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Examples
1
2
History of the displacement vector D
(1791 โ1867)
Michael Faraday
Q
I
Q
Insulator
I?
Conducting current Displacement current
โข Faraday tried many types of insulators but never prevented the displacement of charge
โข The charge displaced was always Q, equal to that placed on the left plate.โข A sketch of the original system used by Faraday:
gold-leaf electroscope
3
The relation between ๐0, ๐, ๐๐ ๐ท,๐ธ in Linear-Isotropic-Homogeneous medium
00 0 0
0
0 0 0 0
N N
S S S
N N N
S S
qE ds E ds E ds q
E D E D D ds q D ds divD
๐ช๐โ๐
๐ธ๐ =๐0,๐๐0+
++++++++++
+++++++++++
-------------
-
+
+
+
-
-
-
๐ฌ๐ท ฮค= ๐ ๐ท ๐บ๐
+๐ ๐,๐ โ๐ ๐,๐โ๐ ๐ท +๐ ๐ท
๐ธ๐=๐ธ๐-๐ธ๐=๐๐
๐บ๐
+๐๐ = ๐ ๐,๐ โ๐ ๐ท โ๐๐= +๐ ๐ท โ ๐ ๐,๐
The polarization per unit volume:
๐
๐ =๐
๐=
๐๐โ๐
๐=
๐๐โ๐ดโ๐
๐=
๐๐โ๐
๐= ๐๐ [1]
[3] 0 0 1 1 ; oi i o
qD E E E
S
[2]0 eP E
The relation between ๐ธ and ๐ is:
And therefore the resultant field in the medium is:
0 0 0
0
0 0 0 0 0 0
0 0 0
0
/
1 1 1
PN P
P e
e e e r
E E E E E
E E E P D P D E
D E E E E
unitless
A privet case of orthogonal fields:
D
4
Boundary condition for the electric field โโ โ 0
1 21 2 1 2
1 2
1. 0
b d
T TT T T T
c a c
D DE dl E dl E dl E E
1 2 1 2 1 2
1 1 2 2
2. ss N N N N s N N
s top bottom
N N s
QQ D ds D ds D ds D D s D D
s
E E
1 2 1 1 2 20 (3) (4)s N N N NWhen D D and E E
An important emphasis:s spis not
Is this is true in the case of time dependent fields?
5
๐๐ ๐1
๐๐ ๐2
๐ธ๐2
๐ธ๐1
๐๐ ๐๐ = ๐๐ ๐1- ๐๐ ๐2
Regarding the charge on the interface plan:
1 1 2 2 0
0 1 1 0 2 2
0 1 1 0 2 2
0 1 0 1 1 0 2 0 2 2 0
0 1 1 0 2 2
0 1 2 2
0 , (4) ; int :
; int 1 :
1 1
; :
; :
s N N r
r N r N r
N N
N N N N
N N N N N sp
N N N
When E E roducing
E E roducing
E E
E E E E recalling that P E
E P E P recalling that P
E E P
1 2 1
2 1
1 2
0 0
:
(5)
N sp sp
sp sp spn
N N
P and therefore
E E
Why is ๐๐ ๐๐ not expressed in Equation 4? ๐1
๐2
As in a capacitor!
The advantage of Eq. 4 over 5 is due to the value of ๐ being known, while ๐๐ ๐๐ is unknown and to be calculated.
6
0 0 01 1 2 2 0 0 0(4) ;N N
r r
E EE E E E E E E
The particular case of EN
2
0 0
2 2
0
4
4 4r r r
q
E r q qE
r r
and for homogeneously charged infinite plane: 0 0
0
s
s s
r r r
EE
and so onโฆ and therefore in all the expressions of electric field one should introduces ๐ instead of ๐0.
As for the electric potential:
0
0
0
0 0
1
b b
ra a
b b
r r
a a
EE dl dl
VVV
VE dl E dl
Is crossing the interface between two media might change the direction of the electric field vector and how?
The electric fields of point charge, infinite wire, charged sphere, infinite charged plane and more, where all calculated for vector field perpendicular to the medium surface. Assuming medium-vaccumeinterface, one get from Eq. 4:
That is to say that the field in vacuum is ๐๐ times greater than that in medium.
Employed this on the case of electric field of a point charge one gets:
7
As for the electric flux:
0
0
0 0 0
1rs s
r r
s s
EE ds ds
E ds E ds
As for the effective charge qโ:
**0 0 0
0 0r r r r
q qq q q qq
And the capacitance:
0
0
r
r
Q QC C C
V V
8
Continuity of the current density ิฆ๐ฝ
๐1, ๐1
๐2, ๐2s
ิฆ๐ฝ๐1
ิฆ๐ฝ๐1
ิฆ๐ฝ๐2
ิฆ๐ฝ๐2
From Kirchhoff Junction law we get: 1 2 1 2 1 20 N N N N N N N
Cylinder Cy top bottom
J ds J ds J ds J ds J J J J
As for ิฆ๐ฝ๐ ? On that we learn from the continuity of ๐ธ๐1 2
1 2
1 2
T TT T
J JE E
๐1
๐2
21
1 2 1 21 2 1
1 2 1 1
22 1
1
tan ; tan tan
tan tan
T
T T
N N N
JJ J
J J J
What happens to าง๐ฝ in the interface conductor-insulator (๐2 โ 0)?
The relation between the direction of J1 and J2 at the interface:
In insulator ๐ฝ = 0 โน ๐ฝ๐1 = 0 = ๐ฝ๐2. On the other hand ๐ธ๐2 โ 0, ๐ฆ๐๐ก ๐๐๐๐ก๐๐๐๐๐๐ก๐ ๐กโ๐ ๐๐๐๐ก ๐กโ๐๐ก ๐ฝ๐2 = 0since ๐ฝ๐2 = ๐2๐ธ๐2 but ๐=0. In any case, at any point within a conductor ๐ธ = ๐ท = 0, ๐. ๐. ๐ธ1 = ๐ท1 = 0, out of which ๐ธ๐2 = 0 ๐๐๐ ๐ท๐2 = 0
9
10
Does this result correlate with what we know about currents and equi-potetial surfaces?
Therefore, for ๐1 โซ ๐2 โน ๐2 โ๐
2and hence for every current arriving the conducting-isolator, it apparently
flow along the interface. However, though current can not exist in insulator, electric field does.
In summary, the only component of the electric field exist is perpendicular to the interface conductor-isolator (see figure)
21
1 2 1 21 2 1
1 2 1 1
22 1
1
tan ; tan tan
tan tan
T
T T
N N N
JJ J
J J J
11
0
1 2
1 1s NJ
The continuity of J in the interface between two conductors:
1 1 2 2
1 21 2 1 2
1 2
1 2
1 2 1 2
; int
;sin
1 1
N N s
N Ns N N N
s N N
JE E roducing E
J Jce J J J
J J
From the continuity of ๐ท โ๐ท๐ = ๐๐ :
In good conductors ๐ โ ๐0 ๐๐๐ โ๐๐๐๐:
Why a voltage difference is developed between the to sides of a resistor?
12
Conductor Insulator
J=ฯE=-ฯ gradV D=ฮตE=-ฮต gradV
div J=0 Div D=0
delta JN =0 delta DN=0
ฮค๐ฝT1 ๐1 = ฮค๐ฝT2 ๐2 ฮค๐ท1๐ ๐1 = ฮค๐ทT2 ๐2
๐ = ฮค๐ ๐๐ด ๐ถ = ฮค๐๐ด ๐
๐ ๐ถ =๐
๐= ฯ
0E dl 0E dl
The duality between J and D (electrostatics):
2/27/2020 13
2 2
4
:
( )
1 ( )
1 10 ( )
0.5Earth
B
B
MKS
Weber Wb
WbB T Tesla
m m
T Gauss cgs
B G
Magnetism: Units
๐พ๐ =๐ฒ๐ โ ๐๐
๐๐๐๐ โ ๐จ= ๐ฝ๐๐๐ โ ๐๐๐ = ๐ป โ ๐๐ =
๐ฑ
๐จ= ๐๐๐๐ด๐
0 0
2
ห 1[1]
4 4l l
I dl r IB dl grad
r r
:ืืืฉืืื ืืืงืืืจื ืืืืข ืืฉืืืื
: [2]From vector analysis curl A curl A A grad A grad curl A curl A
:ืื[ 1]ื ืืชื ืืืืืข ืืช [ 2]ืืืืฆืขืืช
0 0
0
1
4 4
; [3]4
l l
l
I Idl dlB curldl curl curl
r r r
I dlcurl curlA B curlA
r
= 0
2/27/2020 14
BiotโSavart law and the vector potential เดฅ๐จ
:ิฆ๐ดืืืืืช ืืคืืื ืฆืืื ืืืืงืืืจื
0 0 0 0
, ,
ห
4 4 4 4l l s l s V
I dl Jdsdl Jldsdl JdvA
r r r r
:ืืืืื ื, ืืืืื ื ืืชืืืืช ืืืื ืืืจื ืืื ืืื ืืื ืฉื ืืคืื ืฆืืื
V consequently one get the similarity between ๐ ๐๐๐ าง๐ด:
2 2
2 2 2; ;
V
x x y y z z
V A J
A J A J A J
4 4
V
V
dvQV
r r
2/27/2020 15
:ืืืฆืื ืืืื ืืืจืืืช ืืืืืคืจื ืฆืืืืืช ืฉื ืืืง ืืืคืจ
0 f
L
B dl I
2/27/2020 16
Stokes theorem
0 0( )s s
B ds J ds B J
The integral and differential expression of Ampere's law:
Current is the source of magnetic field and charge is the source of electric field
2/27/2020 17
Lorentz force
The magnetic force acting upon a current caring wire is given by:
sin sin sin sinB eF Il B IlB JSlB vVB QvB Qv B
e
s s
I J ds v ds ิฆ๐
ิฆ๐
๐ตฮธ
And in the presence of eclectic field the total force is:
Hendrik Antoon Lorentz 1853 โ1928
Type equation here.เดค๐น = ๐ เดค๐ธ + าง๐ฃ ร เดค๐ต
2/27/2020 18
Bemf
s
d dV B ds
dt dt
Hence the circulation integral should include the contribution of the magnetic field as well:
0 BTotal T emf T
l s l s
d d dV E dl V B ds E dl B ds
dt dt dt
s s
d dBE ds B ds E
dt dt
Faraday law
StokeStokes theorem
Conservative field
Michael Faraday (1791 โ 1867)
2/27/2020 19
, 2
,
0 0 0
: ; ; [ ; ] ; ;f n V V
e f n V
s s
qI E ds divE D ds q divD E gradV V
2
0: 0 ; 0 ; [ 0 ; 0] ; ;s s
II B ds divB H ds divH B curlA A J
: ;B
General
l s
d Ad d dBIII E dl B ds E A E gradV A
dt dt dt dt
0 0
0
: ? ; ?
[ ? ; ?]
l
l
IV B dl I B J
BH H dl I H J
Maxwell Equations :
20
:ืืฉืืืช ืืงืกืืืื
curlH J divcurlH divJ 0 0
:Following the continuity law
0
0
0
0
( ) 0 ( ) ( ) 0
VV V
C D
divJ divE divE
div J E div J D div J J
0 0e
D
s s
d d dI D ds E ds
dt dt dt
: ืขืืืืื ืืงืกืืืื
0 0 0 0 0
0 0 0 0
: ( ) ( ) ( )
( ) ( ) ( )
eC D C C
l s
C D C C
d dIV B dl I I I I E ds
dt dt
B J J J E J D
Relating to the differential aspect of equation VI
H-The magnetic field intensity
0 0
0 0
;f f
L L
f
L
B BB dl I dl I H B H
AH dl I H
m
:ืืืฉืืืืช ืืืฉืืืืืช ืืงืกืืืื
0 0
( ) ( )
0
:f f
L L S L S L
f
B dl I Via StcokesTheorm B dl XB ds J ds
XB J
:ืืืืคื ืืืื ื ืงืืfXH J
:ืฉืืืช
:
: int
B Themagnetic flux density
H Themagnetic field ensity
2/27/2020 21
Types of magnetism:
Diamagnetism Paramagnetism Ferromagnetism
No Yes but weak Yes and constant Magnetic field
Electrons spin orbital Electrons spin Magnetizing regions Source of magnetism
Opposite Identical Varies, Hysteresis Direction of B in respect to H
Cupper, Lead, Diamond, Mercury, Silicon
Aluminum, Calcium, Magnesium, Tungsten
Iron, Nickel, Cobalt Typical materials
โ โ10โ5 โ 10โ5 ๐ ๐ โซ 1 ๐๐Typical
โ 1 โ 1 ๐ ๐ โซ 1 Typical ๐๐
2/27/2020 22
Ferromagnetism
Diamagnetism
Paramagnetism
2/27/2020 23
24
Within each grain are a series of lighter
and darker stripes (imaged by using the
optical Kerr effect) that are ferromagnetic
domains with opposite orientations.
Averaged over the whole sample, these
domains have random orientation so the
net magnetization is zero. Moving
domain exist as well.
๐ฉ
1 2 1 1 2 20 N N N N
s
B ds B B H H
1 2 1 2
1 2
1 2
1 2
1 2
0 : 0
f
f T T f T T f T f
L
T Tf
T Tf T
IH dl I H H L I H H i H i
L
B Bi
B BFor i H and
2/27/2020 25
Boundary conditions for B and H