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Physics 1C
Midterm 2 Review
Maxwells Equations, EM Waves,
Geometric Optics, and Interference
by Nathan Tung
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Maxwells equations in integral &
differential formIntegral Form Differential Form
=
=
= 0
= +
=
=
= 0 = +
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Energy & Momentum in EM Fields
Electromagnetic Energy density: =+
Poynting Vector energy flux density (energy per area per time)
=
Power: = Electromagnetic Momentum Density: =
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Example: Coaxial Cable
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Electromagnetic Waves in VacuumMaxwells Equations in Vacuum
= 0 =
= 0 =
Decouple Maxwells equations in vacuum to get the wave equations:
=
, = , =
1= 3 10
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Monochromatic Sinusoidal Electromagnetic
Plane Wave propagating in z-direction
, , are in phase
= , = cos + ,
, = cos +
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Energy and Momentum in EM
Plane WavesEnergy Density
=
=
cos
+ Poynting Vector
= cos + =
Momentum Density
=1 cos + =1
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Time-Average Values for EM Plane WavesTime-average values are given by the time-average of cos = :
=12
=12 =12
The time-average of the magnitude of is also called the Intensity: =12
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Radiation Pressure in EM Plane Waves
The radiation pressure (average force per unit area) is
==1 =1 =12 =
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Classical Radiation Created by an
Accelerating Charge
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Propagation of Light in Matter
Wave Equations in Matter:
= , = = 1
Index of refraction: =
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The Nature and Treatment of Light
Quantum: Photon
EM Theory: Electromagnetic Wave
Wave Optics: Scalar Light Wave
Geometric Optics: Light Ray
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The Laws of Geometric Optics1. The Plane of Incidence
2. Law of Reflection:
= 3. Law of Refraction (Snells Law): sin = sin
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Sign Conventions of Geometric OpticsSign rule for the object distance:
Objecton side of incoming light
object distance ispositiveSign rule for the object distance:
Image on side of outgoing light
image distance ispositiveSign rule for radius of curvature:
Center of curvature on side of outgoinglight
radius ispositive
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Object-Image Equations for
Spherical Mirrors and Thin Lenses
1+1=1
= ( )
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Focal Point for all Spherical Mirrors
=2 ( )
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The Lens Makers Equation for Thin Mirrors
1=
1
1
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The Principle Rays of Geometric
Ray Tracing
P-ray: Parallel ray; comes in parallel to the optical axisF-ray: Focal ray; proceeds from/towards focal point V-ray: Vertex ray; incident on vertex of mirror/lens
C-ray: Center-of-curvature ray; proceeds from/towardscenter of curvature
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Ray Tracing Rules for Spherical Mirrors
For mirrors, there is only focal point,
=
:
1. P-ray reflects through focal point2. F-ray reflects parallel to axis
3.V-ray
reflects symmetrically (equal angle reflection)
4. C-ray reflects back along original path
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The Focal Points of Thin Lenses
Converging lens havepositive focal lengths.
Diverging lenses havenegative focal lengths.
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Ray Tracing Rules for Thin Lenses
For thin lenses, there are two focal points, and :1. P-ray through focal point 2. F-ray parallel to axis3. V-ray un-deflected; along same line
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Object-Image Equation for
Spherical Refracting Surface (SRS)
+
=
( )
=
( )
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Focal Points of SRSTwo focal points: an object focal point , and an image focal point .
The object focal point is found by setting the image distance to infinityand its sign conventions follow those of the object distance:
+
=
=
The image focal point is found by setting the object distance to infinityand its sign conventions follow those of the object distance:
+
=
=
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Ray Tracing Rules for SRS
1. P-ray through focal point 2. F-ray parallel to axis3. V-ray
refracts according to Snells law
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Scalar Wave OpticsPropagation of light given by scalar light waves; if we plotthe amplitude of the wave it looks like this:
These are also calledplane waves because the wave itselfexists throughout an infinite plane perpendicular to thepropagation direction:
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Propagation of Wave Fronts and
Huygens Principle
Huygenss Principle describes the propagation of wavefronts by the formation and superposition of secondary
wavelets:
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The Dispersion of LightLight as a wave has a speed, wavelength, andfrequency.
In addition, the wavelength of light changes slightly in amaterial of index of refraction :
=
( )
The index of refraction of a material also slightly dependson the frequency of incident light.
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The Phase of a Wave
For a propagating wave: cos = cos + ,the phase is defined as everything inside the cosine:
Phase: = +
Two superimposing waves with different phases will interfere.There are three contributions to the relative phases between waves:
= ..+ +
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Initial Condition Phase Difference
The initial condition phase difference simply occurs whentwo waves begin out of phase.
If these waves travel and interfere at some location, theirrelative phase difference due to those initial conditions willaffect how those waves superimpose.
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The Path Phase DifferenceWhen two waves travel two different paths and to point , there isa relative path phase difference given by
= =2
For two wave sources in phase:
= = 0,1, 2, () = + 12 = 0,1, 2, ()
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Phase Shift by Reflection
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Conditions for Constructive and
Destructive Interference
The interference of coherent waves is completely dependent on therelative phase difference between the waves:
= 2 = 0, 1, 2,
= 2 + 12 = 0, 1,2,
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Two Source Interference Pattern
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Youngs Double Slit Experiment
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Far-Field Approximation of Double
Slit Interference
When , the path difference can be approximated by
= sin
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Far-Field Approximation of Double
Slit Interference
The interference conditions simplify to:
sin = = 0, 1,2, () sin = + = 0, 1, 2, ()
http://upload.wikimedia.org/wikipedia/commons/0/01/Two-Slit_Diffraction.pnghttp://upload.wikimedia.org/wikipedia/commons/0/01/Two-Slit_Diffraction.png8/13/2019 Electrodynamics Review
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Far-Field Approximation of Double
Slit Interference
The positions of the light bands on the screen,for smallangles only, can be determined via the relation,
= ( )
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Thin Film Interference at Near-
Normal Incidence
For thin films at near-normal incidence,
the path difference is simply
= 2
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Thin film interference conditions flip if there
is a phase shift by reflection
If there is no relative phase shift and the reflected waves are in phase:
2 = = 0,1,2, ( )2 = +12 = 0, 1, 2, ( )
If there is a 180 relative phase shift due to reflection:2 = + 12 = 0, 1, 2, ( )
2 = = 0, 1,2, ( )
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The wavelength in thin film interference
depends on the medium of the film
IMPORTANT: The in the previous equations refer to thewavelength of light in the thin film.
Recall, if the thin film has an index of refraction of : =
where is the initial wavelength of light before entering the thinfilm, and is the wavelength of light in the thin film.
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The following set of thin film equations
depend on the incidentwavelengthFor no phase shift:
2 = = 0,1, 2, ( )
2 = + = 0,1,2, ( , )
For 180 relative phase shift:
2 = + 12 = 0,1, 2, ( )2 = = 0,1,2, ( )