Faraday’s and Maxwell-Ampere Laws
• A changing magnetic flux produces a curly
electric field:

 
d 
Edl  
Bnˆ dA
dt

• A changing electric flux produces a (curly)
magnetic field:

 
d 


Bdl 0  I encl   0
Enˆ dA 
dt



A moving “slab” of E and B fields
• Thin region in which there are uniform electric and
magnetic fields.
–
–
–
–
No charges or currents in the vicinity of the slab.
Outside the slab, both fields are zero.
Inside the slab, E and B are perpendicular to each other.
Slab moves in a direction perpendicular to both fields.
E
E
B
v
v
E
B
front view
v
B
top view
Does Slab obey Faraday’s Law?
E
E
B
v
v
B
front view

 
d 
Edl  
Bnˆ dA
dt

Does Slab obey Maxwell-Ampere Law?
top view
E
B
v
E
v
B

 
d 


Bdl 0  I encl   0
Enˆ dA 
dt



Constraints imposed by Faraday’s and
Maxwell-Ampere Laws
E  Bv
B  0 0 Ev
c
 0  8.85 10
12
C2
Nm2
0  4 10
c  3.00  108
m
s
7
Ns 2
C2
1
 0 0
Properties of EM Waves
•
•
•
•
1
Move at the speed of light (in vacuum): c 
 0 0
Magnitudes of fields related by E  Bc.


E and B are perpendicular to each other.
 
Wave moves in direction of E  B , i.e.
perpendicular to both fields.
E
B
v
Dipole Antenna
• To create
electromagnetic
waves, charges
must be
accelerating.
• Recall that in the
steady state, the
fields are constant.
Figure 31.6b
t=0
Later time
Figure 31.7
Plane waves
• Far from the antenna, the EM waves are well
described as plane waves – the fields are
uniform within planes perpendicular to the
direction of propagation.
• Sinusoidal waves – fields are sinusoidal
functions of position and time.
– E.g. wave moving along x axis:

2 x 2 t ˆ
E  E0 sin(

) j  E0 sin(kx  t )ˆj

T

2 x 2 t ˆ
B  B0 sin(

)k  B0 sin(kx  t )kˆ

T
Figure 31.9

2 x 2 t ˆ
E  E0 sin(

) j  E0 sin(kx  t )ˆj

T

2 x 2 t ˆ
B  B0 sin(

)k  B0 sin(kx  t )kˆ

T
c

T
f
Energy density in EM Waves
Electric Field Energy density
uE  12  0 E 2
Magnetic Field Energy density
uB 
1
2
B2
0
E  Bc
c
1
 0 0
Half of the energy in an EM wave is in the electric field and
half is in the magnetic field.