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



Dynamic interaction



Electric field and dielectrics
Magnetic field and magnets
Faraday’s law of induction
Ampère’s circuital law and displacement
current
Maxwell’s equations and wave equation
2
Dielectricity: materials in an
electric field

Dielectric materials (dielectrics) possess electric
polarization P in a static electric field E.
Electric field :
V
E  e z [V/m]
d
Electric Polarizati on : P 
P  E  ε 0 E [C/m 2 ]
Q
ez
S
 : electric polarizabi lity
ε
: vacuum permittivi ty
 : electric susceptibi lity
(non - dimensiona l)3
0
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

ε0: electric permittivity
of vacuum
ε: 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).
5

Note: Why is P proportional to E?

Induced electric polarization might be
proportional to an applied electric field at
weak-field limit, because

Odd function


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.
6
Magnetism: materials in a
magnetic field

Magnetic materials become magnets in a static
magnetic field.
Magnetic field :
H [A/m]
Magnetizat ion : M 
M   mH [A/m ]
md
ez
dS
m : magnetic charge
 m : magnetic susceptibi lity
7
Classifications of magnetic materials

Paramagnetic materials


Ferromagnetic materials


They become weak magnets when they
are subjected to an external magnetic field.
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.
V 
d
dt
V [V] : Induced voltage
  BS  HS [Wb] :
Magnetic flux
B  H [Wb/m 2 ] :
Magnetic flux density
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Induced voltage V

Induced voltage is evaluated from electric field
Induced voltage (right-handed rotation)
V  E x 0x  E y x y
 E y y  x   E y 0 y 
 E x y   E y 0 x
 E y y   E y 0 y
Divided by S=ΔxΔy
V E y x   E y 0 Ex y   Ex 0


S
x
y
Small area limit (S→0⇔Δx, Δy→0)
V dE y dEx


S
dx
dy
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Differential equation of
Faraday’s law

Faraday’s law can be expressed by a
differential equation of electric and
magnetic fields.
d
V 
dt
E y
Ex
H z

 
x
y
t
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)
H
I
2r
H : Magnetic field
I : Electric current
r : radius of the loop
13
Displacement current

Current conservation
The current flowing in a single loop
circuit is the same everywhere.

How is the current inside the capacitor?
V
dQ
, I
d
dt
S
Q  CV , C  
d
E
I  S


dE
 I D Displacement
current
dt
There exists “displacement current”
inside the capacitor instead of current!
Introducing displacement current
density,
I
dE
jD 
D
S

dt
14
Magnetic field induced by
displacement current

Displacement current can induce
magnetic field.
Magnetic field induced by
displacement current
Electric field induced by temporal
change in magnetic field
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Differential equation of
Ampère’s law

Ampère’s law corresponds to Faraday’s law.

Ampère’s law
Hx, Hy
dE
jD  
induces magnetic field
dt
H y
H x
Ez


x
y
t
Faraday’s law
Ex , E y
d
induces electric field
dt
E y
Ex
H z

 
x
y
t
EXERCISE: Derive the above left differential equation.
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
div E   / 
 : electric charge density
 No magnetic monopole
div H  0


Faraday’s induction law
rotE   H / t
Ampère’s circuital law
rotH  j  E / t
jD
Specific coordinates
E y
x

Ex
H z
 
y
t
H y
H x
Ez


x
y
t
17
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
Faraday’s law
H y
E x E z

 
z
x
t
H y
E x
 
z
t
Ampère’s law
E
H z H y

 x
y
z
t

H y
z

E x
t
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Wave equation: separation of
electric and magnetic fields

Electric field

Magnetic field
H y
E x
 
z
t
  E x 
  H y 
2
2
  2 E x   2 E x

    
z  z 
z  t  z
t
H y
E

 x
z
t
  H y 
  E x 
2
2

   
  2 H y   2 H y
z  z 
z  t  z
t
Wave equation
19
Wave equation in general
coordinate system

Specific coordinate system

Faraday’s law
E y
Ex
H z

 
x
y
t

Wave equation
2
2
E x   2 Ex
2
z
t
General coordinate system

Faraday’s law
Wave equation
2

E
2
rotE   H / t
t 2
2


2
2
  2  2  2 Laplacian 
x
y
z
You will learn this in “Electromagnetics I, II”.
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 2 E  
Wave equation and
electromagnetic wave

Wave equations of fields
2
2
E x   2 Ex
2
z
t

2
2
H y   2 H y
2
z
t
Solutions of wave equations
(electromagnetic wave)
E , H : Amplitude
Ex t , z   E 0 coskz  t  0 
 : Angular frequency
H y t , z   H 0 coskz  t  0 
k : Wave number
0
0
0 : Initial phase
EXERCISE: Verify that the solutions satisfy the wave
equations.
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Electromagnetic wave
https://www.nde-ed.org/EducationResources/CommunityCollege/RadiationSafety/theory/nature.htm
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Summary

Static interaction of materials



Dynamic interaction



Electric field and dielectrics
Magnetic field and magnets
Faraday’s law of induction
Ampère’s circuital law and displacement
current
Maxwell’s equations and wave equation
23