ii. electromagnetic waves a.displacement current 1.recall ampere’s law: 2.as we’ve learned it,...
TRANSCRIPT
![Page 1: II. Electromagnetic Waves A.Displacement Current 1.Recall Ampere’s Law: 2.As we’ve learned it, AL is incomplete. We need to add an additional current,](https://reader033.vdocuments.net/reader033/viewer/2022052509/56649cb75503460f9497c890/html5/thumbnails/1.jpg)
II. Electromagnetic WavesII. Electromagnetic Waves
A. Displacement Current1. Recall Ampere’s Law:
2. As we’ve learned it, AL is incomplete. We need to add an additional current, called the displacement current, ID. ID arises from time-varying electric fields (not present in a steady current along an infinite wire):
encILdB 0
dt
dI
dt
EdI
ED
D
;||
||(II.A.1,2)
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II. Electromagnetic WavesII. Electromagnetic Waves
A. Displacement Current3. General form of Ampere’s Law includes terms
due to “conduction current” and “displacement current”:
encE
c dt
dILdB )( 00
(II.A.3)
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II. Electromagnetic WavesII. Electromagnetic Waves
B.B. MAXWELL’S EQUATIONSMAXWELL’S EQUATIONS1. Unified description of E, B:
.)(
.
.0
.
00
0
encE
c
B
enc
dt
dILdB
dt
dLdE
AdB
QAdE
(II.B.1-4)
(Gauss’s Law)
(Gauss’s Law for B)
(Faraday’s Law for B)
(Ampere’s Law for B)
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II. Electromagnetic WavesII. Electromagnetic Waves
B.B. MAXWELL’S EQUATIONSMAXWELL’S EQUATIONS2. Plane Wave
a) As we shall see, the solution to Maxwell’s Equations is a wave of Electric and Magnetic Fields.
b) Plane Wave Definition: Wave in which the transverse components are uniform on a plane perpendicular to the direction of propagation.
V
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II. Electromagnetic WavesII. Electromagnetic Waves
B.B. MAXWELL’S EQUATIONSMAXWELL’S EQUATIONS2. Plane Wave
a) As we shall see, the solution to Maxwell’s Equations is a wave of Electric and Magnetic Fields.
b) Plane Wave Definition: Wave in which the transverse components are uniform on a plane perpendicular to the direction of propagation.
E
B
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II. Electromagnetic WavesII. Electromagnetic Waves
B.B. MAXWELL’S EQUATIONSMAXWELL’S EQUATIONS2. Plane Wave
a) As we shall see, the solution to Maxwell’s Equations is a wave of Electric and Magnetic Fields.
b) Plane Wave Definition: Wave in which the transverse components are uniform on a plane perpendicular to the direction of propagation.
E
B
E
B
![Page 7: II. Electromagnetic Waves A.Displacement Current 1.Recall Ampere’s Law: 2.As we’ve learned it, AL is incomplete. We need to add an additional current,](https://reader033.vdocuments.net/reader033/viewer/2022052509/56649cb75503460f9497c890/html5/thumbnails/7.jpg)
II. Electromagnetic WavesII. Electromagnetic Waves
B.B. MAXWELL’S EQUATIONSMAXWELL’S EQUATIONS2. Plane Wave
a) As we shall see, the solution to Maxwell’s Equations is a wave of Electric and Magnetic Fields.
b) Plane Wave Definition: Wave in which the transverse components are uniform on a plane perpendicular to the direction of propagation.
E
B
E
B
E
B
![Page 8: II. Electromagnetic Waves A.Displacement Current 1.Recall Ampere’s Law: 2.As we’ve learned it, AL is incomplete. We need to add an additional current,](https://reader033.vdocuments.net/reader033/viewer/2022052509/56649cb75503460f9497c890/html5/thumbnails/8.jpg)
II. Electromagnetic WavesII. Electromagnetic Waves
B.B. MAXWELL’S EQUATIONSMAXWELL’S EQUATIONS3. Electromagnetic Wave Properties
a) Transverse waveb) Ratio between E,B:
E/B = c. (II.B.5)
c) Constant speedd) No medium required: E
and B reinforce each other.
E
B
E
B
E
B
E
B
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II. Electromagnetic WavesII. Electromagnetic WavesB.B. MAXWELL’S EQUATIONSMAXWELL’S EQUATIONS
4.4. Derivation of Solution: Plane WaveDerivation of Solution: Plane Wavea) Consider a plane wave with Bz, Ey propagating
in the x-direction with speed v. After time t, the two wave fronts are separated by a distance x.
b) Apply Faraday’s Law to a rectangle
in the xy-plane:E
B
E
B
x x x
y
z B, A
a
x
y
)).,(),((
),(),(
txEtxxEa
atxxEatxE
LdE
yy
yy
![Page 10: II. Electromagnetic Waves A.Displacement Current 1.Recall Ampere’s Law: 2.As we’ve learned it, AL is incomplete. We need to add an additional current,](https://reader033.vdocuments.net/reader033/viewer/2022052509/56649cb75503460f9497c890/html5/thumbnails/10.jpg)
II. EM Waves B.Maxwell’s II. EM Waves B.Maxwell’s EquationsEquations
4.4. Derivation of Solution: Plane WaveDerivation of Solution: Plane Waveb) Apply Faraday’s Law to a rectangle
in the xy-plane: Assume x is small enough that Bz ~ uniform
over surface.xa
t
txB
dt
d zB
),(
E
B
E
B
x x x
y
z
B, A
a
x
y
x
.),(),(),(
)),(),((t
txB
x
txExa
t
txBtxEtxxEa zyz
yy
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II. EM Waves B.Maxwell’s II. EM Waves B.Maxwell’s EquationsEquations
4.4. Derivation of Solution: Plane WaveDerivation of Solution: Plane Waveb) Apply Ampere’s Law to a rectangle
in the zx-plane: Assume x is small enough that Ey ~ uniform
over surface.
E
B
E
B
x x x
y
z
E, A
a
x
z
x
)).,(),((
),(),(
txEtxxEa
atxxBatxB
LdB
yy
zz
![Page 12: II. Electromagnetic Waves A.Displacement Current 1.Recall Ampere’s Law: 2.As we’ve learned it, AL is incomplete. We need to add an additional current,](https://reader033.vdocuments.net/reader033/viewer/2022052509/56649cb75503460f9497c890/html5/thumbnails/12.jpg)
II. EM Waves B.Maxwell’s II. EM Waves B.Maxwell’s EquationsEquations
4.4. Derivation of Solution: Plane WaveDerivation of Solution: Plane Waveb) Apply Ampere’s Law to a rectangle
in the zx-plane: Assume x is small enough that Ey ~ uniform
over surface.xa
t
txE
dt
d yE
),(
E
B
E
B
x x x
y
z
E, A
a
x
.),(),(),(
)),(),(( 0000 t
txE
x
txBxa
t
txEtxBtxxBa yzy
zz
x
z
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II. EM Waves B.Maxwell’s II. EM Waves B.Maxwell’s EquationsEquations
4.4. Derivation of Solution: Plane WaveDerivation of Solution: Plane Wavec) Now take partial time and space derivatives
of both equations:
.),(),(
.),(),(
2
22
2
2
00
2
x
txE
tx
txB
t
txE
xt
txB
yz
yz
E
B
E
B
x x x
y
z z
.),(),(
2
2
002
2
t
txE
x
txE yy
This is the wave equation withv = ()-1/2 = c! (II.B.7)
(II.B.6)
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II. EM Waves B.Maxwell’s II. EM Waves B.Maxwell’s EquationsEquations
5. Sinusoidal Wavesa) A more accurate representation of EM Wavesb) Plane waves can be a good approximationc) For wave propagating in the +x-direction:
d) E,B in phase, follow RHR: c, E, B
ktkxBB
jtkxEE
ˆ)cos(
ˆ)cos(
max
max
(II.B.8)
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C. The Production of Electromagnetic Waves3. Antennae
a) Accelerating charges radiate energy as EM waves
b) Oscillating voltage => accelerates charge => EM radiation
+++
t = 0: Charge placed on metal rods connected to anAC generator.
---
EV
II. EM WavesII. EM Waves
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F. The Production of Electromagnetic Waves3. Antennae
a) Accelerating charges radiate energy as EM waves
b) Oscillating voltage => accelerates charge => EM radiation
++
t = 0 to T/4: Rods neutralize, and E decreases to 0.Note: Initial E propagates away from arrayat speed c.
--
E
II. EM WavesII. EM Waves
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C. The Production of Electromagnetic Waves3. Antennae
a) Accelerating charges radiate energy as EM waves
b) Oscillating voltage => accelerates charge => EM radiation
+
t = 0 to T/4: Rods neutralize, and E decreases to 0.Note: Initial E propagates away from arrayat speed c.
-
E
II. EM WavesII. EM Waves
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C. The Production of Electromagnetic Waves3. Antennae
a) Accelerating charges radiate energy as EM waves
b) Oscillating voltage => accelerates charge => EM radiation
t = 0 to T/4: Rods neutralize, and E decreases to 0.Note: Initial E propagates away from arrayat speed c.
E = 0 at t = T/4.
II. EM WavesII. EM Waves
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C. The Production of Electromagnetic Waves3. Antennae
a) Accelerating charges radiate energy as EM waves
b) Oscillating voltage => accelerates charge => EM radiation-
t = T/4 to T/2: E reverses direction and grows.
+
II. EM WavesII. EM Waves
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C. The Production of Electromagnetic Waves3. Antennae
a) Accelerating charges radiate energy as EM waves
b) Oscillating voltage => accelerates charge => EM radiation
--
t = T/4 to T/2: E reverses direction and grows.
++
II. EM WavesII. EM Waves
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+++
C. The Production of Electromagnetic Waves3. Antennae
a) Accelerating charges radiate energy as EM waves
b) Oscillating voltage => accelerates charge => EM radiation
---
t = T/4 to T/2: E reverses direction and grows.
II. EM WavesII. EM Waves
![Page 22: II. Electromagnetic Waves A.Displacement Current 1.Recall Ampere’s Law: 2.As we’ve learned it, AL is incomplete. We need to add an additional current,](https://reader033.vdocuments.net/reader033/viewer/2022052509/56649cb75503460f9497c890/html5/thumbnails/22.jpg)
C. The Production of Electromagnetic Waves3. Antennae
a) Accelerating charges radiate energy as EM waves
b) Oscillating voltage => accelerates charge => EM radiation
---
+++
II. EM WavesII. EM Waves
![Page 23: II. Electromagnetic Waves A.Displacement Current 1.Recall Ampere’s Law: 2.As we’ve learned it, AL is incomplete. We need to add an additional current,](https://reader033.vdocuments.net/reader033/viewer/2022052509/56649cb75503460f9497c890/html5/thumbnails/23.jpg)
C. The Production of Electromagnetic Waves3. Antennae
c) Oscillating E => Oscillating B wave---
+++
v = c.
B
End result: A transverse wave of E propagatingat speed v = (00)-1/2 = c.
II. EM WavesII. EM Waves
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C. The Production of Electromagnetic Waves3. Antennae
c) Oscillating E => Oscillating B wave
Top View
-
B
II. EM WavesII. EM Waves
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C. The Production of Electromagnetic Waves3. Antennae
c) Oscillating E => Oscillating B wave
Top View
-
II. EM WavesII. EM Waves
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C. The Production of Electromagnetic Waves3. Antennae
c) Oscillating E => Oscillating B wave
Top View
-
II. EM WavesII. EM Waves
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C. The Production of Electromagnetic Waves3. Antennae
c) Oscillating E => Oscillating B wave
Top View
II. EM WavesII. EM Waves
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C. The Production of Electromagnetic Waves3. Antennae
c) Oscillating E => Oscillating B wave
Top View
+
II. EM WavesII. EM Waves
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C. The Production of Electromagnetic Waves3. Antennae
c) Oscillating E => Oscillating B wave
Top View
II. EM WavesII. EM Waves
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C. The Production of Electromagnetic Waves3. Antennae
c) Oscillating E => Oscillating B wave
Top View
-c
* E and B perpendicular to each other.* E and B perpendicular to v.* E and B in phase.
II. EM WavesII. EM Waves
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II. EM WavesII. EM WavesD. Properties of EM Waves
1. Field strengths of EM waveE/B = c.
(II.D.1)2. Poynting Vector: Energy Flow Rate Vector
3. Power and Intensity: P= S per unit area, I = S(avg)
I = EmaxBmax/(20), (II.D.2)
= E2max/(20c) = B2
max(c/20).
BES
1
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D. Properties of EM Waves
4. Radiation Pressurep = I/c (complete absorption) (II.D.3)p = 2I/c (complete reflection) (II.D.4)
5. EM waves in mattern = c/v = “index of refraction”
(II.D.5)
II. EM WavesII. EM Waves
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E. The Electromagnetic Spectrum1. Units
a) Angstrom (Å) = 10-10 mb) Nanometer (nm) = 10-9 mc) Micron (m) = 10-6 m
2. Radio, Microwave, Infrared, Visible, Ultraviolet, X-rays, Gamma rays
3. VISIBLE: “ROYGBIV” = Red, Orange, Yellow, Green, Blue, Indigo, and Violet
(large wavelength to small)
II. EM WavesII. EM Waves
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E. The Electromagnetic Spectrum1. Units
a) Angstrom (Å) = 10-10 mb) Nanometer (nm) = 10-9 mc) Micron (m) = 10-6 m
2. Radio, Microwave, Infrared, Visible, Ultraviolet, X-rays, Gamma rays
3. VISIBLE: “ROYGBIV” = Red, Orange, Yellow, Green, Blue, Indigo, and Violet
(large wavelength to small)
II. EM WavesII. EM Waves
![Page 35: II. Electromagnetic Waves A.Displacement Current 1.Recall Ampere’s Law: 2.As we’ve learned it, AL is incomplete. We need to add an additional current,](https://reader033.vdocuments.net/reader033/viewer/2022052509/56649cb75503460f9497c890/html5/thumbnails/35.jpg)
A. Working Definitions1. Diffraction occurs when light source is not a
perfect point source and wave encounters a sharp edge.
2. Diffraction is essentially an example of interference between a large (continuous) distribution of sources.
3. Limits resolution of instruments—but also can be used to separate multi-chormatic light.
III. DiffractionIII. Diffraction
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4. Spreading of wave from its initial line of travel
No diffraction
III. DiffractionIII. Diffraction
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4. Spreading of wave from its initial line of travel
Diffraction
III. DiffractionIII. Diffraction
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5. Diffraction occurs when light passes through a narrow opening, around obstacles and at sharp edges.
a) Application: Calculating stellar diameters by lunar occultation
Unresolved pointof light
III. DiffractionIII. Diffraction
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5. Diffraction occurs when light passes through a narrow opening, around obstacles and at sharp edges.
a) Application: Calculating stellar diameters by lunar occultation
Unresolved pointof light
III. DiffractionIII. Diffraction
![Page 40: II. Electromagnetic Waves A.Displacement Current 1.Recall Ampere’s Law: 2.As we’ve learned it, AL is incomplete. We need to add an additional current,](https://reader033.vdocuments.net/reader033/viewer/2022052509/56649cb75503460f9497c890/html5/thumbnails/40.jpg)
5. Diffraction occurs when light passes through a narrow opening, around obstacles and at sharp edges.
a) Application: Calculating stellar diameters by lunar occultation
Unresolved pointof light
III. DiffractionIII. Diffraction
![Page 41: II. Electromagnetic Waves A.Displacement Current 1.Recall Ampere’s Law: 2.As we’ve learned it, AL is incomplete. We need to add an additional current,](https://reader033.vdocuments.net/reader033/viewer/2022052509/56649cb75503460f9497c890/html5/thumbnails/41.jpg)
5. Diffraction occurs when light passes through a narrow opening, around obstacles and at sharp edges.
a) Application: Calculating stellar diameters by lunar occultation
III. DiffractionIII. Diffraction
Resolved diffractionpattern: spacing of fringes=> width of star