Plane Wave Equations

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Plane Wave Equations

• Alan Murray

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Maxwell's Equations Completed!
Gauss(D)
Gauss (B)
Ampere
Faraday
Displacement current (L)
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What does this mean?
Faraday
a changing magnetic field causes an electric
field
Ampere
a changing electric field/flux causes an
magnetic field
Question If we put these together, can we get
electric and magnetic fields that, once created,
sustain one another?
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Cross-breed Ampere and Faraday!
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Cross-breed Ampere and Faraday!
Same equation as acquired for E
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Now some simplifications
EY EY0sin(?t-ßx)
E (0,EY,0) only
Align y-axis with electric field and the x-axis
with the direction of (wave) propagation.
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Travelling Waves
Take a time-varying electric field, E, at a point

EY EY0sin(?t)
Add a second one with a smallphase difference,
nearby
Now lets have a lot of them,with a sinusoidal
variationof phase with direction x.
EY EY0sin(?t-ßx)
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Plane Wave
We will also look for a plane wave solution
where the field EY is the same (at an instant in
time) across the entire zy plane.
Here is an animation to see what this means -
looking at the yz plane, down the direction of
travel
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Cross-breed Ampere and Faraday!
And, as we have simplified down to E(0,Ey,0),
with EY constant in the zy plane, this reduces
to
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Cross-breed Ampere and Faraday!
(in 3D)
Becomes the 1D equation

• Plane wave equation for E
• describes the variation in time and space of an
  electric plane wave
• With a y-component only (we have aligned the
  y-axis with E)
• propagating in the x-direction.
• There is an exactly equivalent equation for H
• Eliminate E, not H, from the combination of
  Ampere and Faraday.
• rather a waste of our time in notes, but not
  lectured.
• We can, however, infer that whatever behaviour we
  get for Ey will apply to H, although we do not
  yet know the direction of H.
• Watch this space

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What have we here?
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Start with an insulatorto make life easy (s0)
becomes
Where w and b depend upon m and e the
characteristics of the insulator
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Still dont know what it means

• Travelling wave of the form

In a vacuum, mm04px10-7, ee08.85x10-12
In (eg) glass, mm04px10-7, eere05x8.85x10-12
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This is why lenses work
V3x108m/s
V1.43x108m/s
V3x108m/s
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What is H up to?
Lookie here
And here
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Summary so far Insulator

• H and E both obey ej(wt-bx)
• H and E are in time-phase
• EZiH, Zi is the characteristic impedance
• Zi is real in an insulator
• Zi 377O in free space (air!)
• Zi 150O in glass
• Wave travels at a velocity vvme
• 3x108 m/s in free space

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Now a conductor

• Fields lead to currents
• Currents cause Joule heating (I2R)
• Leads to loss of energy
• Fields still oscillate, but they decay
• Multiply the solution we have already by a term
  e-ax?

HEAT!
HEAT!
HEAT!
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Now a conductor sgt0
X
X
X
X
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Example Good Conductor
s f w e m a b v
6x107 (S/m) 100MHz 6.28x108 8.85x10-12 1.26x10-6 1.54x105 1.54x105 4x103m/s

• Comments
• vltlt speed of light
• a 1.54x105 gtgt1 rapid attenuation via e-ax
• Lets have a look at e-ax

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Example Good Conductor
Amplitude falls by 0.361/e in 6mm i.e. the wave
doesnt get far in copper!
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Example Good Conductor,EZiH . Intrinsic
Impedance
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Example Good Conductor,EZi H. Intrinsic
Impedance
It looks like this
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Poynting Vector (introduction only)

• PExH is called the Poynting Vector
• direction of travel
• power
• Actually power/area

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Reflection at a Boundary
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Reflection at a Boundary