+ + + + + + + + + + + +
- - - - - - - - - - - -
+Qfree on inner surface
-Qfree on inner surface
Interior points electric field must be zero
-qbound
+qbound
E
D
P
0
1( )E D P
Symmetry – fields must be uniform – field lines perpendicular to plates
+ + + + + +
- - - - - - -
+ + + + + + + + + + + +
- - - - - - - - - - - -
+Qfree on inner surface
-Qfree on inner surfaceplate separation d
area of plates A
conductor dielectric+
+
+
+ +
-
-
-
0E
E
freebound
Gauss’s Law
0
free boundE
frequency
dielectricConstant(polarmolecules)
+ + + + + + + + +
+ + + + + + + + +
- - - - - - - - -
dy
F
Fme
+ + + + + + + + + + + +
- - - - - - - - + + + + + + + +
- - - - - - - - - - - -
f
f
b
b
f
f
bb
Electric displacement Electric field Polarization
+q -q
d
ep
A B C
only some of the windings are shown
Integration paths
L
dA3
Bz
dA1
Bz
dA2
Br
1 2 2
0 0
(2 ) 0 0S S S
z z r r
B dA B dA B dA B dA
B A B A B r L B
Z
Y
X
zB
xB
yB
B
rB
B
N
Bz1
s
Bz2
x
I
Br = 0 0d
A
Ienclosed = 0
s
Bz2
x
I
Br = 0 0d C
Ienclosed = n s I
Bz1= 0
Ienclosed = 0
xI
Bz2
Bz1= 0
xx
xx
B
single turn of wire with current I
around integration loop Bdr = 0 and Br = 0
outside loop Bz = 0
BFe HFe
Bgap Hgap
Bair Hair
i
coil windings
gap region
iron core
r
XXXXXX
.
.
.
.
.
.
.
.1
2 3
4
Circulation loop: square of length L
Cross-section through electromagnet
B H M
Current iout of page
Current iinto page
B
width L
thickness t
area
Aq = - e
electrons are the chargecarriers in copper
0r A
d
0r A
d
E
+ - + - + - + -
+ - + - + - + -
+ - + - + - + -
+ - + - + - + -
b b
+ + + + + + + + +
+ + + + + + + + +
- - - - - - - - -
dy
F
+q -q
d
+ + + + + + + + +
- - - - - - - - -
x L-x
V r
C = CA + CB
C
CA CB
Induced dipole moment – helium atom
-e -e+2e
Zero electric field – helium atom symmetric zero dipole moment
-e +2e -e
E
d
A B
effectively charge +2e at A and -2e at B
dipole moment p = 2 e d
p
Induced dipole moment – sulfur atom
-8e -8e+16e
Zero electric field – helium atom symmetric zero dipole moment
-8e +16e -8e
E
d
A B
effectively charge +16e at A and -16e at B
dipole moment p = 16 e d
p
-q +qd
r1 r – (d/2)cos
r2 r + (d/2)cos
r
P
ErE
(d/2)cos
+ + + + + + + + +
- - - - - - - - -
+f
-f
dA
-b
+b
d
+q-q
+f
-b
+b
-f
O
r
S
P
+
dr
Pcos
surface S
++
- - -
E Area of the shaded ring
between and + d 2 sinr rd
sinr
Width of ring r dRadius of ring r sin
+
++
- - -
E
element of charge dqe
electric field at O due to charge dqe
E0
E0 cos
E
a
+Ze
ad
+Ze
d << a
E
F
F
F
d
+Q
- Q
p
0 π/2 π
0
+ p E
- p E
U
+ -E
U = - p ELowest energy state
+-
U = 0
+-
U = + p Ehighest energy state
= 0 = 180o = 90o
1/T
r - 12
slope3 B
n p
k T
0
intercept e i
n
T
Po
p E / k T
1
0 10
oP
n p
slope = 1/3
non-conducting liquid
airconducting
sphere q
a
Gaussian surface S
r
Symmetry field lines must be radial
non-conducting liquid
airconducting
sphere q
Symmetry Eairt = Eliquidt Eair = Eliquid = E
Eairt
Eliquidt
field lines of E field lines of D
+
field lines of E
field lines of D
+
++ ++ +
+ +
++ +
greater concentration of chargeon surface bounded by liquid
+ -
E
induced dipoles due to shift in electron cloud
+
+
-
rotation orientation of polar molecules
-+
shift in atoms due to ionic nature of bond
NS
1 2 3
4
HFe Hair0
airair
BH
2 3 5 6
1 2 4 5
2 6 3 5
1 5 2 4
0Fe air air Fe
Fe Fe air air
Fe air
H dl H dl H dl H dl H dl
H dl H dl H dl H dl
H H
Circulation loop: square side L
56
outsideB
insideBoutsideA
insideA
0inside outside
inside outside
B dA B A B A
B B
B-field lines – form continuous loops
Gauss’s Law for magnetism Cylindrical Gaussian surface
H
M
0M B
H
Bound surface currents im (right hand screw rule) M
B
N pole
im
un-magnetized piece of iron
N
Bar magnet bought near un-magnetized piece of iron
B
N N
Bar magnet will attract the iron that was initially un-magnetizednorth pole attracts
south pole
Fe ramp Cu ramp plastic ramp
N N N
Circulation loop for circulation integration used in applying Ampere’s Law
N
N
Hiron
Hair
d
II
B
H(0,0)
B
d
B, Hgap Mgap = 0
B = Bgap = Biron
Hiron
Miron
B
B, Hgap Mgap = 0
B = Bgap = Biron
Hiron
Miron
B
PERMANENT MAGNET
ELECTROMAGNET
B
IX
Y
Z
thickness t
width w
area A = w t
magnetic field in Z direction
current in X direction
Schematic diagram of a Hall Probe
+ + + + +
- - - - -
+ + + + +
- - - - -
I
X
Y
B .
Z directionout of page
charge carriers electrons (-)eg wire, N-type semiconductor
charge carriers positive (+)eg holes in P-type semiconductor
HE
HE
+_ VHVH
HH
H H
VE
wV E w
width w
HH
H H
VE
wV E w
I
area A
length L
+
_
V
resistance Rresistivity conductivity number density n
J
E
_v
electron
e- e-
e- e-
X
Y
Z
objectimage
electron beam
(0, , )A y zB B B
AvA
+Y
+X
+Z
Bz
By
vy
Fx
Electron at A moving parallel to +Y-axis
Electron acted upon by the radial component of the magnetic field force on electron in +X direction +X-component to the velocity
axis for the motion of the electron beam
radial component of magnetic field
due to Bz
+Y
+X
+Z
Bz
By
vx
Fy
Electron at B has a velocity component in the +X direction
Electron acted upon by the axial component of the magnetic field By force on electron in -Z direction i.e. towards to axis focusing action
axis for the motion of the electron beam
radial component of magnetic field
Fz due to By
due to Bz
B
I
dl
F
..
.
. . ..
.dl
H
ifree
H
0M
0M
external magnetic field
Electrostatic capacitor
Ar
0r A VC E
d d
E
V
Electrolytic capacitor
A
d
r
E
V
d
dx
2 3Al O
Electrostatic capacitor
Ar
0r AC
d
VE
d
E
V
Electrolytic capacitor
A
d
r
E
V
d
dx
2 3Al O
-1max max~ 0.1 F ~ 0.1 W.h.kgC u-1
max max~ 0.01 F ~ 0.01 W.h.kgC u
Electrochemical double layer capacitor
conductive electrode
conductive electrode
separator
activated carbon
-1max max~ 5000 F ~ 30 W.h.kgC u
d
+++++++++
---------
+++++++++
---------
+++++++++
---------
+
-
+
-
-
Electric Field Electric Field
Zero applied stress
Compressive stressInduces a voltage Applied voltage produces
An expansion
+
-
+ - + - + -
+ - + - + -
+ - + - + -
Ferroelectric material
+ - + - + -
+-+-+-
+ - + - + -
Antiferroelectric material
+ - + - + -+-+-+-
+ + + + + + + + + +
- - - - - - - - - -
+Q on inner surface
-Q on inner surface
Interior points electric field must be zero
Symmetry – electric field must be uniform – electric field lines perpendicular to conductive plates
+ +
+0.2Q on outer surfaceInterior points electric field must be zero
+ + + + + + + + + +
- - - - - - - - - -
+Q on inner surface
-Q on outer surface
Interior points electric field must be zero
Symmetry – fields must be uniform – field lines perpendicular to plates
Interior points electric field must be zero
+ + + + + + + + + +
- - - - - - - - - - -Q on inner surface
+Q on outer surfaceE
dl
+V
+q
+q+q
-q-q
-q
Electric field betweenAdjacent plates
0
qE
A
0
q dV E d
A
0 0
2 3
3 3
QC
VQ q q q
q A AC
q d d
... ... ...
Series branch
V
QC
V
221 1 1
2 2 2
QU QV CV
C
1 2 ...totalC C C
Capacitors in series (charge on each plate)
1 2
11 1
...totalC
C C
Capacitors in parallel (voltage across each capacitor is the same)
V
Capacitors in parallel
V
+Q1
+Q2 -Q2
-Q1
Q =Q1+Q2
Capacitors in series
C1
C2
Ceq = C1+C2
VC1 C2
+Q
+Q -Q
-Q
V
Q
1/Ceq = 1/C1+1/C2
fuel
air
w
hl
fuel
- ++Q
r
Induced dipole
pQE
0
5a
f
f
Slab 1
Slab 2
a
a
a
a
a
f
f
/ 2f/ 2f
/ 3f
/ 3f
0E
0E
E
E
2E
1E
E
S1
S2
S3
S4
1r
2r
+ Q
- Q
+ Qb1
- Qb1
+ Qb2
- Qb2
C1
C2
+ Q
+ Q
- Q
- Q
Capacitors in series
E
E = 0
E
E = 0++
+++++ +
+
+
+
+
+++
+ +++
+
+
+
+
+
---
--
-
--
V1
+Q/2 +Q/2
- Q/2- Q/2
C1 = Q / 2V1
Q = 2 C1 V1
V2
+qA +qB
- qB- qA
C1C1
C1
qA = C2 V2 = r C1 V2
qB = C1 V2
Q = qA + qB
= C1 V2 (r + 1) = 2 C1 V1
V2 = 2 V1 / (r + 1)
qA = 2 C1 V1 r / (r + 1)qB = 2 C1 V1 / (r + 1)
C2
+Qf
-Qf
+q
- q
rd t
+++++++
--
--
--
-
P
externalE
depE
dielectricE
n̂
n̂
ˆbP n
cosb P b P
b b b bq q
Dielectric is neutral
0b
Homogenous dielectric – uniformly polarized
The electrical field is reduced in the dielectric material
extE
+-
+
+
-
-
depE
+-0depE
++
++
++
+
-
---
--
-
Flat plate L = 1Max polarization
Thin long rod L = 0Zero polarization Sphere L = 1/3
Concentration of chargesAt surface given by
cosb P
0
1
3depE P
0
1
3sphere extE E P
rod extE E0depE
0
1depE P
0
1plate extE E P
depE
1rSlab 1
Slab 2
d/2
d/22r
- Q
+ Q
+ Q
- Q
+ Qb1
- Qb1
+ Qb2
- Qb2
E1
E2
R1
R2
