Physics II

1
Physics II

• Scattering, rectilinear propagation, and the Law
  of Reflection

2
In a vacuum

• Transmission of e-m radiation through a vacuum.
• S E x B / ?o
• Right hand rule for TEM
• Rectilinear propagation in a vacuum.

3
Reminder

• Phase relationships
• Part of vibration cycle
• Incoherent radiation
• No fixed (in time) relationship of phases
• Intensities add
• Coherent radiation
• Fixed relationship in phases
• Electric fields add
• Irradiance proportional to square of electric
  field
• Constructive Interference
• Destructive Interference

4
Small particles in a tenuous gas

• Ground state scattering
• Positive and negative charge centers form an
  oscillating dipole of , - charge
• Do not absorb photon though because not enough
  energy to excite electron from ground state
• Atom re-radiates because of accelerating charge
• Radiates with electric field half wavelength out
  of phase with incident electric field.
• Destructive interference with incident wave in
  forward direction
• Re-emitted photon has same frequency as incident
  one.

5
Scattering centers far apart in tenuous gas

• No fixed phase relationship so there is no
  interference in scattering transverse to original
  beam direction
• Light scattered out sideways.
• But scattering at each center is quite weak
• But in forward direction
• As incident wave passes, the scattered light is
  180 out of phase with it
• Forward scattering light from all centers is in
  phase then with each other but out of phase with
  incident wave front.
• Constructive interference in forward direction.

6
The symmetry principle involved.

• The direction of the original beam defines a
  unique direction for constructive interference so
  there is a strong forward scattering.
• The asymmetry introduced by the beam itself
  assures that all the scattered wavelets add
  constructively in the forward direction.

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Rayleigh Scattering

• Rayleigh scattering
• Atoms have resonant frequencies as do all
  vibrating systems
• When excited close to resonance frequency the
  vibrations are especially strong.
• For tiny particles, the scattering is
  proportional to the fourth power of the
  frequency.
• Gave first clue that atmosphere was made up of
  particles small compared to wavelength of light.

8
Example

• A beam of white light crosses a large volume
  occupied by a tenuous molecular gas mixture of
  mostly oxygen and nitrogen. Compare the amount of
  scattering occurring fro the yellow (580nm)
  component with that of the violet (400nm)
  component.

9
Example Solution

• Problem statement (molecules) suggests apply
  ideas of Rayleigh scattering.
• Assume roughly equal number of oxygen and
  nitrogen scatterers.
• Amt yellow/amt violet fy4 / fv4 (?v/ ?y)4
• Amt yellow/amt violet (400/580)4
• 22.6
• Only about 1/5 as much yellow is scattered as
  violet.

10
Scattering in denser materials

• For example consider atmosphere at lower
  altitudes, say at STP
• How many molecules in a cube of edge comparable
  to wavelength of visible light?
• PVnRTNkBT (ideal gas law)
• N/V P/(kBT)
• 1.01325x105 / (1.381x10-23 x 273.15)
  particles/m3
• 2.7 x 1025 particles per cubic meter.
• If ?500 nm, ?3 125 x 106 x10-27 1.25x10-19m3 .
• N 2.7 x 1.25 x 106 particles 3.4 x 106
  particles
• Dav (?3 / N)1/3 3.3 nm

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Destructive interference almost everywhere except
forward

• With a wavelength of 500 nm incident on more than
  3 million particles about 3.3 nm apart, we cannot
  ignore the coherence of the scattered waves.
• Suppose a molecule scatters in some direction
• A molecule about half a wavelength away will also
  radiate and the two E-fields will cancel
• It is likely that there will always be pairs of
  molecules that have canceling radiation.
• The more dense, uniform, and ordered the medium
  the more complete the destructive interference
  and the less scattering in all but the forward
  direction!
• Forward scattering is coherentbeam moves
  straight on.

12
Remarks

• On molecule by molecule basis scattering is very
  weak.
• Molecule per molecule liquids scatter much less
  than gases.
• Transparent amorphous solids scatter only very
  weakly.

13
Reflection

• Reflection is a result of scattering.
• Atomic separations on the order of 10-10 meters
  (X-Ray diffraction result)
• Wavelengths of visible light 1000 or so times
  larger than interatomic separation
• Unlike previous case, the discontinuity at
  reflecting surface allows backward scattering (in
  some directions)

14
Internal and External Reflections

• In practice it is the atoms within about ½
  wavelength depth of interface that do the
  reflecting
• For normal incidence from air to glass, about 4
  of light is reflected ((1.5 -1)(1.51))2
• Light incident from less to more optically dense
  medium is
• External reflection
• 180 degree relative phase shift in Electric field
• Light incident from more to less optically dense
  medium is
• Internal Reflection
• No relative phase shift

15
Some definitions redux

• A ray is a line drawn in space corresponding to
  the direction of flow of radiant energy (i.e.,
  parallel to the Poynting vector)
• A wave front is a locus of points in a wave that
  have the same phase
• Wave fronts are surfaces to which the rays are
  perpendicular.

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Huygens Principle

• Each point on a wave front serves as a source of
  secondary spherical wavelets, often considered
  modulated by the term
• (1 cos(?) ) with ? the original ray direction.

?
17
Law of Reflection from Huygens

• Huygens Construction for Reflection

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Proof 1

• CD is the reflection of AB
• Wavelet emitted from A will reach C in phase with
  wavelet emitted from D.
• That is, if AC BD
• Consider triangles ABD and ACD
• But BD AD sin (?i) and AC AD sin (?t)
• So sin (?i) sin (?t)
• ?I ?t
• Angle of incidence angle of reflection

19
Law of reflection.

• The angle of incidence equals the angle of
  reflection.
• The angle made by the wavefront with the
  interface equals the angle made by the ray with
  the normal to the interface.
• The incident ray, the reflected ray, and the
  normal at the point of reflection all lie in the
  same plane.

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Types of reflection

• Specular (as from a mirror)
• Diffuse (as from a rough surface)
• Real reflections lie somewhere between the above
  two extremes

21
Plane mirror

• Forms a virtual image
• Light rays do not pass through the image.
• Image Reverted (left to right) but not inverted
  (top to bottom)
• Image appears same distance behind mirror as
  object is in front of mirror.