10.b. electromagnetic waves -...

10.B. Electromagnetic Waves

◈ Propagation velocity

smc /458,792,2991

00

Characteristics of Electromagnetic Waves

• Direction: (E⊥B)⊥(direction of progress) ⇒ Transverse wave

(E x B)∥(direction of progress)

(in a vacuum)

◈ Electromagnetic field vector

• Phase: E and B are vibration in the form of harmonic functions,

same phase

• Amplitude:

)sin( tkxEE m

)sin( tkxBB m

Generation of Artificial E&M Waves

1885년으로부터 1889년에 독일의 헤르츠는 최초로 실험실에서 인공전자

기파를 발생하여 이를 검출할 수 있었다. 물론 가시광선도 전자기파의 일

종이니까 부싯돌로 불을 피우면 전자기파가 만들어진다고 볼 수도 있지만

여기서는 순수한 전자기 이론에 따른 전자기파를 만드는 것을 말한다.

헤르츠가 1987년 전자기파를 발생시키고 검출한 실험장치로서 왼쪽은 발생장치이고 오른쪽은 검출장치이다.

발생장치는 놋쇠로 만든 둥근 전극을 아주 가까이 두고 이것에 고전압을 걸어서 공기 중에서 방전이 일어나게 한 것이다. 한편 이렇게 해서 만들어진 전자기파를 고리모양의 원형도선에서 잡아 이 결과 위쪽의 조그마한 전극에서 작은 불꽃의 방전이 일어나게 된다.

• Changes in the electromagnetic wave emitted

from the antenna :

— Temporal changes at observation point P

— Spatial changes according

to the direction of progress

• Antenna : E&M wave emission source

≡ Oscillating electric dipole

Generation of E&M Waves by Antenna

• LC Oscillators : Alternating current source

s

AC

4

1

VE

AsE

sEs

ACW

8842

1

2

1 2222

Energy stored in this parallel-plate capacitor …..

i.e.,

the potential energy U of the charge system is E2/8 per unit volume.

i.e., even if no current of charges,

allows solar energy E2/8 per volume element to warm the earth.

dvE

AdsE

dsEs

AdW

8842

1 222

Energy Transfer by E&M Waves (1/2)

▷ Capacitance C of a parallel-plate capacitor,

made up of area A separated by a distance s ?

▷ Energy delivered per volume element (dv) in the space of the earth

from the sun ?

dvBEdvB

dvE 22

22

8

1

88

at t = 0, E2 = Eo2 sin2y , B2 = Bo

2 sin2y

where, |E| = Eo sin(y - vt),

|B| = Bo sin(y - vt)

where, Eo = Bo in vacuum

yBEyByEBEE 22

0

2

0

22

0

22

0

22 sin8

1sinsin

8

1

8

1

yE 22

0 sin4

1

2

0

22

08

1sin

4

1EyEE

2

08

1E

where, sin2y = 1/2

00

2

0

2 1

2

1,

candEEwherecES 2

4

1

Energy Transfer by E&M Waves (2/2)

▷ Energy by electromagnetic wave …..

▷ Energy density per unit volume …..

▷ Average energy …..

▷ Average energy ratio through a unit area (per sec) :

▷ Power density (power, i.e., flowing energy density per unit area)

A standing wave

produced by reflection

at a perfectly

conducting plate

1. 전자기파가 전기 도체를 만나면,

그 전기장 때문에 도체에서 전류

가 흐른다. 이때 전자기파의 에너

지가 소모된다.

2. 도체에 입사된 전자기파가 반사될

때 흡수된 것 만큼 에너지를 잃게

된다.

Energy Transfer by E&M Waves

1/5 Electromagnetic Wave in Vacuum

▷ Maxwell’s Equations for Empty Space

① E = 4 ② B = 0

③ E = (1/c) (∂B/∂t) ④ B = (4/c)J + (1/c)(∂E/∂t)

① E = 0 ② B = 0

③ E = (1/c) (∂B/∂t) ④ B = (1/c)(∂E/∂t)

In the empty space

= 0 and J = 0

Suppose there are electric field E and magnetic field B,

Let the dependence have this particular form;

(y,z) plane wave : z-direction

(x,y) plane wave : x-direction

)sin(ˆ0 vtyEzE

)sin(ˆ0 vtyBxB

2/5

where,

x x-direction unit vector

y y-direction unit vector

z z-direction unit vector

^

^

^

The wave described

by Eqs.

It is shown at three

different times.

It is traveling to the

right, in the positive

y direction

)sin(ˆ0 vtyEzE

)sin(ˆ0 vtyBxB

Electromagnetic Wave in Vacuum

3/5

▷ Properties of Electromagnetic Wave

y

E

x

Ez

x

E

z

Ey

z

E

y

ExE xyzxyz ˆˆˆ

In the empty space and

E = (1/c) (∂B/∂t) and B = (1/c)(∂E/∂t)

)sin(ˆ0 vtyEzE

)sin(ˆ

0 vtyBxB

0

where, zEzvtyEzE ˆ)sin(ˆ0

also, ),()sin(0 tyEvtyEE zz

0 0 0 0

① )cos(ˆˆ0 vtyEx

y

ExE z

and ② )cos(ˆ0 vtyEzv

t

E


4/5

y

B

x

Bz

x

B

z

By

z

B

y

BxB xyzxyz ˆˆˆ

In the empty space and

E = (1/c) (∂B/∂t) and B = (1/c)(∂E/∂t)

)sin(ˆ0 vtyEzE

)sin(ˆ

0 vtyBxB

0

where, xBxvtyBxB ˆ)sin(ˆ0

also, ),()sin(0 tyBvtyBB xx

0 0 0 0

)cos(ˆˆ0 vtyBz

y

BzB x

③

and )cos(ˆ0 vtyBxv

t

B

④


5/5

Then, Eo = Bo and v = ±c

▷ We have now learned that our electromagnetic wave must have the following properties;

1. The filed pattern travels with speed c.

2. At every point in the wave at any instant of time, the electric and magnetic field strengths are equal.

3. The electric field and the magnetic field are perpendicular to one another and to the direction of travel, or propagation.

①

④

)cos(ˆ0 vtyExE

)cos(ˆ0 vtyBxv

t

B

00 Bc

vE

00 Ec

vB

②

③ )cos(ˆ0 vtyBzB

)cos(ˆ0 vtyEzv

t

E

For the Maxwell’s eqs. E = (1/c) (∂B/∂t)

For the Maxwell’s eqs. B = (1/c)(∂E/∂t)


1/4 Electromagnetic Wave in Dielectric

▷ Maxwell’s Equations in a Dielectric

① E = 4 ② B = 0

③ E = (1/c) (∂B/∂t) ④ B = (4/c)J + (1/c)(∂E/∂t)

① E = 0 ② B = 0

③ E = (1/c) (∂B/∂t) ④ B = ( /c)(∂E/∂t)

No free current in the

complete insulation medium

= 0 and J = 0

Insert a dielectric constant

in an infinite dielectric medium.

0

)sin(ˆ0 tkyEzE

)sin(ˆ

0 tkyBxB

Let the dependence have this particular form;

(ky - t) = phase , /k = phase velocity

Phase is constant for moving towards the point +y with wave

velocity /k.

2/4

▷ Properties of Electromagnetic Wave

y

E

x

Ez

x

E

z

Ey

z

E

y

ExE xyzxyz ˆˆˆ

In a dielectric and

E = (1/c) (∂B/∂t) and B = ( /c)(∂E/∂t)

)sin(ˆ0 tkyEzE

)sin(ˆ

0 tkyBxB

0

where, zEztkyEzE ˆ)sin(ˆ0

also, ),()sin(0 tyEtkyEE zz

0 0 0 0

① )cos(ˆˆ0 tkykEx

y

ExE z

and ② )cos(ˆ0 tkyEz

t

E

Electromagnetic Wave in Dielectric

3/4

In a dielectric and

E = (1/c) (∂B/∂t) and B = ( /c)(∂E/∂t)

)sin(ˆ0 tkyEzE

)sin(ˆ

0 tkyBxB

y

B

x

Bz

x

B

z

By

z

B

y

BxB xyzxyz ˆˆˆ

0

where, xBxtkyBxB ˆ)sin(ˆ0

also, ),()sin(0 tyBtkyBB xx

0 0 0 0

)cos(ˆˆ0 tkykBz

y

BzB x

③

and )cos(ˆ0 tkyBx

t

B

④


①

④

)cos(ˆ0 tkykExE

)cos(ˆ0 tkyBx

t

B

00 B

ckE

②

③ )cos(ˆ0 tkykBzB

)cos(ˆ0 tkyEz

t

E

For the Maxwell’s eqs. E = (1/c) (∂B/∂t)

For the Maxwell’s eqs. B = ( /c)(∂E/∂t)

4/4

⑤ 00 E

ckB

⑥ 00 Ec

kB

From equation ⑤/⑥

phase velocity 2

21

kccck

k

2

2

2 c

k

c

k

00000 EEc

cEk

cEck

B

From ⑤

00 EB


Spectrum of Electromagnetic Wave

Thanks

Practice Problem

Previous Tests

10 - 1, 3, 5, 6, 8, 13

2, 5, 10, 13, 16, 17, 20,

21, 23, 25, 28, 29

10.b. electromagnetic waves -...

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