7.1 a Electric field induced by magnetic field b Magnetic field induced by electric field The key features of an electromagnetic wave - The electric field E r is always perpendicular to
Trang 1Chapter 7 ELECTROMAGNETIC WAVES 7.1 Maxwell’s Equations
1) Maxwell’s Equations
Gauss’law for electricity ∫
surface closed
A d E r r
= o
q
Gauss’law for magnetism ∫
surface closed
A d B r r
path closed
s d
r
=
-t
B
∂
Φ
∂
Ampere-Maxwell Law ∫
path closed
s d
r
=
t ε
∂
Φ
∂
2) Vector calculus (or vector analysis)
del operator :
z
i y
i x
∂
∂ +
∂
∂ +
∂
∂
=
• gradient : grad(V) =
z
V i y
V i x
V i
∂
∂ +
∂
∂ +
∂
∂
=
Maps scalar fields to vector fields
Measures the rate and direction of change in a scalar field
• divergence : div(F
r
) = x y z (ixFx iyFy izFz)
z
i y
i x i F
r r r r
r r
r
+ +
∂
∂ +
∂
∂ +
∂
∂
=
∇
=
z
F y
F x
∂
∂ +
∂
∂ +
∂
∂
(7.7)
Maps vector fields to scalar fields
Measures the magnitude of a source or sink at a given point in a vector field
Property : ∫∇ = ∫
surface closed volume
A d F dV
F
r r r
• curl: curl(F
r
) = rot(F
r
) = x x y z x(ixFx iyFy izFz)
z
i y
i x i F
r r r r
r r
r
+ +
∂
∂ +
∂
∂ +
∂
∂
=
∇
Trang 2=
z y x
z y x
F F F
z y x
i i i
∂
∂
∂
∂
∂
∂
r r r
(7.9)
Maps vector fields to vector fields
Measures the tendency to rotate about a point in a vector field
Property : ∫∇ = ∫
path closed surface
s d F A
.d F x
r r r
r
(7.10)
• Laplacian: ∆ V = ∇2.V = ∇ ∇ V = div(grad(V ))
z
i y
i x i z
i y
i x
ix y z x ∂∂ + y ∂∂ + z ∂∂
∂
∂ +
∂
∂ +
∂
r
=
2
2 2
2 2 2
z
V y
V x
V
∂
∂ +
∂
∂ +
∂
∂
(7.11)
Maps scalar fields to scalar fields
A composition of the divergence and gradient operations
F
r
z y
2 2
2 2
∂
∂ +
∂
∂ +
∂
∂
= Fxix Fyiy Fziz
r r
r
∆ +
∆ +
Property : F
r
r
∇ - ∇x∇xF
r
(7.13) 3) Maxwell’s Equations in term of del operator
Gauss’law for electricity ∫ ∇
volume
.dV E
r
surface closed
A d E r r
= o
q
volume
.dV ε
ρ o
It follows that E
r
∇ =
o ε
ρ
(7.14)
Gauss’law for magnetism ∫ ∇
volume
.dV B
r
surface closed
A d B r r
= 0
It follows that B
r
Faraday ‘s law ∫∇
surface
A d E x r r
path closed
s d
r
=
-t
B
∂
Φ
∂
= - dA
t
r
∫ ∂
∂ surface
Trang 3It follows that xE
r
∇ =
-t
B
∂
∂r
(7.16)
Ampere-Maxwell Law ∫∇
surface
A d B x r r
path closed
s d
r
=
dt
o o
Φ ε
µ + µoi
= ∫ ∂ ∂ +
surface
A d J µ t
E ε
µo o o
r r r
=
It follows that xB
r
t
E ε
r r +
∂
∂
(7.17)
4) Wave equation
Applying (7.13) yields
E
r
r
∇ - ∇x∇xE
s
(7.18) With J = 0 and ρ = 0
E
r
∇ = 0 and ∇xE
s
=
-t
B
∂
∂r
(7.19)
We have
E
r
∆ = ∇x
t
B
∂
∂r
=
t
∂
∂
B
x
r
∇ =
t
E ε µ
2
2 o o
∂
(7.20)
Inserting (7.19) and (7.20) into (7.18) we have the wave equation
E
r
t
E ε µ
2
2 o o
∂
7.2 Electromagnetic Waves
An electromagnetic wave consists of oscillating electric and magnetic fields The various possible frequencies of electromagnetic waves form a spectrum, a small part of which is visible light An electromagnetic wave traveling along an x axis has an electric field E
r
and a magnetic field B
r
with magnitudes that depend on x and t
where ω : angular frequency of the wave, k : angular wave number of the wave These two components can not exist independently The two fields continuously create each other via induction : the time varying magnetic field induces the electric field via Faraday ‘s law of induction, the time varying electric field induces the magnetic field via Maxwell ‘s law of induction
Trang 4a) b) Fig 7.1 a) Electric field induced by magnetic field b) Magnetic field induced by electric field The key features of an electromagnetic wave
- The electric field E
r
is always perpendicular to the magnetic field B
r
The electric field E
r
and the magnetic field B
r
are always perpendicular to the direction in which the wave is travelling (the wave
is a transverse wave) The cross product E
r
x B
r
always gives the direction in which the wave travels
- The fields always vary sinusoidally with the same frequency and in phase with each other
- All electromagnetic waves, including visible light, have the same speed c (3x108 m/s) in vacuum The electromagnetic wave requires no medium for its travel It can travel through a medium such as air or glass It can also travel through vacuum
c k
1 B
E
o o m
ε µ
=
= B
E
(7.24)
Fig 7.2 : The electromagnetic spectrum
7.3 Energy Flow
The rate per unit area at which energy is transported via an electromagnetic wave is given by the Poynting vector
Trang 51 S= E x B
µ
(7.25)
Fig 7.3 : Electromagnetic wave The direction of S
r (and thus of the wave’s travel and the energy transport) is perpendicular to the direction of both E
r and B
r Since E
r and B
r are perpendicular
o
2
E EB S
µ
= µ
2 m o
E
cµ sin
2
The time-averaged of S is called the intensity I of the wave
2 m o
E
I =
A point source of electromagnetic waves emits the wave isotropically (i.e with equal intensity in all directions) The intensity of the waves at distance r from a point source of power Ps is
s 2
P
I =
7.4 Radiation pressure
When a surface intercepts electromagnetic radiation, a force and a pressure are exerted on the surface If the radiation is totally absorbed by the surface, the force is
IA
F =
where I is the intensity of the radiation and A is the area of the surface perpendicular to the path of the radiation If the radiation is totally reflected back along its original path, the force is
2IA
F =
The radiation pressure pr is the force per unit area
r
F
p =
Trang 6Problems
7.1) An electromagnetic wave with frequency 4x1014 Hz travels through vacuum in the positive direction of an
x axis The wave has its electric field directed parallel to the y axis with amplitude Em At time t = 0, the electric field at point P on the x axis has a value of Em/4 and is decreasing with time What is the distance along the x axis from point P to the first point with E = 0 if we search in
a) the negative direction of the x axis
a) the positive direction of the x axis
7.2) An airplane flying at a distance of 10km from a radio transmitter receives a signal of intensity 10µW/m2 What is the amplitude of the electric and magnetic component of the signal at the airplane ? If the transmitter radiates uniformly over a hemisphere, what is the transmission power ?
7.3) The maximum electric field 10m from an isotropic point source of light is 2V/m What are the maximum value of the magnetic field and the average intensity of the light there ? What is the power of the source ?
7.4) Sunlight just outside earth’s atmosphere has an intensity of 1.4 kW/m2 Calculate the amplitude of the electric and magnetic field there, assuming it to be a plane wave
7.5) A plane electromagnetic wave, with wave length 3m, travels in vacuum in the positive direction of an x axis The electric field, of amplitude 300V/m, oscillates parallel to the y axis What are the frequency, angular frequency and angular wave number of the wave ? What is the amplitude of the magnetic field component ? Parallel to which axis does the magnetic field oscillates ? What is the time-averaged rate of energy flow associated with this wave ? The wave uniformly illuminates a surface of area 2m2 If the surface totally absorbs the wave, what are the rate at which momentum is transferred to the surface and the radiation pressure on the surface ?
7.6) An isotropic point source emits light at wavelength 500nm, at rate of 200W A light detector is positioned 400m from the source What is the maximum rate B
t
∂
∂ at which the magnetic component of the light changes with time at the detector’s location ?
7.7) The basic equations of electromagnetism is called Maxwell’s equations which are given in the vacuum (J = 0, ρ = 0) as below:
∇ =r (Gauss’s law for magnetism)
0 B
∇ = ur (Gauss’s law for electricity)
B E
t
∂
∇ × = −
∂
ur ur
(Faraday’s law)
D
t
∂
∂
r
(Ampere-Maxwell’s law) Whererj =σEr, D r = εεoE r, B oH
r r
µµ
= Show that from Maxwell’s equation the following wave equation can be derived
2
o o
E E
t
ε µ ∂
∂ r r
Trang 7Homeworks 7
H7.1 A plane electromagnetic wave, with wave length λ [m], travels in vacuum in the positive direction of an x axis The electric field, of amplitude E [V/m], oscillates parallel to the y axis What are the frequency, angular frequency and angular wave number of the wave ? What is the amplitude of the magnetic field component ? Parallel to which axis does the magnetic field oscillates ? What is the time-averaged rate of energy flow associated with this wave ? The wave uniformly illuminates a surface of area 2m2 If the surface totally absorbs the wave, what are the rate at which momentum is transferred to the surface and the radiation pressure on the surface ?
n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
λ 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8
E 100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 850
n 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
λ 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5
E 25 50 75 100 150 200 250 300 350 400 450 500 550 600 650 700
n 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48
λ 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5
E 200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950
n 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64
λ 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 4.5 5 5.5 6 6.5
E 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95
Appendix I
Appendix II Surface of a sphere of radius R : S = 4πR2
Volume of a sphere of radius R : V = 4πR3
/3
Trang 8Circumference of a circle of radius R : C = 2πR
( 2 2)3 / 2 2( 2 2)1 / 2
a x a
x a
x
dx
+
= +
∫
Appendix III Dot product of two vectors is a scalar
A
r
•B
r
= |A
r
|.|B
r
|.cos(α) = AxBx + AyBy + AzBz
Cross product of two vectors is a vector
C
r
= A
r
xB
r
where |C
r
| = |A
r
|.|B
r
|.sin(α) and the direction of C
r
is determined by the right hand rule
The line integral of the vector F
r
along the curve L from A to B is a scalar
∫ •
B
A
L d
F
r r
= ∫B
A
dL
F |cos( )
x
B
A x
xdL
y
y
B
A y
ydL
F +∫z
z
B
A z
zdL F
The surface integral of the vector F
r
through the surface A is a scalar
A
dA n
F r
r
= ∫
A
dA
F | cos( )
A
z z y y x
F
The volume integral of the scalar F over the volume V
∫
V
FdV