winter wk 9 – mon.28.feb.05 energy systems, ejz. maxwell equations in vacuum faraday: electric...
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Winter wk 9 – Mon.28.Feb.05
Energy Systems, EJZ
Gauss' Law (Electric)
Charges make E fields
Gauss' Law (Magnetic)
No magnetic monopoles
Ampere's Law
Currents make B fields (so does changing E)
Faraday's Law
Changing B make E fields
B dAd
E dsdt
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0 0 0E
enc
dB ds i
dt
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0
E dA encE
q
B dA 0B
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Maxwell Equations in vacuum
Faraday: Electric fields circulate around changing B fields
Ampere: Magnetic fields circulate around changing E fields
0 0Ed
B dsdt
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B dAd
E dsdt
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Faraday’s law in differential form
( ) _____
____
B
E ds E dE y E y
dE ds
dtdBdx y dE y
dtdB
dt
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Ampere’s law in differential form
0 0
0 0
0 0
( ) _____
____
E
B ds B z B dB z
dB ds
dtdEdx z dB z
dtdE
dt
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Maxwell’s eqns for postulated EM wave
dx
dB
dt
dE00
dx
dE
dt
dB
0 0Ed
B dsdt
����������������������������B dA
dE ds
dt
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Do wave solutions fit these equations?
Consider waves traveling in the x direction with frequency f=
and wavelength = /k
E(x,t)=E0 sin (kx-t) and
B(x,t)=B0 sin (kx-t)
Do these solve Faraday and Ampere’s laws?
Under what condition?
Differentiate E and B for Faraday
Sub in: E=E0 sin (kx-t) and B=B0 sin (kx-t)
dx
dE
dt
dB
Differentiate E and B for Ampere
dx
dB
dt
dE00
Sub in: E=E0 sin (kx-t) and B=B0 sin (kx-t)
Maxwell’s eqns in algebraic form
0 0B kE 0000 BkE
dx
dB
dt
dE00
dx
dE
dt
dB
Subbed in E=E0 sin (kx-t) and B=B0 sin (kx-t)
Recall that speed v = /k. Solve each equation for B0/E0
Speed of Maxwellian waves?Ampere
B0/E0 = 0 vFaraday
B0/E0 = 1/v
Eliminate B0/E0 and solve for v:
0=xm = xC2 N/m2
Maxwell equations Light
Energy of EM wavesElectromagnetic waves in vacuum have speed c and
energy/volume =
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 =
0
1B
powerS E
area
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2 20
02 2E
E Bu