Physics 7E
Prof. D. Casper
Admin
• Chapter 32 HW is due Thursday, 7 am
• Reading
• Wednesday: Chapter 33.3 – 33.5
• Friday: Veterans’ Day!
• Next Monday: Chapter 33.6 – 33.7
• Chapter 33 HW will be due next Thursday (Nov. 17)
The Poynting Vector
The “Poynting Vector” describes both the
energy flow and direction of propagation:
1
𝑆 = 𝐸×𝐵
𝜇0
The total energy flow out of a closed surface
is the integral of 𝑆 over the area of the
surface:
𝑃 = ∮ 𝑆 ⋅ 𝑑𝐴
Intensity of Electromagnetic Wave
Because 𝐸 and 𝐵 are oscillating, the magnitude of 𝑆 =
1
𝐸
𝜇0
×𝐵
changes with time. The intensity is the average magnitude of 𝑆, which
is equal to
1
2
𝐼 = 𝑆𝑎𝑣 = 𝜖0 𝑐𝐸max
2
This has the same general form as we found for the intensity of sound
wave, but one important difference is that here the intensity does not
depend on the frequency of the wave, 𝜔. (Why?)
Radiation Pressure
Electromagnetic waves carry momentum as well as energy, and this
momentum can exert a force on a surface.
The radiation pressure (force per unit area) for a totally absorbed EM
wave is
𝑆𝑎𝑣 𝐼
𝑝𝑟𝑎𝑑 =
= (absorbed)
𝑐
𝑐
If the wave is totally reflected, the pressure is twice as large:
2𝑆𝑎𝑣 2𝐼
𝑝𝑟𝑎𝑑 =
=
(reflected)
𝑐
𝑐
Electromagnetic Standing Waves
A reflected EM wave can produce a
standing wave pattern, similar to
sound.
At the surface of the reflector (a
conductor) 𝐸 is always zero (node)
and 𝐵 is maximum (anti-node)
𝐸𝑦 𝑥, 𝑡 = −2𝐸max sin 𝑘𝑥 sin 𝜔𝑡
𝐵𝑧 𝑥, 𝑡 = −2𝐵max cos 𝑘𝑥 cos 𝜔𝑡
Q32.8
The drawing shows a sinusoidal
electromagnetic standing wave.
The average Poynting vector in
this wave
A. points along the x-axis.
B. points along the y-axis.
C. points along the z-axis.
D. is zero.
E. none of the above
A32.8
The drawing shows a sinusoidal
electromagnetic standing wave.
The average Poynting vector in
this wave
A. points along the x-axis.
B. points along the y-axis.
C. points along the z-axis.
D. is zero.
E. none of the above
𝐸𝑦 𝑥, 𝑡 = −2𝐸max sin 𝑘𝑥 sin 𝜔𝑡
𝐵𝑧 𝑥, 𝑡 = −2𝐵max cos 𝑘𝑥 cos 𝜔𝑡
𝐸max 𝐵max
sin 2𝑘𝑥 sin 2𝜔𝑡
𝜇0
𝑆𝑎𝑣 = 0
Makes sense: We have two waves with equal
amplitude traveling in opposite directions
𝑆𝑥 𝑥, 𝑡 =
Electromagnetic Standing Waves
Nodes of 𝐸 (anti-nodes of 𝐵)
occur where sin 𝑘𝑥 = 0
𝜆
𝑥 = 0, , 𝜆, …
2
Anti-nodes of 𝐸 (nodes of 𝐵)
occur where sin 𝑘𝑥 = 1
𝜆 3𝜆
𝑥 = , ,…
4 4
Resonant Cavities
Ever wonder why microwave
ovens don’t come in all different
sizes?
They use light with 𝑓 = 2.46 GHz,
which is strongly absorbed by
water
That corresponds to 𝜆 = 12.2 cm;
the dimensions of the interior
have to be a multiple of this:
𝑐
𝑓𝑛 = 𝑛
2𝐿
Chapter 33: Nature and Propagation of Light
In this chapter you’ll learn:
• What light rays are and how they
are related to wave fronts
• The laws that govern reflection and
refraction of light
• Circumstances under which light is
totally reflected at an interface
• How to make polarized light from
ordinary light
• How Huygens’s principle helps
analyze reflection and refraction
The Dual Nature of Light
Maxwell’s Equations (1865) and Hertz’s
detection of electromagnetic waves (1887)
seemed to prove conclusively that light
behaves as a wave
But things aren’t that simple…experiments
in the next 50 years showed that absorption
and emission of light can sometimes only be
understood by treating light as a particle –
the wave model gives incorrect predictions!
Light: Production Mechanisms
Accelerated electric charges emit light:
• Thermal radiation is one example
Excited or disturbed atoms and
molecules also emit light
• Fluorescent lights
• Lasers
Unique Properties of Laser Light
In most cases, light is produced by
an extended source where every
atom emits light independently
In lasers, a huge number of atoms
are arranged to emit light coherently
This results in a very narrow,
extremely intense and almost
monochromatic (single frequency)
beam
Wave Fronts and the Ray Model
A wave front is a set of adjacent
points at which the phase of the wave
is the same
• Crests are wavefronts
• Troughs are wavefronts too
Rays are imaginary lines along the
wave’s direction of travel
For waves traveling in a homogeneous
material, rays are straight lines normal
to the wave fronts
(At a boundary between materials, the
direction of rays may change)
Point source emitting spherically
expanding wave fronts (crests)
Reflection and Refraction
Just as a wave on a string can be
partially reflected and partially
transmitted at a boundary, so can
electromagnetic waves
The transmitted part of the light
wave is called “refraction” because
unlike the one-dimensional wave on
the string, it is bent into a different
direction
Law of Reflection
Specular reflection (reflection from a
smooth surface) obeys very simple laws:
• Incident and reflected rays line in same
plane with normal to surface
• Angle of reflection = Angle of incidence
• 𝜃𝑟 = 𝜃𝑎
Specular
reflection
Diffuse
reflection
Index of Refraction
Light always moves at the same, fixed speed in vacuum: 𝑐
In a material, the speed of light can and will be different (smaller): 𝑣
The index of refraction 𝑛 specifies the speed of light in a given material:
𝑐
𝑣=
𝑛
𝑛 = 1 for vacuum, and 𝑛 > 1 for materials
For air, 𝑛 = 1.00029, which we usually approximate as 𝑛 = 1
The frequency of waves is the same across a boundary; the wavelength
changes:
𝜆vacuum
𝜆=
𝑛
Q33.1
When light passes from vacuum (index of refraction n = 1) into water (n = 1.333),
A. the wavelength increases and the frequency is unchanged.
B. the wavelength decreases and the frequency is unchanged.
C. the wavelength is unchanged and the frequency increases.
D. the wavelength is unchanged and the frequency decreases.
E. both the wavelength and the frequency change.
A33.1
When light passes from vacuum (index of refraction n = 1) into water (n = 1.333),
A. the wavelength increases and the frequency is unchanged.
B. the wavelength decreases and the frequency is unchanged.
C. the wavelength is unchanged and the frequency increases.
D. the wavelength is unchanged and the frequency decreases.
E. both the wavelength and the frequency change.
Law of Refraction (Snell’s Law)
The law of refraction at a boundary depends on the indices of refraction of
the two materials:
𝑛1 sin 𝜃1 = 𝑛2 sin 𝜃2
Note the incident and refracted rays are also in the same plane as the normal
The angles of incidence and reflection are measured from the normal
Q33.2
Light passes from vacuum (index of refraction n = 1) into water (n = 1.333).
If the incident angle qa is in the range 0° < qa < 90°,
A. the refracted angle is greater than the incident angle.
B. the refracted angle is equal to the incident angle.
C. the refracted angle is less than the incident angle.
D. the answer depends on the specific value of qa .
A33.2
Light passes from vacuum (index of refraction n = 1) into water (n = 1.333).
If the incident angle qa is in the range 0° < qa < 90°,
A. the refracted angle is greater than the incident angle.
B. the refracted angle is equal to the incident angle.
C. the refracted angle is less than the incident angle.
D. the answer depends on the specific value of qa .