Electromagnetic Waves - Wake Forest Universityusers.wfu.edu/ucerkb/Phy114/L12-EM_Waves.pdf ·...
Transcript of Electromagnetic Waves - Wake Forest Universityusers.wfu.edu/ucerkb/Phy114/L12-EM_Waves.pdf ·...
Electromagnetic WavesAmpere’s Law UpdatedHertz’s DiscoveriesPlane Electromagnetic WavesEnergy in an EM WaveMomentum and Radiation PressureThe Electromagnetic Spectrum
Problem with Ampère’s
LawId 0. μ=∫ sB rr
Ampère’s Law
For S1 , Id 0. μ=∫ sB rr
For S2 , 0. 0 ==∫ Id μsB rr
(No current in the gap)
But there is a problem if I varies with time!
Consider a parallel plate capacitor
Solution -
Displacement CurrentCall I as the conduction current.
Specify a new current that exists when a change in the electric field occurs. Call it the displacement current.
dtdI E
dΦ
= 0ε
Electrical flux through S2 :
00 εεQA
AQEAE =⎟⎟
⎠
⎞⎜⎜⎝
⎛==Φ
IdtdQ
dtdI E
d ==Φ
= 0ε
Ampère-Maxwell Law
Magnetic fields are produced by both conduction currents and by
time-varying electrical fields.
( )dt
dIIId Ed
Φ+=+=∫ 0000. εμμμsB rr
Now the generalized form of Ampère’s Law, or Ampère-Maxwell Law becomes:
Maxwell’s Equations
BvEFrrrr
×+= qq Lorentz Force Law
Faraday’s Lawdtdd BΦ
−=∫ sE rr.
Ampère-Maxwell LawdtdId EΦ
+=∫ 000. εμμsBrr
Gauss’ Law for Magnetism – no magnetic monopoles
0. =∫ ABrr
d
Gauss’ Law0
.εQd =∫ AE
rr
Hertz’s Radio WavesBy supplying short voltage bursts from the coil to the transmitter electrode we can ionize the air between the electrodes.
In effect, the circuit can be modeled as a LC circuit, with the coil as the inductor and the electrodes as the capacitor.
A receiver loop placed nearby is able to receive these oscillations and creates sparks as well.
Waves
Circular (or Spherical in 3D) Waves
Plane Waves
Electric and Magnetic Fields in Free Space
dtdd BΦ
−=∫ sE rr.
dtdId EΦ
+=∫ 000. εμμsBrr
Free Space: I = 0, Q = 0
dtdd EΦ
=∫ 00. εμsBrr
tE
xB
tB
xE
∂∂
−=∂∂
∂∂
−=∂∂
00εμ
2
2
002
2
002
2
tE
xE
tE
txB
ttB
xxE
∂∂
=∂∂
⎟⎠⎞
⎜⎝⎛
∂∂
−∂∂
−=⎟⎠⎞
⎜⎝⎛∂∂
∂∂
−=⎟⎠⎞
⎜⎝⎛∂∂
∂∂
−=∂∂
εμ
εμ
2
2
002
2
tB
xB
∂∂
=∂∂ εμ
( )( )tkxBB
tkxEEωω
−=−=
coscos
max
max
EM Wave Oscillation
Some Important Quantities
λπ2
=k Wavenumber
00
1εμ
=c Speed of Light
fπω 2= Angular Frequency
fc
=λ Wavelength
ck=
ω
( )( )tkxBB
tkxEEωω
−=−=
coscos
max
max
( )( )tkxBB
tkxEEωω
−=−=
coscos
max
max
tE
xB
tB
xE
∂∂
−=∂∂
∂∂
−=∂∂
00εμ
ckB
EBE
BkE
===
=ω
ω
max
max
maxmax
Properties of EM WavesThe solutions to Maxwell’s equations in free space are wavelikeElectromagnetic waves travel through free space at the speed of light.The electric and magnetic fields of a plane wave are perpendicular to each other and the direction of propagation (they are transverse).The ratio of the magnitudes of the electric and magnetic fields is c.EM waves obey the superposition principle.
An EM Wave
MHzf 40=
a) λ=?, T=?
sf
T
mfc
86
6
8
105.21040
11
5.71040103
−×=×
==
=××
==λ
b) If Emax =750N/C, then Bmax =?
Tc
EB 68
maxmax 105.2
103750 −×=×
==
Directed towards the z-direction
c) E(t)=? and B(t)=?
( ) ( )( ) ( ) )1051.2838.0cos(105.2cos
)1051.2838.0cos(/750cos86
max
8max
txTtkxBB
txCNtkxEE
×−×=−=
×−=−=−ω
ωsradf
mradk
/1051.2104022
/838.05.7
22
86 ×=×==
===
ππω
πλπ
Energy Carried by EM WavesBESrrr
×=0
1μ Poynting Vector (W/m2)
For a plane EM Wave: 2
00
2
0
Bcc
EEBSμμμ
===
2max
00
2max
0
maxmax
222Bc
cEBESI av μμμ
====The average value of S is called the intensity:
S is equal to the rate of EM energy flow per unit area (power per unit area)
0
2
2μBuB =
2
20EuEε
=
( ) 20
2
0
00
0
2
21
22EEcEuB ε
μεμ
μ===
0
22
0 221
με BEuu BE ===
For an EM wave, the instantaneous electric and magnetic energies are equal.
0
22
0 με BEuuu BE ==+=
( )0
2max2
max02
0 221
μεε BEEu avav ===
avav cuSI ==
Total Energy Density of an EM Wave
The intensity is c times the total average energy density
The Energy Density
Which gives the largest average energy densityat the distance specified and thus, at least qualitatively, the best illumination
1. a 50-W source at a distance R.2. a 100-W source at a distance 2R.3. a 200-W source at a distance 4R.
Concept Question
Fields on the Screen
cE
AI av
0
2max
2μ==
P
P lamp = 150 W, 3% efficiencyA = 15 m2
mVAcE av /42.182 0
max ==Pμ
Wlampav 5.403.0 == PP
Tc
EB 8maxmax 1014.6 −×==
A
Momentum and Radiation Pressure
cTp ER= Momentum for complete absorption
dtdp
AAFP 1==
( )A
dtdTcc
Tdtd
AP ERER 11
=⎟⎠⎞
⎜⎝⎛=
cSP =
Pressure on surface
For complete reflection:cTp ER2
=cSP 2
=
Pressure From a Laser PointerP laser = 3 mW, 70% reflection, d = 2 mm
( )2
2 /955001.0003.0 mW
AS ===
πP
268 /104.5
1039557.1
7.17.0
mNP
cS
cS
cSP
−×=×
=
=+=
AntennasThe fundamental mechanism for electromagnetic radiation is
an accelerating charge
Half-Wave Antenna
Electromagnetic Spectrum
For Next ClassReading Assignment
Chapter 35: Nature of Light and Laws of Geometric Optics
WebAssign: Assignment 12