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Transcript
Discussion Notes and Class Agenda
Physics 240
December 5, 2005
Problem Assignments
Problem
Groups
32.52
1, 3
32.17
5, 7
32.10
2, 4
32.13
6, 8
Special Event
The U of M Society of Physics Students Presents:
The Golden Age of Ballooning: Particle
Astrophysics Aboard High Altitude
Balloons
Dr. Andrew Tomasch
Tuesday December 6, 2005
7:00 pm in the Don Meyer Commons
(Next to 335 West Hall)
Refreshments will be served
Class Agenda
Physics 240
December 7, 2005
Problem Assignments
Problem
Groups
32.7
1, 3
32.15
5, 7
32.51
2, 4
32.41
6, 8
Discussion: Electromagnetic Waves
Motivation: Displacement Current and Maxwell’s Equations
It was known in Maxwell’s time that
1
 0 0
 c  speed of light in vacuum=3  108 m/s
Maxwell first realized that formulation of Ampere’s Law was
incomplete and required the addition of a term proportional to the time
derivative of the electric field called the displacement current. With the
displacement current included in Ampere’s Law, it is possible for a time
varying electric field to induce a magnetic field in a manner analogous
to Faraday Induction, where a changing magnetic field induces an
electric field. This process is now called Maxwell Induction. Once the
displacement current is included, the two induction equations become
symmetric and imply that electric and magnetic fields can propagate in
the absence of charges and currents. Once the displacement current had
been included in the equations governing electromagnetism, Maxwell
was able to derive a traveling wave equation from Faraday’s Law and
2
Ampere’s Law in the absence of charges or currents due to moving
charges which are shown below
 E  dl  
d
d
B

dA


B
dt 
dt
d
1 d
 d

E

dA

E

dA




0 0
E   0 I Disp
dt 
c 2 dt 
dt


 B dl   0 0
 E  dl  
d
 B (Faraday)
dt
 B dl 
1 d
 E (Maxwell)
c 2 dt
With the displacement current I Disp included in Maxwell’s equations,
the combination
 0 0 
1
c2
naturally appears in the equation for
Maxwell induction. We have studied Maxwell’s equation in integral
form. To derive a traveling wave equation, Maxwell actually worked
with the equations in differential form. While we will not study this
formulation in this course, it is still informative to display Maxwell’s
equations in differential form
 E 
E  

B
t

0
 B  0

E 
  B  0  J   0
t 

J the current density. We have
already mentioned the divergence operation (   ), for example (   E ),
where
is the charge density and
which measures how the flux of a vector field flows into or out of a
3
region of space. A region with a nonzero divergence contains a source
or sink for the field (charge in the case of electrostatic fields). The curl
operation (   ), for example (   B ), similarly measures the
circulation of a vector field--its tendency to form closed loops. If you
think of the velocity field for water flowing in a river, a region with a
nonzero divergence contains either a spring (source) of water or a drain
(sink). A region of a flowing river with a nonzero curl will cause a
paddle wheel to turn if placed in the flow. Whirlpools have a nonzero
curl. It is very easy to obtain the integral form of Maxwell’s equations
from the differential form by integrating both sides of each equation
over a volume (Gauss’ Law for E and B ) or over a surface (Faraday
induction and Ampere’s Law with displacement current) and then
applying two theorems from vector calculus, the Divergence Theorem
and Stokes’ Theorem respectively. You will soon learn this approach in
your study of vector calculus. The point to emphasize is that this is very
straightforward to learn and offers powerful new techniques and
insights for the further study of Maxwell’s equation and
electromagnetism.
Electromagnetic Waves
Having stated that a wave equation can be derived from Maxwell’s
equations and that solutions to this equation propagate in vacuum at the
speed of light, we will now study the properties of these waves. The
discussion thus far has been restricted to waves propagating in vacuum.
We will also include the correct prescription to describe electromagnetic
waves traveling in matter.
Section 32.2 of Young and Freedman demonstrates the derivation of the
wave equation from the integral formulation of Maxwell’s equations.
The derivation involves taking limits where the paths over which the
loop integrals for Faraday’s Law and Ampere’s Law approach zero
size. This amounts to employing the differential form of Maxwell’s
equations. The waves are assumed to be plane polarized, traveling in
the +x direction. The electric field oscillates in the +y direction and the
magnetic field in the +z direction. For this set of assumptions, the
resulting wave equations for E and B are
4
2 E y
2 E y
2
1  Ey
  0 0
 2
x 2
t 2
c t 2
 2 Bz
 2 Bz 1  2 Bz
  0 0 2  2
x 2
t
c t 2
which have solutions
E  x, t   Emax cos(kx  t ) yˆ
B  x, t   Bmax cos(kx  t ) zˆ
as illustrated below in Figure 32.10
This solution describes transverse traveling wave with linear
polarization in the y direction, that is, the electric field is always aligned
in the y direction. Other polarization states are possible, for example z
polarization, with the electric field oriented in the z direction and the
magnetic field in the –y direction. In fact, any general electromagnetic
wave can be described as a superposition of these two linear polarization
states. The quantity k is called the wave number and has value 2/
where  is the wavelength. The wave number can also be expressed as a
5
vector, called the wave vector, which is oriented in the direction of
propagation. For the y-polarized wave
k
2 ( Eˆ  Bˆ )
2



xˆ
The electric and magnetic fields are related to each other as
E  cB B   0 o cE
The assigned problems demand complete familiarity with the
relationships between frequency, period, wavelength, angular
frequency, speed of propagation and wave number for traveling
waves. For electromagnetic waves propagating in vacuum
c  f   (2 f )
2
  2 f 
 

2 k
k

2

1
c
 0 0
For waves propagating in matter, the speed of light in vacuum must be
replaced with the propagation speed in matter v
v  f   (2 f )
  2 f 
v
1


2
 

2 k
k

2
1
KK M  0 0
n
c
v
6


c
c

KK M n
where K and KM are the relative permiativity and permeability of the
material. The maximum electric and magnetic field are then related by
E  vB B  vB
The quantity n is called the refractive index of the material. The fact
that electromagnetic waves travel with a lower speed in matter than in
vacuum as characterized by the refractive index gives rise to many of
the effects in the science of optics. It is the refractive index that enables
lenses to focus light and produce images.
Energy and Momentum for Electromagnetic Waves
The Poynting vector describes the instantaneous power per unit area
transmitted by a traveling electromagnetic wave (energy/area-time)
S
1
0
EB
S 
EB
0
The total energy flow per time exiting a closed surface is therefore
P

S  dA
Closed
Surface
Thus the flux of the Poynting vector out of a closed surface yields the
total power (energy/time) exiting the surface in the form of
electromagnetic radiation.
The Poynting vector is time dependant, and we must substitute the time
varying forms of the electric and magnetic fields to evaluate it at any
instant in time. The time average of the Poynting vector is called the
intensity and is given in vacuum by
7
2
Emax Bmax Emax
1 0 2
1
2
I  S  Sav 


Emax   0 cEmax
2 0
2 0 c 2 0
2
To describe waves propagating in matter, the quantities
replaced by  ,  ,v
 0 , 0 ,c are
2
Emax Bmax Emax
1  2
1
2
I


Emax   vEmax
2
2v 2 
2
1
1 K  0c 2
1  0c 2
2
I   vEmax

Emax 
Emax
2
2 KK M
2 KM
2
2
Emax
KK M  0 0 1 K
Emax
I


2v
2 K M 0
2 KM
2
2
Emax
KK M
Emax
I


2v
2 K M 0 c
0 2
Emax
0
2
K Emax
K M 2 0 c
Finally, electromagnetic waves carry momentum as well as energy. The
momentum density (momentum per unit volume) is
dp
EB
S


dV 0 c 2 c 2
The momentum transferred per unit area per unit time is
8
1 dp S EB
 
A dt c 0 c
The average rate of momentum transfer per area is obtained by
substituting the intensity (average value of S)
S
1 dp
I
 av 
A dt
c
c
Since momentum transfer per time is equivalent to a force, the traveling
electromagnetic wave will exert a radiation pressure
(force/area=momentum transfer/time-area) on a surface if it is absorbed
or reflected on a surface. For a surface perpendicular to the
propagation direction
Sav I

 total absorbtion 
c
c
2S
2I
 av 
 total reflection 
c
c
prad 
prad
The factor of two difference between reflection and absorption is easy to
see. Recall from mechanics that in a perfectly inelastic collision a
particle with momentum p transfers its total momentum to the target.
If however the collision is perfectly elastic, the projectile recoils from the
target with momentum –p and the momentum transferred to the target
is -(pf - pi)=2p and twice the momentum is transferred in the collision.
Total absorption is therefore equivalent to a perfectly inelastic collision,
and the total reflection is equivalent to a perfectly elastic collision.
9