Faraday’s and Maxwell-Ampere Laws
• A changing magnetic flux produces a curly
electric field:
d
Edl
Bnˆ dA
dt
• A changing electric flux produces a (curly)
magnetic field:
d
Bdl 0 I encl 0
Enˆ 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
Edl
Bnˆ dA
dt
Does Slab obey Maxwell-Ampere Law?
top view
E
B
v
E
v
B
d
Bdl 0 I encl 0
Enˆ 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.
Faraday’s and Maxwell-Ampere Laws
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