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Physics 203/204
1: Electromagnetic Waves
•Maxwells Equations
•Integral form
•Differential form
•Solutions to Maxwells equation
•Transport of Energy in em waves
•Antennas.
Maxwells Equations
•Gauss’ Law
•For Electric
Fields:

Q
E  dA    E
0

BdA  0B
surface enclosing electric charge
•Gauss’ Law
•For
Magnetic
Fields:
surface enclosing magnetic cha
Ampere Maxwell Law

dE
Bdsm0I+0m0
dt
path enclosing current I
and changing electric field
Faradays Law

Eds

dB
dt
path enclosing changing magnetic flux
Changing
Magnetic
Flux
Produces
Induced
Electric
Field
Changing
Electric
Flux
Produces
Induced
Magnetic
Field
Changing Electric Field
Fluctuating
electric and
magnetic fields
electromagnetic
Radiation
Changing Magnetic Field
1
c
 0m 0
Differential
Form of
Maxwells
equations
Maxwells
Equations
 r 
 E 

 B  0
B
E 
t
E 

B m J+ 

t 
Integral to Differential

E dA 

B dA 
closed surface
closed surface

r 
dV

enclosed volume
0


B

 dA
t
E ds 
closed path
Area enclosed by path




as  B  B dA 



B ds 
closed path
  E 
m J+ 
 dA

t  
area enclosed by path



as I  J dA
 
 
Vector
Calculus
Gauss Divergence Theorem


F  dA 
closed surface
   F dV
volume enclosed by surface

 
 
     

    i +   j +   k 
  x  
  z    
  y  


closed path
Stokes
F  ds 
Theorem

  F   d A
area enclosed by path


EdA
Eds 
closed path
area enclosed by path


B
dA
t
Area enclosed by path
B
 E
t

Bds
closed path

BdA
area enclosed by path

  E 
m J+ dA
  t  
area enclosed by path
  E 
 Bm J+
  t 


EdA 
closed surface
EdA 
enclosed volume

r
dV

enclosed volume
r
  E 


B dA 
closed surface

 BdA 0
enclosed volume
  B 0
Maxwells Equations in empty space
:
 = 0; J  0
gives
2E 
1

2E 
c2 t2  
Wave Equation for light


2B
1

2B  2 2  
c t  
where the Laplacian
2
2
2



2   2 + 2 + 2
x y z
These fields can be expanded as
Ex, y, z,t   Ex x, y, z,t i + Ey x, y, z,t  j + Ez x, y, z,t k
Bx, y, z,t   B x  x, y, z,t i + B y x, y, z,t j + B z  x, y, z,t k
and we get general linear wave equations
for the components,
e.g.
 2E x  2E x  2E x 1  2E x
+ 2 + 2  2 2
2
x
y
z
c t
The solution of these equations
MUSTalso satisfy Maxwells equations.
The simplest solution has the form
Er, t   E x, t ; The field has the same,
constant,value for all y, z at any time t and
propagates in the positive- x direction.
Then
E x, t   Emax sinkx  wt 
B x, t   Bmax sinkx  wt 
E2  c 2B2 ;
2p
wave number k  ; angular frequency
w = 2pn
l
w
ln = c   c
k
One set of solutions to Maxwells Equations are
Er, t   E y max j sin kx  wt 
Br, t   Bz max k sin kx  wt 
Which are PLANE waves propagating in the xdirection
z
y
E
x
B
•Waves are made up of propagating
vibrations in some medium. In the case
of em waves:
•the medium is NON MATERIAL i.e. does
not have MASS
•the medium is electric and magnetic
fields
•the vibrations are fluctuations of the
electric and magnetic fields
•the medium creates itself as
•changing magnetic fields produce
•changing electric fields etc....
•Energy is propagated through the
medium
•Energy Density in :
•Electric Field
1 2
uE  0 E
2
•Magnetic Field
1 2
uB  B
2m0
 0 2  0 2 2  0c 2
uE 
E 
c B 
2m 0 uB
2
2
2
1
2
as uB 
B
2m0
1
 uE   0 c m 0 uB   0
m 0 uB  uB
 0m 0
2
thus total energy density is
2
B
u  uE + uB   0 E 
m0
2
In order to describe the energy propagated we need to
consider the transport of energy per unit time across
a unit area.
energy per unit time is power
During a very small time interval
D t,
A ( c D t),
the energy contained in the volume
will cross the area
A , this energy is
 A ( c D t) u
given
Thus the energy and per unit area per unit time
is by
S 
A ( c D t) u
EB
= uc 
m0
ADt
energy flow is in direction of propagation of wave,
the vector describing this is called the
Poynting vector
S 
1
E B
m
The average power transported over
each cycle for a sinusodial plane em
wave is called the WAVE INTENSITY,
denoted by
I.
2
2
Emax Bmax Emax
c Bmax
I  Sav =
=
=
= cuav
2m 0
2m 0 c 2m 0
The SI units of S are
S = W m 2
Electromagnetic waves transport linear
momentum as well as energy, therefore
produce pressure
*If the radiation makes a perfect inelastic collision
with a surface (i.e. all the radiation is absorbed
- -a black
body surface) then all the momentum of the radiation
to surface)
is transferred to the surface,
(consider component normal
then momentum delivered to the surface
U
Dt
DP = ; where U is the total energy transported during time
c
*If all the radiation is reflected the total momentum
2U
DP =
Change of surface during this time will be
c
Rate of momentum transfer
= Force

dP  
 Newtons


2nd Law
F=

dt  
F
 p , pressure
Force per unit area
A
dP
\ radiation pressure
p = dt A
1 dU S
* for perfect absorber

p
=
Ac dt
c
1 2dU 2S
* for perfect reflector
p
=
Ac dt
c
for sunlight
p  5  10 6 N
m
2

Accelerating charge produces
em waves
e.g. oscillating charge
varying current
produces varying
magnetic field
varying position of
charge produces
varying electric
field
radiation “far” from “dipole antenna”