Transcript EM waves

Winter wk 9 – Mon.28.Feb.05
Gauss' Law (Electric) Gauss' Law (Magnetic)
E 
 E  dA 
qenc
0
B 
 B  dA  0
Charges make E fields
No magnetic monopoles
Ampere's Law
Faraday's Law
dE
 B  ds  0 0 dt  0ienc
Currents make B fields
(so does changing E)
d
B  dA    E  ds

dt
Changing B make E fields
Energy Systems, EJZ
Maxwell Equations in vacuum
Faraday: Electric fields circulate around
changing B fields
d
  E  ds   B  dA
dt
Ampere: Magnetic fields circulate around
changing E fields
dE
B

ds



0
0

dt
Faraday’s law in differential form
 E  ds  ( E  dE )y  Ey  _____
dB

  E  ds
dt
dB

dxy  dE y
dt
dB
 ____
dt
Ampere’s law in differential form
 B  ds  Bz  ( B  dB)z  _____
dE
 0 0
  B  ds
dt
dE
 0 0
dxz  dBz
dt
dE
 0 0
 ____
dt
Maxwell’s eqns for postulated EM wave
d
B  dA    E  ds

dt
dB
dE

dt
dx
dE
 0 0
  B  ds
dt
dE
dB
 0 0

dt
dx
Do wave solutions fit these equations?
Consider waves traveling in the x direction
with frequency f= w/2p
and wavelength l= 2p/k
E(x,t)=E0 sin (kx-wt) and
B(x,t)=B0 sin (kx-wt)
Do these solve Faraday and Ampere’s laws?
Under what condition?
Differentiate E and B for Faraday
dB
dE

dt
dx
Sub in: E=E0 sin (kx-wt) and B=B0 sin (kx-wt)
Differentiate E and B for Ampere
dE
dB
 0 0

dt
dx
Sub in: E=E0 sin (kx-wt) and B=B0 sin (kx-wt)
Maxwell’s eqns in algebraic form
dB
dE

dt
dx
dE
dB
 0 0

dt
dx
Subbed in E=E0 sin (kx-wt) and B=B0 sin (kx-wt)
w B0   kE0
0 0w E0  k B0
Recall that speed v = w/k. Solve each equation for B0/E0
Speed of Maxwellian waves?
Faraday
B0/E0 = 1/v
Ampere
B0/E0 = 0 0 v
Eliminate B0/E0 and solve for v:
0 = 4 p x 107 Tm/A
0 = 8.85 x 1012 C2 N/m2
Maxwell equations  Light
Energy of EM waves
Electromagnetic waves in vacuum have speed c and
energy/volume =
0 E 2 B2
uE 

2
20
E and B vectors point (are polarized) perpendicular to
the direction the wave travels.
EM energy travels in the direction of the EM wave.
Poynting vector =
power
1
 S  EB
area
0