Electromagnetic Waves Hertz’s Discoveries Plane Electromagnetic Waves Energy in an EM Wave Momentum and Radiation Pressure The Electromagnetic Spectrum
Maxwell: Problem with Ampère’s Law
Id 0. sBAmpère’s Law
For S1, Id 0. sB
For S2, 0. 0 Id sB
(No current in the gap)
But there is a problem!
Consider a parallel plate capacitor
Contradiction! What can we do to fix
it? Ampere’s Law works
well – most of the time.
What is special about the region between the capacitors?
.o oQ AE
o o
QE
A
.o
dQ dI
dt dt
Let’s try to relate E to the current I.
Try fudging Ampere’s Law:
0 0 0. .Edd I
dt
B s
Solution - Displacement Current
Call I as the conduction current.
Specify a new current that exists when a change in the electric field occurs. Call it the displacement current.
dt
dI Ed
0
Electrical flux through S2:
00 Q
AA
QEAE
Idt
dQ
dt
dI Ed
0
Maxwell Gets Excited!
dt
dd B
sE.
dt
dId E 000. sB
Free Space: I = 0, Q = 0
dt
dd E
00. sB
t
E
x
Bt
B
x
E
00
2
2
002
2
002
2
t
E
x
E
t
E
tx
B
tt
B
xx
E
2
2
002
2
t
B
x
B
tkxBB
tkxEE
cos
cos
max
max
Speed:
1.
o o
v
Waves
Circular (or Spherical in 3D) Waves
Plane Waves
Ampère-Maxwell Law
Magnetic fields are produced by both conduction currents and by
time-varying electrical fields.
dt
dIIId E
d
0000. sB
Now the generalized form of Ampère’s Law, or Ampère-Maxwell Lax becomes:
Maxwell’s Equations
dt
dd B
sE.
dt
dId E 000. sB
0. AB d
0
.Q
d AE
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.
EM Wave Oscillation
Some Important Quantities
2
k Wavenumber
00
1
c Speed of Light
f 2 Angular Frequency
f
c Wavelength
ck
tkxBB
tkxEE
cos
cos
max
max
tkxBB
tkxEE
cos
cos
max
max
t
E
x
Bt
B
x
E
00
ckB
E
B
E
BkE
max
max
maxmax
Properties of EM Waves The solutions to Maxwell’s
equations in free space are wavelike
Electromagnetic 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) =?, T=?
sf
T
mf
c
86
6
8
105.21040
11
5.71040
103
b) If Emax=750N/C, then Bmax=?
Tc
EB 6
8max
max 105.2103
750
Directed towards the z-direction
c) E(t)=? and B(t)=?
)1051.2838.0cos(105.2cos
)1051.2838.0cos(/750cos86
max
8max
txTtkxBB
txCNtkxEE
sradf
mradk
/1051.2104022
/838.05.7
22
86
Energy Carried by EM Waves
BES 0
1
Poynting Vector (W/m2)
For a plane EM Wave: 2
00
2
0
Bc
c
EEBS
2max
00
2max
0
maxmax
222B
c
c
EBESI av
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
2B
uB 2
20EuE
20
2
0
00
0
2
2
1
22EE
cEuB
0
22
0 22
1
BEuu BE
For an EM wave, the instantaneous electric and magnetic energies are equal.
0
22
0 BEuuu BE
0
2max2
max02
0 22
1
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
c
E
AI av
0
2max
2
P
P lamp= 150 W, 3% efficiencyA = 15 m2
mVA
cE av /42.18
2 0max
P
Wlampav 5.403.0 PP
Tc
EB 8maxmax 1014.6
A
Momentum and Radiation Pressure
c
Up Momentum for complete absorption
dt
dp
AA
FP
1
A
dtdU
cc
U
dt
d
AP
11
c
SP
Pressure on surface
For complete reflection:c
Up
2
c
SP
2
Pressure From a Laser Pointer
P laser= 3 mW, 70% reflection, d = 2 mm
22
/955)001.0(
003.0mW
AS
P
268
/104.5103
9557.1
7.17.0
mNP
c
S
c
S
c
SP
Review of key points: Partial derivatives Relationship between E and B Poynting vector Omit section 34.4. Section 34.5: only responsible for ideas
presented next.
tkxBB
tkxEE
cos
cos
max
max
t
E
x
Bt
B
x
E
00
ckB
E
B
E
BkE
max
max
maxmax
Properties of EM Waves The solutions to Maxwell’s
equations in free space are wavelike
Electromagnetic 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.
Energy Carried by EM Waves
BES 0
1
Poynting Vector (W/m2)
For a plane EM Wave: 2
00
2
0
Bc
c
EEBS
2max
00
2max
0
maxmax
222B
c
c
EBESI av
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)
Antennas
The fundamental mechanism for electromagnetic radiation is an accelerating charge
Half-Wave Antenna
AM radio antenna
FM and TV broadcast antennas?
Problem: Very roughly, what is the frequency of cell
phones? Are CB radio frequencies higher or lower?
Dorm room FM Antenna http://www.wfu.edu/~matthews/misc/
dipole.html
Electromagnetic Spectrum
