Introduction to materials physics #2

Week 2: Electric and magnetic interaction and

electromagnetic wave

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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.

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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

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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

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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

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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.

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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.

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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

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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

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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.

dt

dV

t

H

y

E

x

Ezxy

EXERCISE: Derive the above right differential equation.

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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

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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

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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

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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

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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

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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

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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