Download Maxwell`s Equations

Chapter
34
Warmup 20
Ampere’s Law is Not Complete!
•Maxwell realized Ampere’s Law is not self-consistent
•This isn’t an experimental argument, but a theoretical one
•Consider a parallel plate capacitor getting charged by a wire
•Consider an Ampere surface between the plates B  ds  0 I1  0
•Consider an Ampere surface in front of plates
B  ds   0 I 2   0 I
•But they must give the same answer!


•There must be something else that creates
B-fields
B  ds  0  I  ? 
I
•Note that for the first surface, there is also
an electric field accumulating in capacitor
•Maybe electric fields?
•Take the time derivative of this formula
•Speculate : This replaces I for first surface

E
Q
0 A
 0 E  Q
I
0
d  E dQ
I

dt
dt
Ampere’s Law (New Recipe)
dE 

B

d
s


I


0
0


dt


 B  ds  0 I   0 0
dE
dt
Maxwell’s Equations
We now have four formulas that describe how to get electric and
magnetic fields from charges and currents
E  nˆ dA  qin  0
•Gauss’s Law
S
•Gauss’s Law for Magnetism
B  nˆ dA  0
•Ampere’s Law (final version)
S
•Faraday’s Law
dE
•Collectively, these are called
B  ds  0 I in  0 0
Maxwell’s Equations
dt



There is also a formula for forces on
charges
•Called Lorentz Force
F  q E  v  B
 E  ds  
dB
dt
All of electricity and magnetism
is somewhere on this page
Warmup 20
Wave solutions
•We can solve Maxwell’s Equations
(take my word for it) and come up with two
“simple” differential equations.
E  x, y, z , t   E0 sin  kx  t 
B  x, y, z , t   B 0 sin  kx  t 
2E
2E
 0 0 2
2
x
t
2B
2B
 0 0 2
2
x
t
with 0 0  1 2
c
•I could have used cosine instead, it makes no difference
•I chose arbitrarily to make it move in the x-direction
•These are waves where we have .   2 f   f  c
k
2 
http://www.ariel.ac.il/sites
/cezar/ariel/ANIMATION
S/physics3.html
Wave Equations summarized:
E  x, y, z , t   E0 sin  kx  t 
•Waves look like:
B  x, y, z , t   B 0 sin  kx  t 
  ck
•Related by:
•Two independent solutions to these equations:
E0  cB0
B0
E0
E0
B0
•Note E  B is in direction of motion
E y 0  cBz 0
Ez 0
or
 cE y 0
•Note that E, B, and
direction of travel are all
mutually perpendicular
•The two solutions are
called polarizations
•We describe polarization by telling which
way E-field points
Understanding Directions for Waves
E0  cB0
•The wave can go in any direction you want
•The electric field must be perpendicular to the wave direction
•The magnetic field is perpendicular to both of them
•Recall: E  B is in direction of motion
A wave has an electric field given by E = j E0 sin(kz – t).
What does the magnetic field look like?
A) B = i (E0/c) sin(kz - t) B) B = k (E0/c) sin(kz - t)
C) B = - i (E0/c) sin(kz - t) D) B = - k (E0/c) sin(kz - t)
•The magnitude of the wave is B0 = E0 / c
•The wave is traveling in the z-direction, because of sin(kz - t).
•The wave must be perpendicular to the E-field, so perpendicular to j
•The wave must be perpendicular to direction of motion, to k
•It must be in either +i direction or –i direction
•If in +i direction, then E  B would be in direction j  i = - k, wrong
•So it had better be in the –i direction
CT – 2 A planar electromagnetic wave is propagating through space. Its electric
field vector is given by E = Eo cos(kz – wt) î . Its magnetic field vector is
A. B = Bo cos(kz – wt) ˆj
B. B = Bo cos(ky – wt) k̂
C. B = Bo cos(ky – wt) î
D. B = Bo cos(kz – wt) k̂
E. None of the above
The meaning of c:
  ck
E  x, y, z , t   E0 sin  kx  t 
•Waves traveling at constant speed B  x, y, z , t   B 0 sin  kx  t 
•Keep track of where they vanish
kx  t  0
x

k
t  ct
•c is the velocity of these waves
c
1
 0 0
 2.99792458 108 m/s
c  3.00 108 m/s
•This is the speed of light
•Light is electromagnetic waves!
•But there are also many other types of EM waves
•The constant c is one of the most important fundamental constants of
the universe
Wavelength and wave number
•The quantity k is called the
wave number
•The wave repeats in time
•It also repeats in space f  1 T
k  2
E  E0 sin  kx  t 
B  B 0 sin  kx  t 
  2 f

  ck
•EM waves most commonly described
in terms of frequency or wavelength

c
 2 f
k
2

cf
•Some of these equations must be modified when inside a material
Ex- (Serway 34-19) In SI units, the electric field in an electromagnetic wave is
described by Ey = 100 sin(1.00 x 107x -t). (a) Calculate the amplitude of the
corresponding magnetic field. (b) Find the wavelength , (c) Find the frequency f.
Also find an expression for the magnetic field.
Solve on Board
Energy and the Poynting Vector
•Let’s find the energy density in the wave
E0  cB0
u E  12  0 E 2  12  0E02 sin 2  kx  t   12  0c2 B02 sin 2  kx  t 
1 0

B02 sin 2  kx  t 
2
B
2  0 0
2
0
u

sin
kx  t 
2
2

B0
B
0
uB 

sin 2  kx  t 
2 0
2 0
1
•Now let’s define the Poynting vector:
S
E  B
1
0
2
S
E0 B0 sin  kx  t   cu
0
•It is energy density times the speed at which the wave is moving
•It points in the direction energy is moving
•It represents the flow of energy in a particular direction
•Units:
J m
W
S
uc
m3 s
m2
Intensity and the Poynting vector
•The time-averaged Poynting vector is called the Intensity
•Power per unit area
2
S c u 
cB0
0
sin
2
S  cB02 20
 kx  t 
In Richard Williams’ lab, a laser can (briefly) produce
50 GW of power and be focused onto a region 1 m2
in area. How big are the electric and magnetic fields?
P 5.0 1010 W
22
2
S 


5.0

10
W/m
2
6
A
10 m 
7
22
2
2
4


10
T

m/A
5.0

10
W/m



2 0 S
2

B0 
3 108 m/s
c
 4.2 108 T 2
12
E

cB
0
0
E

6.1

10
V/m
B  20 kT
0
0
The Electromagnetic Spectrum
•Different types of waves are classified
by their frequency (or wavelength)
 Increasing
f Increasing
cf
Radio Waves
Microwaves
Infrared
Visible
Ultraviolet
X-rays
Gamma Rays
•Boundaries are arbitrary
and overlap
•Visible is 380-740 nm
Red Which of the
Vermillion
following
waves
Orange
hasSaffron
the highest
Yellow speed in vacuum?
Chartreuse
A)
Green Infrared
B)
Orange
Turquoise
Blue C) Green
Indigo
Violet D) Blue
E) It’s a tie
F) Not enough info
Sources of EM waves
+
•A charge at rest produces no EM waves
•There’s no magnetic field
•A charge moving at uniform velocity produces no EM waves
•Obvious if you were moving with the charge
•An accelerating charge produces electromagnetic waves
•Consider a current that changes suddenly
•Current stops – magnetic field diminishes
•Changing B-field produces E-field
•Changing E-field produces B-field
•You have a wave
–