Chapter 32 Maxwell’s Equations and Electromagnetic Wavess Eqn.pdf · MFMcGraw-PHY 2426...
Transcript of Chapter 32 Maxwell’s Equations and Electromagnetic Wavess Eqn.pdf · MFMcGraw-PHY 2426...
![Page 1: Chapter 32 Maxwell’s Equations and Electromagnetic Wavess Eqn.pdf · MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 4 ∫ ∫µ µ i i o o C C S B dl = J dA= I Something](https://reader036.fdocuments.us/reader036/viewer/2022062602/5e9f038681f46b79b4168757/html5/thumbnails/1.jpg)
Chapter 32
Maxwell’s Equations and
Electromagnetic Waves
![Page 2: Chapter 32 Maxwell’s Equations and Electromagnetic Wavess Eqn.pdf · MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 4 ∫ ∫µ µ i i o o C C S B dl = J dA= I Something](https://reader036.fdocuments.us/reader036/viewer/2022062602/5e9f038681f46b79b4168757/html5/thumbnails/2.jpg)
MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 2
Maxwell’s Equations and EM Waves
• Maxwell’s Displacement Current
• Maxwell’s Equations
• The EM Wave Equation
• Electromagnetic Radiation
![Page 3: Chapter 32 Maxwell’s Equations and Electromagnetic Wavess Eqn.pdf · MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 4 ∫ ∫µ µ i i o o C C S B dl = J dA= I Something](https://reader036.fdocuments.us/reader036/viewer/2022062602/5e9f038681f46b79b4168757/html5/thumbnails/3.jpg)
MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 3
µ µ∫ ∫� �� �i i� o o C
C S
B dl = J dA = I
Something is Missing From Ampere’s Law
The surface S in the integral above can be
any surface whose boundary is C.
If the surface S2 is chosen for
use in the above integral the
result will be that the magnetic
field around C is zero. But there
is current flowing through the
wire so we know there is a
magnetic field present.
![Page 4: Chapter 32 Maxwell’s Equations and Electromagnetic Wavess Eqn.pdf · MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 4 ∫ ∫µ µ i i o o C C S B dl = J dA= I Something](https://reader036.fdocuments.us/reader036/viewer/2022062602/5e9f038681f46b79b4168757/html5/thumbnails/4.jpg)
MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 4
µ µ∫ ∫� �� �i i� o o C
C S
B dl = J dA = I
Something is Missing From Ampere’s Law
The surface S2 has the same boundary as S1 but there is no current
passing through S2. The charge is accumulating on the capacitor .
Maxwell noticed this deficiency in
Ampere’s law and fixed it by
defining the Displacement Current Id.
He began by taking surfaces S1 and
S2, putting them together and treating
them like one closed surface S
![Page 5: Chapter 32 Maxwell’s Equations and Electromagnetic Wavess Eqn.pdf · MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 4 ∫ ∫µ µ i i o o C C S B dl = J dA= I Something](https://reader036.fdocuments.us/reader036/viewer/2022062602/5e9f038681f46b79b4168757/html5/thumbnails/5.jpg)
MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 5
µ µ∫ ∫� �� �i i� o o C
C S
B dl = J dA = I
The Displacement Current
Charge is building up on the disk within the closed surface S.
Therefore there are electric field lines, E, that are crossing the
surface S. We can use Gauss’s Law here.
∂ ∂
∂ ∂
∂
∂
∫��i�
enclosed
e
S o
e
d
o o
e
d o
Qφ = E dA =
ε
φ 1 Q 1 = = I
t ε t ε
φI =ε
t
![Page 6: Chapter 32 Maxwell’s Equations and Electromagnetic Wavess Eqn.pdf · MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 4 ∫ ∫µ µ i i o o C C S B dl = J dA= I Something](https://reader036.fdocuments.us/reader036/viewer/2022062602/5e9f038681f46b79b4168757/html5/thumbnails/6.jpg)
MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 6
∫ ∫� �� �i i� o o C
C S
B dl = µ J dA = µ I
The Displacement Current
Maxwell fixed the problem with Ampere’s law by adding
another current to the right hand side of the equation below
∂
∂e
d o
φI =ε
t
∂
∂∫ ∫� �� �i i�
e
o o C o d o C o 0
C S
φB dl = µ J dA = µ I + µ I = µ I + µ ε
t
![Page 7: Chapter 32 Maxwell’s Equations and Electromagnetic Wavess Eqn.pdf · MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 4 ∫ ∫µ µ i i o o C C S B dl = J dA= I Something](https://reader036.fdocuments.us/reader036/viewer/2022062602/5e9f038681f46b79b4168757/html5/thumbnails/7.jpg)
MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 7
Displacement Current Example
In calculating the displacement
current we will be making the
approximation that the electric
field is everywhere uniform.
This requires that the plate
separation be much smaller than
R, the radius of the plate.
The surface S must not extend
past the edge of the capacitor
plates. So r must be less than R.
![Page 8: Chapter 32 Maxwell’s Equations and Electromagnetic Wavess Eqn.pdf · MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 4 ∫ ∫µ µ i i o o C C S B dl = J dA= I Something](https://reader036.fdocuments.us/reader036/viewer/2022062602/5e9f038681f46b79b4168757/html5/thumbnails/8.jpg)
MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 8
Displacement Current Example
In calculating the displacement
current we will need to compute
the electric flux across the
surface S.e
d o
dφI =ε
dt
ˆ∫�ie
S
φ = E ndA = EA
o o o
Qσ QAE = = =ε ε ε A
( )
d o o o
o
d EA dE d Q dQI = ε = ε A = ε A =
dt dt dt ε A dt
![Page 9: Chapter 32 Maxwell’s Equations and Electromagnetic Wavess Eqn.pdf · MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 4 ∫ ∫µ µ i i o o C C S B dl = J dA= I Something](https://reader036.fdocuments.us/reader036/viewer/2022062602/5e9f038681f46b79b4168757/html5/thumbnails/9.jpg)
MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 9
B-Field from the Displacement Current
In calculating the B-Field from the
displacement current we will be
making the same approximations
that were made in the last example:
the electric field is everywhere
uniform.
∂
∂∫��i�
e
o C o 0
C
φB dl = µ I + µ ε
t
( )∫��i�
C
B dl = B 2πr
There is no current through S so IC is zero
( )∫��i�
eo o
C
dφB dl = B 2πr = 0 + µ ε
dt
![Page 10: Chapter 32 Maxwell’s Equations and Electromagnetic Wavess Eqn.pdf · MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 4 ∫ ∫µ µ i i o o C C S B dl = J dA= I Something](https://reader036.fdocuments.us/reader036/viewer/2022062602/5e9f038681f46b79b4168757/html5/thumbnails/10.jpg)
MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 10
B-Field from the Displacement Current
( )∫��i�
eo o
C
dφB dl = B 2πr = 0 + µ ε
dt
2 2
e
o
σφ = AE = πr E = πr
ε2
2 2
e 2 2
o o o
σ Q Qrφ = πr = πr =
ε ε πR ε R
The size of S will vary so φe will depend on r
( )
=
2 2
o o o2 2
o
o o
2 2
d Qr r dQB 2πr = µ ε = µ
dt dt ε R R
µ µr dQ rB = I
2π dt 2πR R
![Page 11: Chapter 32 Maxwell’s Equations and Electromagnetic Wavess Eqn.pdf · MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 4 ∫ ∫µ µ i i o o C C S B dl = J dA= I Something](https://reader036.fdocuments.us/reader036/viewer/2022062602/5e9f038681f46b79b4168757/html5/thumbnails/11.jpg)
MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 11
Maxwell’s Equations
![Page 12: Chapter 32 Maxwell’s Equations and Electromagnetic Wavess Eqn.pdf · MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 4 ∫ ∫µ µ i i o o C C S B dl = J dA= I Something](https://reader036.fdocuments.us/reader036/viewer/2022062602/5e9f038681f46b79b4168757/html5/thumbnails/12.jpg)
MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 12
Maxwell’s Equations
( )∂
∂∫��i�
e
o C d o C o 0
C
φB dl = µ I + I = µ I + µ ε
tAmpere’s Law
Faraday’s Law
∫� insiden
0S
QE dA =
εGauss’s Law
∫� n
S
B dA = 0 No name - there are no
magnetic monopoles
[ ] [ ]∂ ∂
∂ ∂∫ ∫��i�
m n
C
φ B= E dl = 0 - = 0 - dA
t tε
![Page 13: Chapter 32 Maxwell’s Equations and Electromagnetic Wavess Eqn.pdf · MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 4 ∫ ∫µ µ i i o o C C S B dl = J dA= I Something](https://reader036.fdocuments.us/reader036/viewer/2022062602/5e9f038681f46b79b4168757/html5/thumbnails/13.jpg)
MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 13
Maxwell’s Equations
( )∂
∂∫ ∫��i�
n
o C d o C o 0
C S
EB dl = µ I + I = µ I + µ ε dA
tAmpere’s Law
Faraday’s Law
∫ ∫� �inside
n
S V0 0
Q1E dA = ρdV =
ε εGauss’s Law
∫� n
S
B dA = 0 No name - there are no
magnetic monopoles
∂ ∂
∂ ∂∫ ∫��i�
m n
C
φ B= E dl = - = - dA
t tε
![Page 14: Chapter 32 Maxwell’s Equations and Electromagnetic Wavess Eqn.pdf · MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 4 ∫ ∫µ µ i i o o C C S B dl = J dA= I Something](https://reader036.fdocuments.us/reader036/viewer/2022062602/5e9f038681f46b79b4168757/html5/thumbnails/14.jpg)
MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 14
EM Wave Equation
![Page 15: Chapter 32 Maxwell’s Equations and Electromagnetic Wavess Eqn.pdf · MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 4 ∫ ∫µ µ i i o o C C S B dl = J dA= I Something](https://reader036.fdocuments.us/reader036/viewer/2022062602/5e9f038681f46b79b4168757/html5/thumbnails/15.jpg)
MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 15
Conservative Forces and Potentials
from Vector Analysis
Work around a closed loop = 0
Stokes Theorem
Therefore a potential function V exists for a conservative force.
( )
( )
( )
⋅
⋅ = ∇ ⋅
∇ ⋅ ⇒ ∇
∇ ∇ ∇
∫
∫ ∫
∫
��
�� � � �
� � � ��
� � � �
�
�
C
C S
S
W = F dl = 0
F dl × F da
× F da = 0 × F = 0
F = - V since × V = 0
![Page 16: Chapter 32 Maxwell’s Equations and Electromagnetic Wavess Eqn.pdf · MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 4 ∫ ∫µ µ i i o o C C S B dl = J dA= I Something](https://reader036.fdocuments.us/reader036/viewer/2022062602/5e9f038681f46b79b4168757/html5/thumbnails/16.jpg)
MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 16
Vector Analysis
ψ��
φ and are scalar functions
F and G are vector functions
∇
∇ ⋅
∇
�
� � � �
� � � �
φ= grad φ= gradient of φ
F = div F = divergence of F
× F = curl F = curl of F
![Page 17: Chapter 32 Maxwell’s Equations and Electromagnetic Wavess Eqn.pdf · MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 4 ∫ ∫µ µ i i o o C C S B dl = J dA= I Something](https://reader036.fdocuments.us/reader036/viewer/2022062602/5e9f038681f46b79b4168757/html5/thumbnails/17.jpg)
MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 17
Vector Analysis
Gradient
Divergence
Curl
ˆˆ ˆφ φ φ∂ ∂ ∂
∇∂ ∂ ∂
�φ= i + j + k
x y z
∂∂ ∂∇ ⋅
∂ ∂ ∂
� �yx z
FF FF = + +
x y z
ˆˆ ˆ
∂ ∂ ∂∇ ×
∂ ∂ ∂
� �
x y z
i j k
F =x y z
F F F
![Page 18: Chapter 32 Maxwell’s Equations and Electromagnetic Wavess Eqn.pdf · MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 4 ∫ ∫µ µ i i o o C C S B dl = J dA= I Something](https://reader036.fdocuments.us/reader036/viewer/2022062602/5e9f038681f46b79b4168757/html5/thumbnails/18.jpg)
MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 18
Vector Identities
( )
( ) ( ) ( ) ( ) ( )
( ) ( ) ( )
∇ ∇ ∇
∇ ∇ ∇ ∇ ∇
∇ ∇ ∇
� � �
� � � � �� � � � � � � � � �i i i
� � �� � �i i i
fg = f g + g f
A B = B A+ A B + B× × A + A× × B
fA = f A+ f A
( ) ( ) ( )
( ) ( ) ( )( ) ( ) ( ) ( ) ( )
( )
∇ ∇ ∇
∇ ∇ ∇
∇ ∇ ∇ ∇ ∇
∇ ∇ ∇ ∇ ∇
� � �� � � � � �i i i
� � �� � �
� � � � �� � � � � � � � � �i i i i
� � �� � � �i
2
A× B = B × A - A × B
× fA = f × A+ f × A
× A× B = B A - A B + B A - A B
× × A = A - A
![Page 19: Chapter 32 Maxwell’s Equations and Electromagnetic Wavess Eqn.pdf · MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 4 ∫ ∫µ µ i i o o C C S B dl = J dA= I Something](https://reader036.fdocuments.us/reader036/viewer/2022062602/5e9f038681f46b79b4168757/html5/thumbnails/19.jpg)
MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 19
Vector Identities
( ) ˆ
ˆ
ˆ
∂ ∂ ∂∇ ∂ ∂ ∂
∂ ∂ ∂ + ∂ ∂ ∂
∂ ∂ ∂+ ∂ ∂ ∂
� � �i
x x xx y z
y y y
x y z
z z zx y z
B B BA B = A + A + A i
x y z
B B BA + A + A j
x y z
B B BA + A + A k
x y z
![Page 20: Chapter 32 Maxwell’s Equations and Electromagnetic Wavess Eqn.pdf · MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 4 ∫ ∫µ µ i i o o C C S B dl = J dA= I Something](https://reader036.fdocuments.us/reader036/viewer/2022062602/5e9f038681f46b79b4168757/html5/thumbnails/20.jpg)
MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 20
Maxwell’s Equations:
Integral Form to Differential Form
Stokes Theorem
Divergence Theorem
ˆ∇∫ ∫�� � �i i�
C S
E dl = × E ndA
ˆ ∇∫ ∫� � �i i�
S V
F ndA = FdV
![Page 21: Chapter 32 Maxwell’s Equations and Electromagnetic Wavess Eqn.pdf · MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 4 ∫ ∫µ µ i i o o C C S B dl = J dA= I Something](https://reader036.fdocuments.us/reader036/viewer/2022062602/5e9f038681f46b79b4168757/html5/thumbnails/21.jpg)
MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 21
Maxwell’s Equations
∂
∂∫ ∫��i�
n
o C o 0
C S
EB dl = µ I + µ ε dA
tAmpere’s Law
Faraday’s Law
∫ ∫� �n
S V0
1E dA = ρdV
εGauss’s Law
∫� n
S
B dA = 0 No name - there are no
magnetic monopoles
∂
∂∫ ∫��i�
n
C
BE dl = - dA
t
![Page 22: Chapter 32 Maxwell’s Equations and Electromagnetic Wavess Eqn.pdf · MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 4 ∫ ∫µ µ i i o o C C S B dl = J dA= I Something](https://reader036.fdocuments.us/reader036/viewer/2022062602/5e9f038681f46b79b4168757/html5/thumbnails/22.jpg)
MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 22
Maxwell’s Equations
ˆ =∫ ∫ ∫�i� � �n
S S V0
1E dA = E ndA ρdV
εGauss’s Law
Use the Divergence Theorem to recast the surface
integral into a volume integral
ˆ ρ∇ =∫ ∫ ∫� � �i i�
oS V V
1E ndA = EdV dV
ε
ρ
ρ
∇ =
∇ =
∫� �i
� �i
oV
o
E - dV 0ε
E - 0ε
![Page 23: Chapter 32 Maxwell’s Equations and Electromagnetic Wavess Eqn.pdf · MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 4 ∫ ∫µ µ i i o o C C S B dl = J dA= I Something](https://reader036.fdocuments.us/reader036/viewer/2022062602/5e9f038681f46b79b4168757/html5/thumbnails/23.jpg)
MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 23
Maxwell’s Equations
∂
∂∫ ∫��i�
n
o C o 0
C S
EB dl = µ I + µ ε dA
tAmpere’s Law
Faraday’s Law
∫ ∫� �n
S V0
1E dA = ρdV
εGauss’s Law
∫� n
S
B dA = 0
∂
∂∫ ∫��i�
n
C
BE dl = - dA
t
ρ∇� �i
o
E =ε
∇ =� �iB 0
∂∇ ×
∂
�� � B
E + = 0t
0 0µ ε
∂∇ ×
∂
�� � �
o m
1 EB - = µ J
t
Integral form Differential form
![Page 24: Chapter 32 Maxwell’s Equations and Electromagnetic Wavess Eqn.pdf · MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 4 ∫ ∫µ µ i i o o C C S B dl = J dA= I Something](https://reader036.fdocuments.us/reader036/viewer/2022062602/5e9f038681f46b79b4168757/html5/thumbnails/24.jpg)
MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 24
Wave Eqn from Maxwell’s Eqn
The differential form of Maxwell’s equations brings out
the symmetry and non-symmetry of the E and B fields
We will use the following vector identity with the E-field
( )∇ ∇ ∇ ∇ ∇� � �� � � �
i2
× × A = A - A
( )∇ ∇ ∇ ∇ ∇� � � � � � �
i2
× × E = E - E ∂∇ ×
∂
�� � B
E + = 0t
( ) ∂∇ ∇ ∇ ∇
∂
� � � � � �i
2E - E = × B
t
( ) ∂ ∂
∇ ∇ ∇ + ∂ ∂
�� � � � �
i2
f
EE - E = µJ εµ
t t
![Page 25: Chapter 32 Maxwell’s Equations and Electromagnetic Wavess Eqn.pdf · MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 4 ∫ ∫µ µ i i o o C C S B dl = J dA= I Something](https://reader036.fdocuments.us/reader036/viewer/2022062602/5e9f038681f46b79b4168757/html5/thumbnails/25.jpg)
MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 25
Wave Eqn from Maxwell’s Eqn
( ) ∂ ∂
∇ ∇ ∇ + ∂ ∂
�� � � � �
i2
f
EE - E = µJ εµ
t t
These are the source
terms
ρ
ε
∂∂ ∇ + ∇
∂ ∂
��� �2
f2
2
JEE -εµ = µ
t t
∂∇
∂
∂∇
∂
��
��
22
o o 2
22
2 2
o o
EE - ε µ = 0
t
1 E 1E - = 0; where c =
c t ε µ
In free space there are no sources
This is the form of a wave equation
is the speed of light
![Page 26: Chapter 32 Maxwell’s Equations and Electromagnetic Wavess Eqn.pdf · MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 4 ∫ ∫µ µ i i o o C C S B dl = J dA= I Something](https://reader036.fdocuments.us/reader036/viewer/2022062602/5e9f038681f46b79b4168757/html5/thumbnails/26.jpg)
MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 26
Solutions of the Wave Equation
In free space the solutions of the wave equations show
that E and B are in phase.
These equations describe plane waves that are uniform
through out any plane perpendicular to the x-axis.
( )
x xo
y yo
E E= sin kx -ωt
B B
2π 2πk = ; ω= = 2πf
λ T
![Page 27: Chapter 32 Maxwell’s Equations and Electromagnetic Wavess Eqn.pdf · MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 4 ∫ ∫µ µ i i o o C C S B dl = J dA= I Something](https://reader036.fdocuments.us/reader036/viewer/2022062602/5e9f038681f46b79b4168757/html5/thumbnails/27.jpg)
MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 27
Plane Polarized Waves
( )
x xo
y yo
E E= sin kx -ωt
B B
2π 2πk = ; ω= = 2πf
λ T
Examining the E and B components
show that this represents a plane
polarized wave.
The E vector is oriented in the x
direction and the B vector is
oriented in the y direction.
![Page 28: Chapter 32 Maxwell’s Equations and Electromagnetic Wavess Eqn.pdf · MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 4 ∫ ∫µ µ i i o o C C S B dl = J dA= I Something](https://reader036.fdocuments.us/reader036/viewer/2022062602/5e9f038681f46b79b4168757/html5/thumbnails/28.jpg)
MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 28
Relationships Between E and B Vectors
oo o o
Ek 1B = E = E =
ω c c
E = cB
The Poynting Vector describes the propagation of the
electromagnetic energy � ��
o
E× BS =
µ
With E in the x-direction and B in the y-direction the
energy flows in the z-direction.
![Page 29: Chapter 32 Maxwell’s Equations and Electromagnetic Wavess Eqn.pdf · MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 4 ∫ ∫µ µ i i o o C C S B dl = J dA= I Something](https://reader036.fdocuments.us/reader036/viewer/2022062602/5e9f038681f46b79b4168757/html5/thumbnails/29.jpg)
MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 29
Relationships Between E and B Vectors
The Poynting Vector describes the propagation of the
electromagnetic energy � ��
o
E× BS =
µ
( ) ( )
( )
ˆ ˆ
ˆ
� �
� �
o o
2
o o
E× B = E sin kx -ωt i× B sin kx -ωt j
E × B = E B sin kx -ωt k
The energy is proportional to E and B and is flowing
in the z-direction, perpendicular to E and B.
![Page 30: Chapter 32 Maxwell’s Equations and Electromagnetic Wavess Eqn.pdf · MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 4 ∫ ∫µ µ i i o o C C S B dl = J dA= I Something](https://reader036.fdocuments.us/reader036/viewer/2022062602/5e9f038681f46b79b4168757/html5/thumbnails/30.jpg)
MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 30
The Principle of Invariance
The laws of Physics should be the same for all
non-accelerated observers.
Einstein’s fundamental postulate of relativity can be stated:
“It is physically impossible to detect the uniform motion of a
frame of reference from observations made entirely within
that frame.”
![Page 31: Chapter 32 Maxwell’s Equations and Electromagnetic Wavess Eqn.pdf · MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 4 ∫ ∫µ µ i i o o C C S B dl = J dA= I Something](https://reader036.fdocuments.us/reader036/viewer/2022062602/5e9f038681f46b79b4168757/html5/thumbnails/31.jpg)
MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 31
The Principle of Invariance
The laws of Physics should be the same for all
non-accelerated observers.
If two observers watch the motion of an object from
two different inertial reference systems (no
acceleration), moving at a relative velocity v, they
should find the same laws of Physics
F1 = m1a1 and F2=m2a2
![Page 32: Chapter 32 Maxwell’s Equations and Electromagnetic Wavess Eqn.pdf · MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 4 ∫ ∫µ µ i i o o C C S B dl = J dA= I Something](https://reader036.fdocuments.us/reader036/viewer/2022062602/5e9f038681f46b79b4168757/html5/thumbnails/32.jpg)
MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 32
Galilean and Lorentz Transformations
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MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 33
Galilean and Lorentz Transformations
The inertial reference frames are related by a Galilean transformation.
Newton’s laws are invariant under these transformations but not Maxwell’s
Equations
Prior to Einstein’s Theory of Special Relativity it was determined that a
Lorentz transformation kept Maxwell’s equation invariant. However, no one
knew exactly what they meant.
![Page 34: Chapter 32 Maxwell’s Equations and Electromagnetic Wavess Eqn.pdf · MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 4 ∫ ∫µ µ i i o o C C S B dl = J dA= I Something](https://reader036.fdocuments.us/reader036/viewer/2022062602/5e9f038681f46b79b4168757/html5/thumbnails/34.jpg)
MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 34
Galilean and Lorentz Transformations
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MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 35
Electromagnetic Radiation
![Page 36: Chapter 32 Maxwell’s Equations and Electromagnetic Wavess Eqn.pdf · MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 4 ∫ ∫µ µ i i o o C C S B dl = J dA= I Something](https://reader036.fdocuments.us/reader036/viewer/2022062602/5e9f038681f46b79b4168757/html5/thumbnails/36.jpg)
MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 36
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MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 37
These are all different forms of
electromagnetic radiation.
Anytime you accelerate or
decelerate a charged particle it
gives off electromagnetic radiation.
Electrons circulating about their
nuclei don't give off radiation
unless they change energy levels.
Thermal motion gives off continuous EM radiation. Example –
Infrared radiation which peaks below the visible spectrum.
Electromagnetic Radiation
![Page 38: Chapter 32 Maxwell’s Equations and Electromagnetic Wavess Eqn.pdf · MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 4 ∫ ∫µ µ i i o o C C S B dl = J dA= I Something](https://reader036.fdocuments.us/reader036/viewer/2022062602/5e9f038681f46b79b4168757/html5/thumbnails/38.jpg)
MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 38
Electric Dipole Radiation
![Page 39: Chapter 32 Maxwell’s Equations and Electromagnetic Wavess Eqn.pdf · MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 4 ∫ ∫µ µ i i o o C C S B dl = J dA= I Something](https://reader036.fdocuments.us/reader036/viewer/2022062602/5e9f038681f46b79b4168757/html5/thumbnails/39.jpg)
MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 39
Electric Dipole Radiation
![Page 40: Chapter 32 Maxwell’s Equations and Electromagnetic Wavess Eqn.pdf · MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 4 ∫ ∫µ µ i i o o C C S B dl = J dA= I Something](https://reader036.fdocuments.us/reader036/viewer/2022062602/5e9f038681f46b79b4168757/html5/thumbnails/40.jpg)
MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 40
Dipole Antenna - Radiation Distribution
Note the different orientation
of the angle measurement∝2
2
sinθI(θ)
r
![Page 41: Chapter 32 Maxwell’s Equations and Electromagnetic Wavess Eqn.pdf · MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 4 ∫ ∫µ µ i i o o C C S B dl = J dA= I Something](https://reader036.fdocuments.us/reader036/viewer/2022062602/5e9f038681f46b79b4168757/html5/thumbnails/41.jpg)
MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 41
Dipole Antenna - Radiation Distribution
2
o 2
sin θI(θ) = I
r
In these problems you will need to determine the value of
Io or else take a ratio so that the Io factor will cancel out.
![Page 42: Chapter 32 Maxwell’s Equations and Electromagnetic Wavess Eqn.pdf · MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 4 ∫ ∫µ µ i i o o C C S B dl = J dA= I Something](https://reader036.fdocuments.us/reader036/viewer/2022062602/5e9f038681f46b79b4168757/html5/thumbnails/42.jpg)
MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 42
Dipole Antenna - Radiation Distribution
2
o 2
sin θI(r,θ) = I
r
(a.) Find I1 at r1 = 10m and θ = 90o
(b.) Find I2 at r2 = 30m and θ = 90o
(c.) Ratio of I2 / I1
=2
o o1 o 2
I1I = I(r = 10,θ = 90 ) = I
10 100
=2
o o2 o 2
I1I = I(r = 30,θ = 90 ) = I
30 900
(a.)
(b.)
(c.)o
2
o1
II 1900= =
II 9100
![Page 43: Chapter 32 Maxwell’s Equations and Electromagnetic Wavess Eqn.pdf · MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 4 ∫ ∫µ µ i i o o C C S B dl = J dA= I Something](https://reader036.fdocuments.us/reader036/viewer/2022062602/5e9f038681f46b79b4168757/html5/thumbnails/43.jpg)
MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 43
![Page 44: Chapter 32 Maxwell’s Equations and Electromagnetic Wavess Eqn.pdf · MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 4 ∫ ∫µ µ i i o o C C S B dl = J dA= I Something](https://reader036.fdocuments.us/reader036/viewer/2022062602/5e9f038681f46b79b4168757/html5/thumbnails/44.jpg)
MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 44
Electric - Dipole Antenna
http://www.austincc.edu/mmcgraw/physics_simulations.htm
![Page 45: Chapter 32 Maxwell’s Equations and Electromagnetic Wavess Eqn.pdf · MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 4 ∫ ∫µ µ i i o o C C S B dl = J dA= I Something](https://reader036.fdocuments.us/reader036/viewer/2022062602/5e9f038681f46b79b4168757/html5/thumbnails/45.jpg)
MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 45
http://www.falstad.com/mathphysics.html
Oscillating Ring Antenna
![Page 46: Chapter 32 Maxwell’s Equations and Electromagnetic Wavess Eqn.pdf · MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 4 ∫ ∫µ µ i i o o C C S B dl = J dA= I Something](https://reader036.fdocuments.us/reader036/viewer/2022062602/5e9f038681f46b79b4168757/html5/thumbnails/46.jpg)
MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 46
http://www.falstad.com/mathphysics.html
Oscillating Ring Pair Antenna
![Page 47: Chapter 32 Maxwell’s Equations and Electromagnetic Wavess Eqn.pdf · MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 4 ∫ ∫µ µ i i o o C C S B dl = J dA= I Something](https://reader036.fdocuments.us/reader036/viewer/2022062602/5e9f038681f46b79b4168757/html5/thumbnails/47.jpg)
MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 47
Electric - Dipole Antenna
Plane wave – Far from source antenna - “Far Field”
![Page 48: Chapter 32 Maxwell’s Equations and Electromagnetic Wavess Eqn.pdf · MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 4 ∫ ∫µ µ i i o o C C S B dl = J dA= I Something](https://reader036.fdocuments.us/reader036/viewer/2022062602/5e9f038681f46b79b4168757/html5/thumbnails/48.jpg)
MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 48
Magnetic - Loop Antenna
Plane wave – Far from source antenna - “Far Field”
![Page 49: Chapter 32 Maxwell’s Equations and Electromagnetic Wavess Eqn.pdf · MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 4 ∫ ∫µ µ i i o o C C S B dl = J dA= I Something](https://reader036.fdocuments.us/reader036/viewer/2022062602/5e9f038681f46b79b4168757/html5/thumbnails/49.jpg)
MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 49
Magnetic - Loop Antenna
( )
( )
( )
( )
∂
∂
∂
∂
∂
∂
∂ ∂
2m
2
rms
o
o
o
o rmsrms
rms
rms
d BAdφ B= - = - = - πr
dt dt t
B= πr
t
B = B sin kx -ωt
B= -ωB cos kx -ωt
t
ωBB= ωB -cos kx -ωt = = ωB
t 2
ε
ε
c
∂=
∂
= =
2 2
rms
rms
22 2 2rms
rms rms
rms
rms
B= πr πr ωB
t
E 2π= πr ωB πr ω r fE
c
ε
ε
Find εrms ?Find εrms ?
![Page 50: Chapter 32 Maxwell’s Equations and Electromagnetic Wavess Eqn.pdf · MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 4 ∫ ∫µ µ i i o o C C S B dl = J dA= I Something](https://reader036.fdocuments.us/reader036/viewer/2022062602/5e9f038681f46b79b4168757/html5/thumbnails/50.jpg)
MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 50
Magnetic - Loop Antenna
22
rmsrms
2π= r f E
cε
rms
r = 10.0cm
N = 1
E = 0.150 V/m
f : (a.) 600 kHz; (b.) 60.0 MHz
( ) ( ) =2
2 3
rms
2π= 0.10 600x10 0.150 59.2µV
cε(a.)
(b.) ( ) ( ) =2
2 6
rms
2π= 0.10 60x10 0.150 5.92mV
cε
![Page 51: Chapter 32 Maxwell’s Equations and Electromagnetic Wavess Eqn.pdf · MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 4 ∫ ∫µ µ i i o o C C S B dl = J dA= I Something](https://reader036.fdocuments.us/reader036/viewer/2022062602/5e9f038681f46b79b4168757/html5/thumbnails/51.jpg)
MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 51
Energy and Momentum in an
Electromagnetic Wave
![Page 52: Chapter 32 Maxwell’s Equations and Electromagnetic Wavess Eqn.pdf · MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 4 ∫ ∫µ µ i i o o C C S B dl = J dA= I Something](https://reader036.fdocuments.us/reader036/viewer/2022062602/5e9f038681f46b79b4168757/html5/thumbnails/52.jpg)
MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 52
Energy and Momentum in an
Electromagnetic Wave
![Page 53: Chapter 32 Maxwell’s Equations and Electromagnetic Wavess Eqn.pdf · MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 4 ∫ ∫µ µ i i o o C C S B dl = J dA= I Something](https://reader036.fdocuments.us/reader036/viewer/2022062602/5e9f038681f46b79b4168757/html5/thumbnails/53.jpg)
MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 53
The Poynting Vector describes the propagation of the
electromagnetic energy � ��
o
E× BS =
µ
avg avg
avg avg
avg
avg
U u LAP = = = u Ac
∆t L c
P I = = u c
A
Uavg is the total energy and uavg is the energy density.
I is the intensity, the average power per unit area.
E-M Energy and Momentum
![Page 54: Chapter 32 Maxwell’s Equations and Electromagnetic Wavess Eqn.pdf · MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 4 ∫ ∫µ µ i i o o C C S B dl = J dA= I Something](https://reader036.fdocuments.us/reader036/viewer/2022062602/5e9f038681f46b79b4168757/html5/thumbnails/54.jpg)
MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 54
E-M Energy and Momentum
22
e o m
o
1 Bu = ε E and u =
2 2µ
These are the electric and magnetic energy densities
Since E = cB
( )2
2 22
m o e2
o o o
E cB E 1u = = = = ε E = u
2µ 2µ 22µ c
Therefore the energy density can be expressed in different ways.
=2
2
e m o
o o
B EBu = u + u = ε E =
µ µ c
![Page 55: Chapter 32 Maxwell’s Equations and Electromagnetic Wavess Eqn.pdf · MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 4 ∫ ∫µ µ i i o o C C S B dl = J dA= I Something](https://reader036.fdocuments.us/reader036/viewer/2022062602/5e9f038681f46b79b4168757/html5/thumbnails/55.jpg)
MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 55
E-M Energy and Momentum
The energy density
=2
2
e m o
o o
B EBu = u + u = ε E =
µ µ c
�rms rms o o
avg avg
o o
E B E B1I = u c = = = S
µ 2 µ
The intensity I is the energy/(m2 sec) = power/m2;
� ��
o
E× BS =
µ
This is the Poynting vector, its magnitude is the intensity.
![Page 56: Chapter 32 Maxwell’s Equations and Electromagnetic Wavess Eqn.pdf · MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 4 ∫ ∫µ µ i i o o C C S B dl = J dA= I Something](https://reader036.fdocuments.us/reader036/viewer/2022062602/5e9f038681f46b79b4168757/html5/thumbnails/56.jpg)
MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 56
Radiation Pressure pr
=
ir avg
2 2
o o rms rms o or 2
o o o o
Momentum Ip = = u
Unit Area Unit Time c
E B E B E BIp = = = = =
c 2µ c µ c 2µ c 2µ
![Page 57: Chapter 32 Maxwell’s Equations and Electromagnetic Wavess Eqn.pdf · MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 4 ∫ ∫µ µ i i o o C C S B dl = J dA= I Something](https://reader036.fdocuments.us/reader036/viewer/2022062602/5e9f038681f46b79b4168757/html5/thumbnails/57.jpg)
MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 57
Radiation Pressure pr - Example
A lightbulb emits spherically symmetric electromagnetic waves in al
directions. Assume 50W of electromagnetic radiation is emitted. Find (a)
the intensity, (b) the radiation pressure, (c) the electric and magnetic field
magnitudes at 3.0m from the bulb.
The energy spreads out uniformly over a sphere of radius r.
The surface area of the sphere is 4πr2.
2 2
Power 50 WI = Intensity = = = 0.442
Area 4πr m(a.)
(b.)-9
r 8
I 0.442p = = = 1.47x10 Pa
c 3.0x10
![Page 58: Chapter 32 Maxwell’s Equations and Electromagnetic Wavess Eqn.pdf · MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 4 ∫ ∫µ µ i i o o C C S B dl = J dA= I Something](https://reader036.fdocuments.us/reader036/viewer/2022062602/5e9f038681f46b79b4168757/html5/thumbnails/58.jpg)
MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 58
Radiation Pressure pr - Example
A lightbulb emits spherically symmetric electromagnetic waves in al
directions. Assume 50W of electromagnetic radiation is emitted. Find (a)
the intensity, (b) the radiation pressure, (c) the electric and magnetic field
magnitudes at 3.0m from the bulb.
(c.) Remember
2 2
o or o o2
o o
E BIp = = = and E = cB
c 2µ c 2µ
( )( )( )
( )
-9 -7
o o r
-8
o
8 -8
0 o
0
B = 2µ p = 1.47x10 2 4πx10
B = 6.08x10 T
E = cB = 3.0x10 6.08x10
VE = 18.2
m
![Page 59: Chapter 32 Maxwell’s Equations and Electromagnetic Wavess Eqn.pdf · MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 4 ∫ ∫µ µ i i o o C C S B dl = J dA= I Something](https://reader036.fdocuments.us/reader036/viewer/2022062602/5e9f038681f46b79b4168757/html5/thumbnails/59.jpg)
MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 59
Extra Slides
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MFMcGraw-PHY 2426 Chap32-Maxwell's Eqn-Revised: 6/24/2012 60