐๐๐ข๐๐๐ Surface vectorิฆs โ
7
The gradient (1)
๐1๐2 ๐ธ โ แ๐ dl= ๐1
๐2 ๐ธ๐๐๐ ๐๐๐ = ๐1๐2 ๐ธ๐๐๐
= โ ๐1๐2 ๐๐= ๐1
๐2 โ๐๐
๐๐d๐
and therefore:
๐ธ โ แ๐ = ๐ธ๐๐๐ ๐ = โ๐๐
๐๐โก โ
๐๐ฝ
๐๐directional derivative โ ๐ธ =
๐๐
๐๐ max
๐1 ๐ฅ, ๐ฆ๐2 ๐ฅ, ๐ฆ
x
y
๐๐
๐๐ฅ
๐๐ฆ
The gradient essence
๐๐
๐๐=๐๐
๐๐ฅ
๐๐ฅ
๐๐+๐๐
๐๐ฆ
๐๐ฆ
๐๐=๐๐
๐๐ฅ๐๐๐ ๐ผ +
๐๐
๐๐ฆ๐๐๐ ๐ฝ =
๐๐
๐๐ฅเท๐ฅ โ แ๐ +
๐๐
๐๐ฆเท๐ฆ โ แ๐=
๐
๐๐ฅเท๐ฅ +
๐
๐๐ฆเท๐ฆ ๐ ๐ฅ, ๐ฆ โ แ๐ โก ๐๐๐๐ ๐ โ แ๐
๐ผ
๐ฝ
๐ธ= - ๐๐๐๐๐
โ๐๐
๐๐1=
๐2 ๐ธ โ ๐๐ = โ[๐ ๐2 โ ๐ ๐1 ]The work per unit charge is
The total derivative of ๐ ๐ฅ, ๐ฆ with respect to ๐๐:
Follow the yellow highlights:
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The gradient (2): direction
๐
โ๐1
โ๐2
โ๐3
โ๐
โ๐1=โ๐
โ๐2=โ๐
โ๐3= 0 โ
โ๐๐
โ ๐ธ=โ ๐๐๐๐๐ โฅ เท๐ข ๐
0 = limโ๐โ0
โ๐
โ๐=
๐๐
๐๐๐= ๐๐๐๐๐ โ เท๐ข ๐ = 0
The electric field lines (flux) is always perpendicular to an equipotential surface and hence work done in moving a charge between two points on an equipotential surface is zero.
9
Some other relations:
0 0
.
1. ; ;
2. 0 ; ???
e
S
cl tr
L
qE ds E divE
V E dl E V gradV
Combining the right expressions of 1 and 2 yield the del squared (Laplacian) โ2 :
Hence, ๐ธ can be calculated via:
1. Coulombโs law2. Gaussโs law
3. The potential: ๐ธ = โ๐๐๐๐๐ (reason for potential continuity)
2
0
( )E V divgradV V
๐ป 2๐ =๐
๐0๐๐๐๐ ๐ ๐๐โฒ๐ ๐๐๐ข๐๐ก๐๐๐
๐ป 2๐ = 0 ๐ฟ๐๐๐๐๐๐โฒ๐ ๐๐๐ข๐๐ก๐๐๐
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๐๐
๐๐
๐๐
๐๐
๐๐
๐๐
๐๐
๐
๐
๐
๐ ๐
๐
๐
๐ง
๐ฅ
๐ฆ
๐ฆ = ๐ฆ0
๐ฅ๐
๐ฅ๐ + ๐๐ฅ
๐ฌ
Circulation path integral of E along closed
Micro curve around area ds: ืฏ๐ ๐ฌ โ ๐ ๐๐
Circulation path integral of ๐ฌ along closed
Macro-curve L around area S: ืฏ๐ณ ๐ธ โ ๐๐ฟ
๐ง๐
๐ง๐ + ๐๐ง
(1) The differential aspect of the circulation theorem - The Rotor
4 0(x , y , z)( d )z cdw E dl E z
3 0(x, y , z dz)( d )x cdw E dl E x
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1 0(x, y , z )dx cdw E dl E x ๐ฅ
๐ฆ
๐ง
๐ฆ = ๐ฆ0
2 0(x dx, y , z)dz cdw E dl E z
0 0 0 0
( )
(x dx, y ,z) (x , y ,z) ( , , dz) ( , , )Infin
z c z c x c x c
l ditesim sal
E dl E E dz E x y z E x y z dx
xEdz dx
z
๐ ๐ = ๐๐ง๐๐ฅ๐ฌ
๐ฅ๐ ๐ฅ๐ + ๐๐ฅ
๐ง๐
๐ง๐ + ๐๐ง
๐
zEdx dz
x
(2) The circulation integral of ๐ฌ over infinitesimal path ๐ (line integral) which defines the area (region) ๐๐
And the work done along ๐:
12
หxzy y
yx z plane
EEE dl dxdz XE ds XE ds y
x z
( )l ds
E dl XE ds
(3) Line integral over infinitesimal path
And in general, regarding line integral around infinitesimal region ๐๐ which has a three-component projection on x-y, x-z, and y-z planes:
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๐๐
๐๐
๐๐
๐๐
๐๐
๐๐
๐๐
๐
๐
๐
๐ ๐
๐
๐
๐ฌCirculation path integral of E along closed
Micro curve around area ds:
๐ืฏ ๐ธ โ ๐๐=๐ป๐๐ธ โ ๐ ิฆ๐ แ= 0,๐ค๐๐กโ๐๐ ๐บ
โ 0 ๐๐ ๐กโ๐ ๐๐๐๐๐ข๐๐๐๐๐๐๐๐ ๐ณ
๐
Circulation path integral of ๐ฌ along closed
Macro-curve L around area S: ืฏ๐ณ ๐ธ โ ๐๐ฟ( ) ( ) 0E dl E dl
๐๐
๐บ
เถฑ
๐
๐ธ โ ๐๐ =
๐ ิฆ๐
๐บ
๐ป๐๐ธ๐โ ๐ ิฆ๐ = เถป
๐ณ
๐ธ โ ๐๐ฟ
เถป
๐ณ
๐ธ โ ๐๐ฟlim๐ ิฆ๐ โ0
๐ ิฆ๐
๐บ
๐ป๐๐ธ๐โ ๐ ิฆ๐ =เถฑ
๐บ
๐ป๐๐ธ โ ๐ ิฆ๐ =
Stokesโ theorem
Stokesโ theorem
The sum of the private microscopic circulation integrals (along ๐๐ ๐๐๐๐ข๐๐ ๐ ิฆ๐ ๐) over the entire
macro plane ๐บ, arrested by the macroscopic path ๐ฟ is:
๐
๐๐ ๐ณืฏ ๐ธ โ ๐๐ฟ = ๐ป๐๐ธHence, the line integral per unit area is:
14
0 = เถป
๐ณ(๐บ)
๐ธ โ ๐๐ = เถฑ๐บ(๐ณ)
๐ป๐๐ธ โ ๐ ิฆ๐
เถป
๐ณ(๐บ)
๐ธ โ ๐๐ = 0
Hence, according to Stokesโ theorem:
๐ป๐๐ธ=0And therefore:
All the electrostatic fields are conservative:
21 1 1 1
ห ( ) X ( ) ( ) 0n n n n
i i i ii
i i i ii i i i
q q q qE K r K grad E K curl grad K curlgrad
r r r r
2 2 2 2 2 2
ห ห ห
ห ห ห
x y z
x y xx y z y z z y z x x z x y y x
x y z
curlgrad
=0 =0 =0
0
For conservative electric vector field:
15
The meaning of Curl
๐น๐ฅ
๐ฅ
๐ฆ โข A flow is traveling in the ๐ฅ โ ๐ฆ ๐๐๐๐ ๐ก๐๐ค๐๐๐๐ เท๐ฅ
โข Where does torque exist? (upper/lower panels)
โข How this can be mathematically expressed?
๐๐น๐ฅ๐๐ฆ
= 0
๐๐น๐ฅ๐๐ฆ
> 0
๐ฅ
๐ฆ๐น๐ฆ
๐๐น๐ฆ
๐๐ฅ< 0
โข Does any other fundamental heterogeneous situation exist?
๐น๐ฅ
๐ฅ
๐ฆ
โข Within the flow, a torque testing profiled wheel with teeth is placed. The flow acts upon the wheel (ืืืื ืฉืื ืืื)circumference by a torque of the force ๐น๐ฅ.
๐น๐ฅ
๐ฅ
๐ฆ๐น๐ฆ
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๐๐น๐ฆ
๐๐ฅโ
๐๐น๐ฅ
๐๐ฆโก ๐ปX ิฆ๐น
๐งโ 0 since โ
โ ๐ปX ิฆ๐น๐ง= โ
๐๐น๐ฆ๐๐ฅ
โ๐๐น๐ฅ๐๐ฆ
ฦธ๐ง = โ๐๐น๐ฆ๐๐ฅ
+๐๐น๐ฅ๐๐ฆ
ฦธ๐ง โ 0
for the general case in ๐ฅ โ ๐ฆ plan, regarding our example:
Therefore, in the present example the direction of ๐ปX ิฆ๐น๐ง
is
โ ฦธ๐ง which presents a clockwise rotation and since ๐ปX ิฆ๐น๐งโ 0
โน ิฆ๐น is non-conservative.
Examples
๐ฅ
๐ฆ
17
continuity equation
๐
โน ืฏ ิฆ๐ฝ โ ๐ ิฆ๐ = -๐๐
๐๐ก
๐๐๐ฃ ๐ฝ ๐๐= -๐
๐๐ก๐๐๐
Divergence theorem
โน ๐๐๐ฃ ๐ฝ= -๐๐
๐๐กโน
๐๐๐ฃ ๐ฝ+๐๐
๐๐ก= 0Continuity equation via ๐:
ิฆ๐ฝ
+q
โน ๐๐๐ฃ ิฆ๐ฝ+๐ ๐ ๐๐๐ฃ๐ธ
๐๐ก= 0 โน๐๐๐ฃ (ิฆ๐ฝ+
๐ ๐ ๐ธ
๐๐ก)= div ( าง๐ฝ๐ถ+
๐ เดฅ๐ท
๐๐ก)= div ( าง๐ฝ๐ถ+ าง๐ฝ๐ท)= 0
But:0
0
divE divE
and introducing it into the above equation yield the expression of the continuity equation via the electric field.
Conducting and displacement currents
18
Kirchhoffโs current law (junction rule)
๐๐๐ฃ ๐ฝ+๐๐
๐๐ก= 0We get:
In steady state ๐๐
๐๐ก= 0
๐๐๐ฃ ๐ฝ= 0 divergence 0 = เถฑ๐๐๐ฃ ๐ฝ ๐๐ = เถป ิฆ๐ฝ โ ๐ ิฆ๐ =0
ฯ๐ ๐ผ๐=0 Kirchhoff's junction rule
Junction
The right integral teaches that the sum of the currents through the closed shell is zero. The
practical meaning of that is that (regarding electric circuits) the sum of currents in (through) a junction must be zero (at zero or at very low frequency).
๐ผ1
๐ผ2
๐ผ3
๐ผ4
๐ผ5
Explain the significance of ๐๐
๐๐กโ 0
19
Relaxation time of the charge density in conductor having conductivity ๐ and permittivity ๐0
0
( )d
divJ div E divEdt
0
0
0 0 00 0
00
( ) ;t
tt td l s
dt t e e RCs l
J E 0divE
Silver: ฯ = 6.2 x 107S๐โ1 ; ๐ โ ๐0 = 8.85 x 10โ12๐โ1๐โ2๐ถ2
๐๐๐๐๐ฃ๐๐ =8.85 x 10โ12
6.2 x 107โ 10โ19sec
5
3
10 sec
4 10 sec
10 20
50
DDW
anber
mica
quartz
hours
days
Insulators:
20
Insulators Have no free charges. In the presence of electric field they are electrically polarized. The resultant electric
field/charge within the insulator is never zero.
-q
+q
๐ธ๐๐ฅ
โข In the figure the centers of the positive and negative charges are coinsized
โข An electric field acts upon the atom upwards
โข As a result, mostly the negative cloud-charge moves downwards while the positive nuclei upwards.
โข Though subjected to electric field, it is weak enough to keep both the negative cloud shape and distribution (symmetry) spherical, and the density constant, ๐ = ๐0.
๐
โข Now, the distance between the centers is ิฆ๐
โข Hence, the electric field from the negative charge at ิฆ๐ i:
3
0
3
0 0 0
3
0
4 3( ) ( )
3 3 4
4 ( )
P ex
ex
qr
r qaE r r E r
a
a E r qr p
โก ๐ผ
This relation is sufficiently correct (by factor 4) for simple atoms. 4๐๐3 is defined as the atomic contribution and called the
atomic polarizability ๐ถ. Next, if in a dielectric unit volume there exist ๐ atoms, then the polarization per unit volume ิฆ๐of that substance is:
Susceptibility is the level (measure) of how much substance is electrically polarized in the presence of electric field.
๐ = ๐0๐๐ผ๐ธ๐๐ฅ = ๐0๐๐๐ธ๐๐ฅ ; ๐๐ โก ๐ธ๐๐๐๐ก๐๐๐ ๐ ๐ข๐ ๐๐๐๐ก๐๐๐๐๐๐ก๐ฆ
21
๐ฌ
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
(a)
๐ฌ
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
(b)
Nonpolar The electric field both induces and orients the electric dipole.
+-
Polar
Si
Sex
Within the dialectic, over macroscopic volumes (Si), the net charge is zero, while
on the surface of this volume (Sex) a net charge is accumulated (induced) and
causes the electric dipole of the entire substance.
23
Units of ๐, ๐๐, and the displacement vector ๐ท :
3 3 3 2 (SCD)
P qr Cm CP
m m m msurface charge density
From Gauss's law:
2
0 0
0
; SCDS S
qE ds E ds q E D Cm
2 2
0 0
e ex e ex
e
P Cm E E Cm
Dimensionless quantity
But we show that:
0 ( )E D displacement vector
The displacement vector ๐ท is defined as: