Physics

Electromagnetic Waves
The Lorentz Force

F = qE + qv  B
Basic Equations Of Electromagnetism

1. Gauss' law of electricity

q
 E  dA = 0 dS
Basic Equations Of Electromagnetism

2. Gauss' law of magnetism

 B  dA = 0
Basic Equations Of Electromagnetism

3. Faraday's law of induction

dB
 E  ds = −
dt
dS
 
B =  B  dS B B
Basic Equations Of Electromagnetism

4. Ampere's law

 B  ds =  0 I
 Maxwell’s Displacement Current
If the surface is chosen as 1, 2 or 4, the
enclosed current = I
circuit
 Maxwell’s Displacement Current
• If the surface 3 is chosen, enclosed current = 0
( there is no current between capacitor pates )
circuit
James Clerk Maxwell realized around
1865 that the equations of electricity &
magnetism were inconsistent.

 B  ds   B  ds 
 (1,2,4)  
 3
Maxwell generalized Ampere's law,

 B  ds =  ( I + I )
0 d

dE Maxwell’s
Id =  0
dt displacement current
dQ d
As I = = (  0 EA ) circuit

dt dt
d ( EA)
= 0
dt
dE
= 0  ID
dt
 Maxwell’s Equations
  Q   d B
 E  dS =  E  d = −
0 dt
     dE 
 B  dS = 0  B  d  = 0  I +  0 
 dt 

Lorentz Force: (
F = q E + v B )
wavelength

Amplitude

wavelength
Node
Example: Red light has l = 700 nm.
Calculate the frequency, n.

8
c 3.00 x 10 m/s 14
n= = = 4.29 x 10 Hz
l -7
7.00 x 10 m
• No medium is required

• Speed of propagation is c for all

waves

• No limit to amplitude

• No limit to frequency
 Microwave Ovens

Typical wavelength l  6cm

 Formation of Electric and
Magnetic Fields around an
Antenna
E-line

Mag field
e- e- e-

Electric field _|_ Magnetic Field _|_ Direction of

Propagation.
Traveling Electromagnetic Fields

consider a long wire

at t = 0 turn on
y current in – y direction

x
produces magnetic field close to wire
B B
 I • view from top

 •
I
 • 
B
 •
 •  B  ds =  I
0

 • 0 I
B=
2 r
after some time

EBE B B EB E
  I • •
  • •
  • •
  • •
  • •
  • •
vt
Ey y

x
vt
By
y

v
x
vt
 Watch the electric field
ANIMATE!!
+ −
+ + − −
+ −
E
E
− +
− − + +
− +
 If charges oscillate back and forth, you
get time varying magnetic fields too !
ANIMATE
+ −
−
+ + − −
+

+ +
+
−
− −
− +
 A chain of events in free space

→ a changing B field causes a changing E field

dB
 E  ds = − dt self-sustaining

→ a changing E field causes a changing B field

dE
 B  ds = 0 0 dt
 x

y
 Plane Harmonic Wave
Ex = E0 sin( kz − t )  = kc
By = B0 sin( kz − t )
where:
E0 = cB0

Note: the direction of propagation ŝis given by the cross product

sˆ = eˆ  bˆ
( )
where eˆ, bˆ are the unit vectors in the (E,B) directions.
The total electric flux
and total magnetic flux
through the surface are
both zero.
Field directions in an
electromagnetic
wave
• By is in phase with Ex
• B0 = E0 / c
The power radiated by a single
charge q with acceleration a is:
2 2 2
P= 3q a
3c
Production of Electromagnetic waves
An electromagnetic wave can be generated by
oscillating the electric field along an antenna.
 +
V ( t ) = V0 cos t +

−
−
After some time

−
−

+
+
Radiation from oscillating dipole

 

Wave propagates outward at speed of light

Electric field lines from oscillating dipole
A traveling electromagnetic wave produced by
an ac generator attached to an antenna.

ANIMATE THIS
Loop:Alternating magnetic field produces an AC
current in the circuit.
Dipole: Alternating electric field produces an AC
current in the circuit.
Far Away
Radiation Pattern:

sin 2 
I ( )  2
r
 Dipole radiation pattern
proportional to sin(t)

• Oscillating electric dipole generates e-m radiation that

is linearly polarized in the direction of the dipole
• Radiation pattern is doughnut shaped & outward
traveling
– zero amplitude above and below dipole
– maximum amplitude in-plane
 Amplitude Modulation (AM)
• Amplitude Modulation (AM) uses changes
in the signal strength to convey information

Carrier

Signal

pressure modulation (sound)

 Output

Electromagnetic Wave Modulation

 Frequency Modulation
• Frequency Modulation (FM) uses changes in
the wave’s frequency to convey information

Carrier

Signal

pressure modulation (sound)

 Output

Electromagnetic Wave Modulation

Receiving radio waves
 Converting back to sound: AM
• AM is easy: just pass the AC signal from the antenna into a diode
– or better yet, a diode bridge
– then use capacitor to smooth out bumps
• but not so much as to smooth out audio bumps

B
radio signal

amplifier/
D
speaker
POLARIZED LIGHT
 Unpolarized Light
Most sources – a candle, the sun, any light bulb –
produce light that is unpolarized.

• No definite direction of the electric field

• No definite phase between orthogonal
components
• The atomic or molecular dipoles that emit
the light are randomly oriented in the
source
 Polarization
A typical plane electromagnetic wave:

Ex = E0 sin(kz − t ) Ey = 0 Ez = 0
Bx = 0 E0 Bz = 0
By = sin(kz − t )
c
This wave is an example of a linearly polarized wave.

The direction of polarization is the direction of the

oscillation of the electric field vector, in this case x.
 Polarization
A wave traveling in the z direction can be polarized at an
arbitrary angle  in the x-y plane.

y
Ex = E0 cos   sin(kz − t ) ê

E y = E0 sin   sin(kz − t ) 
x
Ez = 0
 Linear Polarization
How to produce linearly polarized e-m waves?
–Absorption/reflection of vector component of wave
⊥ to polarizer.
TA
(transmission axis)
LP
microwave
source slotted
metal
(polarized) plate

The E-field component parallel to the slots is absorbed and/or reflected.

The E-field component perpendicular to the slots is transmitted.

 Linear Polarization

polaroid sunglasses:
Long molecules absorb E-field TA
parallel to molecule. (transmission axis)

Absorption produces linearly polarized waves but also

reduces intensity.