introduction to materials physics #2 week 2: electric and magnetic interaction and electromagnetic...
TRANSCRIPT
Introduction to materials physics #2
Week 2: Electric and magnetic interaction and
electromagnetic wave
1
Electric and magnetic interaction and EM wave Static interaction of materials
Electric field and dielectrics Magnetic field and magnets
Dynamic interaction Faraday’s law of induction Ampère’s circuital law and
displacement current Maxwell’s equations and wave
equation2
Dielectricity: materials in an electric field Dielectric materials (dielectrics) possess
electric polarization P in a static electric field E.
[V/m]
:field Electric
zd
VeE
][C/m
:onPolarizati Electric2
0EEP
eP z
εS
Q
l)dimensiona-(non litysusceptibi electric :typermittivi vacuum:
litypolarizabi electric :
0
ε
3
Electric susceptibility An intrinsic constant of a dielectric
material which describes its electric property.
The values are different among the materials.
By measuring the electric susceptibility, we can investigate the electric property of a material.
4
Electric permittivity of material
Due to the electric polarization P, the density of the electric force line, D, is decreased. In order to keep the density of the electric force line, we must add P to that of vacuum (ε0E).
ε0: electric permittivity of vacuum
ε: electric permittivity of material
5
Note: Why is P proportional to E? Induced electric polarization might be
proportional to an applied electric field at weak-field limit, because
1. Without a field, the polarization should be 0.
2. A reversal field induces a reversal polarization with the same magnitude.
A sufficiently strong field can violate the above linearity. If you pull a spring with an enormous force, the spring can not be extended any more.
Odd function
6
Magnetism: materials in a magnetic field Magnetic materials become magnets in a
static magnetic field.
[A/m] :field Magnetic
H
][A/m
:ionMagnetizat
HM
eM z
m
dS
md
litysusceptibi magnetic :charge magnetic :
m
m
7
Classifications of magnetic materials
Paramagnetic materials They become weak magnets when
they are subjected to an external magnetic field.
Ferromagnetic materials They can be magnets without an
external magnetic field. Paramagnetic materials
They become weak magnets in the opposite direction with respect to the magnetic field when they are subjected to an external magnetic field.
8
Magnetic susceptibility An intrinsic constant of a magnetic
material which describes its magnetic property.
The values are different among the materials.
By measuring the magnetic susceptibility, we can investigate the magnetic property of a material.
9
Dynamic interaction 1: Faraday’s induction law
A voltage is induced in a coil when magnetic flux crossing the coil is temporally changed.
tV
d
d
densityflux Magnetic :][Wb/m
flux Magnetic :[Wb]
voltageInduced :[V]
2HB
HSBSV
10
Induced voltage V Induced voltage is evaluated from electric field
yEyE
xEyEyExyE
yxExEV
yy
yx
yy
yx
0 0
0 0
Induced voltage (right-handed rotation)
Divided by S=ΔxΔy
y
EyE
x
ExE
S
V xxyy
00
Small area limit (S→0⇔Δx, Δy→0)
y
E
x
E
S
V xy
d
d
d
d
11
Differential equation of Faraday’s law Faraday’s law can be expressed by
a differential equation of electric and magnetic fields.
dt
dV
t
H
y
E
x
Ezxy
EXERCISE: Derive the above right differential equation.
12
Dynamic interaction 2: Ampère’s circuital law Infinite straight electric current
induces magnetic field in the form of closed loop around the current.
Induced magnetic field (right-handed rotation)
r
IH
2
loop theof radius :current Electric :
field Magnetic :
rIH
13
Displacement current Current conservation
How is the current inside the capacitor?
There exists “displacement current” inside the capacitor instead of current!
Introducing displacement current density,
The current flowing in a single loop circuit is the same everywhere.
d
SCCVQ
t
QI
d
VE
,
d
d ,
Dd
dI
t
ESI Displacement
current
t
E
S
Ij
d
dDD
14
Magnetic field induced by displacement current Displacement current can induce
magnetic field.
Magnetic field induced bydisplacement current
Electric field induced by temporal change in magnetic field 15
Differential equation of Ampère’s law
Ampère’s law corresponds to Faraday’s law. Ampère’s law Faraday’s law
t
H
y
E
x
Ezxy
EXERCISE: Derive the above left differential equation.
field magnetic induces d
d ,
D t
Ej
HH yx
t
E
y
H
x
Hzxy
field electric induces d
d ,
t
EE yx
16
Maxwell’s equations Maxwell’s equations are the electric and
magnetic laws given in the form of differential equation in arbitrary reference coordinate system. Coulomb’s law
No magnetic monopole
Faraday’s induction law
Ampère’s circuital law
density charge electric : /div E
0div H
t /rot HE t
H
y
E
x
Ezxy
t /rot EjH t
E
y
H
x
Hzxy
Specific coordinates
Dj17
Wave equation: field configuration Consider a specific case where
magnetic and electric fields are at right angle to each other. Direction of electric field → x-axis Direction of magnetic field → y-axis
t
H
x
E
z
E yzx
t
H
z
E yx
t
E
z
H
y
H xyz
t
E
z
Hxy
Faraday’s law
Ampère’s law
18
Wave equation: separation of electric and magnetic fields Electric field
Magnetic fieldxx
yx
yx
Et
Ezt
H
zz
E
z
t
H
z
E
2
2
2
2
yyxy
xy
Ht
Hzt
E
zz
H
z
t
E
z
H
2
2
2
2
Wave equation
19
Wave equation in general coordinate system
Specific coordinate system Faraday’s law Wave equation
General coordinate system Faraday’s law Wave equation
t
H
y
E
x
Ezxy
xx E
tE
z 2
2
2
2
t /rot HE
Laplacian 2
2
2
2
2
22
2
22
zyx
t
EE
You will learn this in “Electromagnetics I, II”.20
Wave equation and electromagnetic wave Wave equations of fields
Solutions of wave equations (electromagnetic wave)
xx Et
Ez 2
2
2
2
yy H
tH
z 2
2
2
2
00
00cos,
cos,
tkzHztHtkzEztE
y
x
phase Initial :number Wave:
frequencyAngular :Amplitude : ,
0
00
k
HE
EXERCISE: Verify that the solutions satisfy the wave equations. 21
Electromagnetic wave
https://www.nde-ed.org/EducationResources/CommunityCollege/RadiationSafety/theory/nature.htm
22
Summary Static interaction of materials
Electric field and dielectrics Magnetic field and magnets
Dynamic interaction Faraday’s law of induction Ampère’s circuital law and
displacement current Maxwell’s equations and wave
equation23