PES 2130 Fall 2014, Spendier
Lecture 15/Page 1
Lecture today: Chapter 33 Electromagnetic Waves
1) Electromagnetic waves
2) Wave equation for EM waves
3) Speed of EM waves
4) Relationship between c, Emax and Bmax
5) EM spectrum
Announcements:
- Exam 2 in two weeks (Wednesday October 29)
Last time:
- Doppler Effect for sound waves
The general equation accounting for any motion is:
 v  vD 
f ' f
 v  vS 
- Doppler Effect for Electromagnetic waves
- Shock Waves (Supersonic Speeds):
Mechanical vs Electromagnetic Waves
Mechanical Waves
The existence of medium is essential for propagation. Energy and momentum propagates
by motion of particles of medium. But medium remains at previous position. The
Propagation is possible due to property of medium like elasticity and inertia. Examples:
vibration of string, vibration of string, the surface wave produced on the surface of solid
and liquid, sound waves, tsunami waves, earthquake P-waves, ultra sounds, vibrations in
gas, and oscillations in spring, internal water waves, and waves in slink etc.
Electromagnetic Waves
The existence of medium is not essential for propagation. The Periodic changes takes
place in electric and magnetic fields hence it is called Electromagnetic Wave. The waves
are defined as the disturbance through any medium of substance. The electromagnetic
waves are the waves which are generated by coupling of magnetic field with electric
field.
A mechanical wave is a wave that propagates as an oscillation of matter, and therefore
transfers energy through a medium (such as Air). Electromagnetic radiation (travelling as
waves): radiation consisting of waves of energy associated with electric and magnetic
fields resulting from the acceleration of an electric charge
PES 2130 Fall 2014, Spendier
Lecture 15/Page 2
1) Electromagnetic Waves
Light is an electromagnetic wave. Electromagnetic waves (EM Waves) are produced by
charged particles when they vibrate. As the charged particles execute SHM, a sinusoidal
electric field and a sinusoidal magnetic field are simultaneously produced. These two
fields are mutually perpendicular to each other and constitute an electromagnetic wave.
An e.m wave is able to propagate through vacuum without the presence of any medium.
The figure below shows an electromagnetic wave.
EM waves exhibit the following properties:
1. They consist of two sinusoidal fields – the Electric-field and Magnetic-field, which are
oscillating in phase and at right angles to each other.
E  x, t   Emax cos  kx  t  yˆ
[V/m]
B  x, t   Bmax cos  kx  t  zˆ
[T]
2. They are transverse waves. Since the Electric-field and Magnetic-field are
perpendicular to each other and to the direction of propagation, they are also called
"plane waves"
3. The direction of propagation for EM wave points in the direction of the cross product
For our example her, the EM waves travels in the positive x-direction:
4. All electromagnetic waves can travel through vacuum (or free space).
5. In vacuum (or air in approximation), they travel with the same speed
c = 3.00 x 108 ms-1.
6. All EM waves exhibit properties such as reflection, refraction, interference, diffraction
and polarization.
PES 2130 Fall 2014, Spendier
Lecture 15/Page 3
2) Wave equation for EM waves
Like waves on a string, EM waves must be a solution to some "wave" equation of
motion. One can show (after a lot of steps) the Maxwell's equations (Physics 2: Faraday's
law of induction, Maxwell's law of induction, Gauss' Law, Gauss' Law of magnetic
fields) for a EM wave propagating in the x-direction, in vacuum, give
 2 Bz  x, t 
x 2
 0 0
 2 E y  x, t 
 0 0
x 2
 2 Bz  x, t 
t 2
 2 E y  x, t 
t 2
μ0 = permeability of free space
ε0 = permittivity of free space
which is in the general form of a one-dimensional wave equation is given by
 2   x, t 
2
1    x, t 
v2
t 2

x 2
3) Speed of EM wave in vacuum:
v2  c2 
c
1
0  0
1
0  0
 2.998 108 m / s
Speed of EM wave in a material
material has dielectric constant κ (unitless)
v
1
0 0

c


c
n
where n =  = refractive index of a material
typically n  1
for vacuum n = 1.
for air n = 1.000293 (at 0oC and 1 atm)
PES 2130 Fall 2014, Spendier
Lecture 15/Page 4
4) Relationship between c, Emax and Bmax
Maxwell's equations also give us that for a wave propagating in the x-direction that the
electric and magnetic fields are coupled.
E y  x, t 
x

Bz  x, t 
t
using
E  x, t   Emax cos  kx  t  yˆ
B  x, t   Bmax cos  kx  t  zˆ
LHS:
E y  x, t 
 Emax

 cos  kx  t   kEmax   sin  kx  t  
x
x
RHS:
B  x, t 

 z
  Bmax  cos  kx  t        Bmax   sin  kx  t     Bmax sin  kx  t 
t
x
LHS = RHS gives
E y  x, t 

Bz  x, t 
x
t
kEmax sin  kx  t    Bmax sin  kx  t 
kEmax   Bmax
Emax 
 c
Bmax k
(in a vacuum)
Since c is so large, Emax >> Bmax
Example:
Emax = 5 V/m
hence
Bmax = Emax /c = 5 /(3 x 108) = 1.7 x 10-8T
PES 2130 Fall 2014, Spendier
Lecture 15/Page 5
5) The Electromagnetic Spectrum
c

k
 f
(in a vacuum)
higher frequency ==> higher energy
The scale is open-ended; the wavelengths of electromagnetic waves have no inherent
upper or lower bound.
There are no gaps in the electromagnetic spectrum—and all electromagnetic waves, no
matter where they lie in the spectrum, travel through free space (vacuum) with the same
speed c.
Note: In a material the frequency of light is unaltered which means that the wavelength is
altered

 f
k
c
v   f'
n
c 
'  f 
n n
c
So the speed and wavelength are reduced in a material compared to vacuum.
PES 2130 Fall 2014, Spendier
Lecture 15/Page 6
Example 1:
A carbon dioxide laser emits a sinusoidal wave that travels in vacuum in the negative xdirection. The wavelength is 10.6 µm and the E-field is parallel to the z-axis, with
maximum magnitude of 1.5 MV/m. Write vector equations for the E-field and B-field as
functions of time and position.