UNIT 4 : PHYSICS FOR ENGINEERS

MODULE: WAVE MOTION AND LIGHTS

INTRODUCTION:
Wave motion, propagation of disturbances—that is, deviations from a state of rest or
equilibrium—from place to place in a regular and organized way. Most familiar are surface
waves on water, but both sound and light travel as wavelike disturbances, and the motion of all
subatomic particles exhibits wavelike properties. The study of waves therefore forms a topic of
central importance in all physical science and engineering.

The simplest types of wave motion are vibrations of elastic media, such as air, crystalline solids,
or stretched strings. If, for example, the surface of a metal block is struck a sharp blow, the
deformation of the surface material compresses the metal in the vicinity of the surface, and this
transmits the disturbance to the layers beneath. The surface relaxes back to its initial
configuration, and the compression propagates on into the body of the material at a speed
determined by the stiffness of the material. This is an example of a compression wave. The
steady transmission of a localized disturbance through an elastic medium is common to many
forms of wave motion.

In most systems of interest, two or more disturbances of small amplitude may be superimposed
without modifying one another. Conversely, a complicated disturbance may be analyzed into
several simple components. In radio transmission, for example, a high-frequency signal can be
superimposed on a low-frequency carrier wave and then filtered out intact on reception.

In the simplest waves, the disturbance oscillates periodically with a

fixed frequency and wavelength. These sinusoidal oscillations form the basis for the study of
almost all forms of linear wave motion. In sound, for instance, a single sine wave produces a
pure tone, and the distinctive timbre of different musical instruments playing the same note
results from the admixture of sine waves of different frequencies. In electronics, the natural
rhythmic oscillations of electric currents in tuned circuits are used to produce sinusoidal radio
waves.

OBJECTIVES:
By the end of this section, you will be able to:
 Compare particle motion and wave motion in different types of waves.
 Distinguish between pulse waves and periodic waves.
 Distinguish between Longitudinal and Transverse waves.
 Relate the speed, frequency and length of a wave.
 Relate the energy carried by a wave to the amplitude of the wave.

What is Wave Motion?

Wave motion is the transfer of energy and momentum from one point of the medium to another
point of the medium without actual transport of matter between two points. Wave motion is
classified into three different ways they are,
  The medium of propagation,
 The dimensions in which a wave propagates energy,
 The energy transfer

Table of Content

 Classification of Wave Motion

 Mechanical Waves
 Non-Mechanical Waves
 Characteristics
 Terminologies

Classification of Wave Motion

Based on the Medium of Propagation

Classification of Wave Motion Based on the Medium of Propagation

Number of Dimensions a Wave Propagates Energy

Classification of Wave Motion Based on the Number of Dimensions a Wave Propagates
Energy

Based on the Transfer of Energy

 Standing waves (or stationary waves)

 Progressive wave
Standing waves remain confined to a region without any transfer of energy and momentum
whereas the progressive waves transfer energy and momentum between the particles of the
medium.

Mechanical waves (Elastic waves)

Waves that require a medium for their propagation are called mechanical waves or elastic waves.
The particles of the medium execute periodic motion about a mean position when the wave
propagates through the medium.
For Example, waves on a string
A mechanical wave is produced due to a disturbance at a point in a medium.

 The disturbed particle interacts with the neighbouring particle and its energy is handed
over to the next particle (due to the inertia of the medium).
 The disturbed particles return to the equilibrium position (due to the elasticity of
medium).

Properties of Medium for Mechanical Wave Propagation

 The medium must possess inertia so that its particles can store kinetic energy.
 The medium must possess elasticity.
 The minimum frictional force between the particles of the medium.

Non-Mechanical Waves
Waves which do not require a medium for their propagation are called a non-mechanical wave.
These types of waves can propagate through vacuum also. These are transverse in nature. For
example, electromagnetic waves and matter waves.

Transverse Wave Motion

The particles of the medium vibrate in a direction perpendicular to the direction of propagation
of the wave. The region of maximum upward displacement is called the crest, the region of
maximum downward displacement is called trough.
Transverse wave motion occurs only through a medium which has rigidity modulus or shape
conservation. For example, string waves.
Longitudinal Wave Motion
The particles of the medium vibrate about their equilibrium position in a direction parallel to the
direction of propagation of the wave is called a longitudinal waves.
Longitudinal waves require a medium with only elasticity of volume (or Bulk modulus) for its
propagation. In this type of wave motion, the waves travel through a medium in the form of
compression and rarefaction.

The region of high pressure is called compression and the region of low pressure is called
rarefaction. For example, Sound waves in the tube.

Periodic Wave Motion

1. If the disturbance is continuous and is periodic in nature, then the wave produced is
termed as a periodic wave.
2. A periodic wave that is varying sinusoidally is called a sinusoidal periodic wave.
 3. The particles of the medium execute simple harmonic motion (SHM) when a sinusoidal
periodic wave passes through the medium.

Characteristics of Wave Motion

 In wave motion, the disturbance travels through the medium due to repeated periodic
oscillations of the particles of the medium about their mean position (or) Equilibrium
position.
 Energy and momentum are transferred from one point to another without any actual
transfer of the particles of the medium.
 There is a regular phase difference between the particles of the medium because each
particle receives disturbance little later than its preceding particle.
 The velocity with which wave travels is different from the velocity of the particles with
which they vibrate about their mean (or) equilibrium position.
 For a given medium the velocity of the wave motion remains constant, while the particle
velocity changes continuously during its vibration about their equilibrium position.
 The velocity of the particle is maximum at the mean position and zeroes at the extreme
position.

Terminologies in Progressive Wave Motion

 Amplitude
 Period
 Wavelength
 Frequency
 Wave velocity
 Phase or phase angle (O)
 Phase difference
 Path difference
 Time difference

Explanation:
Amplitude (A): The amplitude of a wave is the maximum displacement of any particle of the
medium from its equilibrium position.
Period (T): Period (T) of a wave is the time taken by any particle of the medium to complete
one vibration during a period (T).
Wavelength (λ): Wavelength (λ) is equal to the distance between two consecutive particles of
the medium which are in the same state of vibration. It is equal to the distance travelled by the
wave by its time period (T).
Frequency (f): It is the number of vibrations made per second by any particles of the medium (f
= 1/T). Since the frequency of a wave is a characteristic property of the source which is
producing the wave motion, hence, the frequency of a wave does not change when a wave travels
from one medium to another medium.
Phase or Phase Angle (Φ): It represents the state of vibration of the particle of a medium with
respect to its mean position.
Phase Difference Δ(Φ): It represents the different state of vibration of a particle at two different
instants (or) any pair of particles at the same instant. ΔΦ = Φ2 – Φ1.
Wave Velocity (v): It is the distance travelled by the wave in one second (v = λ/T). It is
determined by the mechanical properties of the medium through which the wave propagates. The
velocity of wave motion is measured with respect to the medium, the wave velocity changes
when the medium is in motion i.e. speed of sound through air changes when the wind is blowing.
⇒ Check: Sound Waves
There are two velocities associated with a wave. One is the wave velocity and the other one is
particle velocity (which is the speed with which the particle of the medium vibrate when the
wave passes through the medium).
Path Difference (Δx) or (x): It indicates the distance between two points measured along the
direction of propagation of the wave through the medium.
Time Difference (ΔT): It indicates the time taken by the wave to travel from one point to
another through the medium.

The relationship between Path Difference and Phase Difference

Consider a progressive wave motion advancing in the positive direction of the x-axis

Path Difference vs Phase Difference

Let A and B be two points in the medium through which the wave passes.
The path difference between A and B is, x = x2 – x1
By the time the wave reaches B from A the phase of vibration of A has changed. The difference
between the states of vibration of A and B is called phase difference (ΔO).
From this wave motion, if we consider any two consecutive crests c 1 and c2, the path difference
between them is λ, the time difference is T and the phase difference is 2π.
A path difference of (λ) corresponds to a phase difference of 2π, thus, a path difference (x)
corresponds to the phase difference 2πr/λ.
Δϕ = (2πx)/λ = 2π/λ (path difference)
Where k = 2π/λ is called wave number or propagation constant of the wave motion.
A path difference (λ) corresponds to a time difference (T), therefore, a path difference (x)
corresponds to a time difference of (x/λ)T.
The relations connecting the path difference, phase difference and time difference are given
in the below table.

Path Difference Phase Difference Time Difference

X [2πX]/λ XT/λ

λ × [Δϕ/2π] Δϕ [Δϕ/2π] × T

λ × [ΔT/T] 2π × [ΔT/T] ΔT

Example 1
A frequency generator with fixed frequency of 343 Hz is allowed to vibrate above a 1.0 m high
tube. A pump is switched on to fill the water slowly in the tube. In order to get resonance, what
must be the minimum height of the water?. (speed of sound in air is 343 m s−1)

Solution

Let the length of the resonant columns be L1, L2 and L3. The first resonance occurs at length L1
The second resonance occurs at length L2

and so on.

Since total length of the tube is 1.0 m the third and other higher resonances do not occur.
Therefore, the minimum height of water Hmin for resonance is,

Hmin = 1.0 m − 0.75 m = 0.25 m

Example 2
How long does it take a wave of frequency 0.2 Hz and wavelength 2 m to travel along a rope of
length 4 m?
solution:

Let the wavelength be λ, frequency f and wave velocity v

λ=vT=v/f

v = f λ = 0.2 (2) = 0.4 m/s

time = length / velocity = 4 m / 0.4 m/s = 10 s

Example 3
What is the frequency of a pendulum that swings at the rate of 45 cycles per minute.

Solution:
1 minute = 60 seconds

frequency = number of cycles / total time = 45 / 60 = 0.75 Hz

Example 4
A standing wave is produced along a string of 100 cm whose ends are fixed. What is the
wavelength of the wave if there are 3 nodes between the fixed ends of the string?

Solution:

n = total number of nodes = 3 + 2 (fixed ends do not move and are counted as nodes)

wavelength = λ length of string = L = 100 cm

L = (5 - 1) (λ / 2)

Hence λ = 2 L / 4 = 50 cm

LIGHTS

Introduction
Light is all around us. It not only lets us see in the dark, but the properties of light are
important to many aspects of our lives. Reflections in rear-view mirrors of cars help to keep us
safe. Refraction through lenses of eyeglasses or contact lens’ helps some people see better.
More generally, electromagnetic waves (of which visible light is one example) are transmitted
as a signal that our radios pick up so we can listen to music. Pulses of infrared light are
transmitted as signals so we can communicate with our TVs. This backgrounder is all about
visible light and how we interact with it.
Objectives:

 Develop a general understanding of what light is and how it behaves

 Determine the path of light rays using the laws of reflection and refraction
 Determine the conditions under which total internal reflection occurs, and apply total
internal reflection to fiber optic and similar materials
 Understand that the index of refraction of a material is wavelength-dependent

Light and its Properties

In a vacuum (a container with no air), light travels at the speed of approximately 299 792 458
metres per second (m/s). This is known as the speed of light. It is the fastest that anything in
the universe is able to move! For comparison, the speed of sound is only approximately 300
m/s. This is why during a storm you always see lightning before hearing thunder.
An important thing to know about light is that it travels in a straight line through a material.

Waves and the Spectrum of Light

Light has the properties of waves. Like ocean waves, light waves have crests and troughs. The
distance between one crest and the next, which is the same as the distance between one trough
and the next, is called the wavelength. The frequency of a wave is the number of crests (or
troughs) that pass a point in one second. The wavelength multiplied by the frequency equals
the speed at which the wave travels.
The colours of visible light are red, orange, yellow, green, blue, indigo, and violet. These
different colours of light have different wavelengths and frequencies. Red light has the longest
wavelength, and the lowest frequency of the visible spectrum. Violet has the shortest
wavelength, and the highest frequency of the visible spectrum. Look at the two waves in the
picture below. You can imagine how, if they were both moving to the right at the same speed,
the number of violet crests passing the edge of the box in one second would be higher than the
number of red crests.

There is also light that is not visible to humans. Ultraviolet light and x-rays are also light, but
have too small a wavelength and too high a frequency to be visible to us. Infrared light which
can be detected by night-vision goggles, and radio waves, which are picked up by your radio
so you can hear music, have wavelengths which are too long and frequencies which are too
low to be seen by the human eye.

Chrysanthemum flower as seen using visible light (top), ultraviolet light (middle) and infrared
light (bottom) (Source: Dave Kennard [CC BY-SA] via Wikimedia Commons).

Visible light, together with these invisible types of lights, make up what is known as
the electromagnetic spectrum (EMS).

Primary Colours of Light

You will remember from art class that the primary colours are red, yellow and blue. These can
mix to form the secondary colours orange, green and purple. Light has primary colours as well.
But these are different colours than the colours we use in paint and markers. The primary
colours of light are red, green, and blue. The secondary colours of light are cyan (made by
combining blue and green), magenta (made by combining blue and red) and yellow (made by
combining green and red). Computer screens use various amounts of red, blue, and green light
to make all the colours that you see. When the primary colours of light are combined, they
make white light (see below).
What is Law of Reflection?
Definition:

1. The law of reflection defines that upon reflection from a smooth surface, the angle of the
reflected ray is equal to the angle of the incident ray, with respect to the normal to the
surface that is to a line perpendicular to the surface at the point of contact.
2. The reflected ray is always in the plane defined by the incident ray and the normal to the
surface at the point of contact of the incident ray.

The images produced by plane mirrors and curved mirrors can be understood by the law of
reflection.
Law of reflection is defined as:
The principle when the light rays falls on the smooth surface, the angle of reflection is equal to
the angle of incidence, also the incident ray, the reflected ray, and the normal to the surface all
lie in the same plane.

What is Reflection of Light?

The process through which light rays fall on the surface and gets bounced back is known as
a reflection of light.

Types of Reflection:

Regular Reflection:
The plane mirrors with a smooth surface produce this type of reflection. In this case, the image is
clear and is very much visible. The images produced by plane mirrors are always virtual, that is
they cannot be collected on a screen.
In the case of curved mirrors with a smooth surface, we can see the images of reflection either
virtually or really. That is, the images produced by curved mirrors can be either real (collected on
a screen and seen), or virtual (cannot be collected on a screen, but only seen).

Irregular Reflection:
Unlike mirrors, most natural surfaces are rough on the scale of the wavelength of light, and, as a
consequence, parallel incident light rays are reflected in many different directions irregularly, or
diffusely. Hence, diffuse reflection helps in seeing the objects and is responsible for the ability to
see most illuminated surfaces from any position.

In both regular and irregular reflections, the laws of reflection are followed.

Law of Reflection Formula:

The law of reflection formula is given as:

θi = θr

Where,

 θi is the angle of incidence

 θr is the angle of reflection
What is Angle of Reflection?
The angle of reflection Θr of a ray is the angle measured from the reflected ray to the normal
surface.

Calculation of Angle of Incidence and Angle of Reflection

The angle of incidence and the angle of reflection are calculated by drawing a normal line that is
perpendicular to the reflecting surface.

Examples of Laws of Reflection

EXAMPLE 1
A ray of light is incident towards a plane mirror at an angle of 30° with the mirror surface. What
will be the angle of reflection?
SOLUTION:
Since the angle of incidence is measured between the incident ray and the normal, so, here the
angle of incidence is not 60°
According to the Law of Reflection,
θi = θr
Hence,
Angle of Reflection = 60°
EXAMPLE 2
A light ray strikes a reflective plane surface at an angle of 54° with the surface.
(i) Calculate the angle of incidence.
(ii) Calculate the angle of reflection.
(iii) Calculate the angle made by the reflected ray and the surface.
(iv) Calculate the angle made by the incident and reflected rays.
SOLUTION:
(i) Angle of incidence, θi = 90° – 54°=36°
(ii) Angle of Reflection, θr = 36° (As per Law of Reflection)
(iii) Angle made by the reflected ray and the surface,
q=90° – r = 90° – 36° = 54°
(iv) Angle made by the incident and reflected rays,
θi = θr =36° + 36° = 72°
EXAMPLE 3

Find angle α made by the system of the two mirrors shown in the figure below so that the
incident ray at A and the reflected ray at B are parallel.
SOLUTION:

We first complete the given diagram with the angles of incidence and reflection as shown below
and also labelling the incident and reflected rays.
For the incident ray at A and the reflected ray at B to be parallel, angles i + r and i’ + r’ have to
be supplementary. (geometry: parallel lines cut by a transversal).
Hence,
i + r + i’+ r’ = 180 °
by law of reflection : r = i and r’ = i’
Substitute to obtain
i + i + i’ + i’ = 180 °
i + i’ = 90
In triangle AOB, we have
α + (90 – r) + (90 – i’) = 180 °
α = r + i’ = i + i’ = 90 °
If α = 90 °, the incident ray at A and the reflected ray at B are parallel.

Concave Mirrors:
Concave mirrors give real, inverted images if the object is beyond the focus and a virtual, erect,
enlarged image if the object has a distance less than the focal length from the pole of the mirror.
Uses of Concave Mirrors:

1. Concave mirrors are used in torches, searchlights, and headlights of vehicles to get
powerful parallel beams of light.
2. Concave mirrors are also used as shaving mirrors to see a larger image of the face.
3. Dentists use concave mirrors to see bigger images of the teeth of the patients.
4. 4) Large concave mirrors are used to focus sunlight to produce heat in the solar furnaces.

Convex Mirrors:
Convex Mirrors always give a virtual, erect, diminished image of the object behind the mirror.
Uses of Convex Mirrors:

1. The convex mirror is used as a side-view mirror in vehicles to give a smaller view of the
vehicles coming from behind.
2. They are used in shops and supermarkets and any other place where there is a
requirement for detecting burglars.
3. Convex mirrors are used in making lenses of sunglasses.
4. Convex mirrors are used in magnifying glasses, and telescopes.
5. Convex mirrors are used to reflect street light; because they can reflect over a wide area.
6. Convex mirrors are kept at the street corners to avoid collisions.

Total Internal Reflection:

When light passes from denser medium to lighter medium at an angle more than the critical
angle required for refraction, then the light is reflected back into the denser medium. This is a
phenomenon called Total Internal Reflection. The light undergoing the total internal reflection
also follows the ordinary laws of reflection for light as shown below:
The phenomenon, total internal reflection, is taken advantage in piping light in a curved path.
The light directed down a narrow fiber of glass or plastic repeatedly reflects from the fiber-air
interface at larger than the critical angle. Optical fibers can transmit light over long distances
without any loss of intensity.
Sequences of light pulses are used to transmit information through an optical fiber network with
the help of this total internal reflection. Medical instruments like ‘endoscopes’ use the total
internal reflection of light through an optical fiber bundle to image internal organs.

Uses of Reflection:

1. Reflection is used in periscopes to view advancing enemies on the battlefield from a safe
position.
2. Reflection is the reason why we see objects.
3. Reflection by a concave mirror and a convex mirror has many uses as listed above.
4. Reflection helps in medical diagnosis and optical communications.
5. Light and Sound both follow the law of reflection, both being waves.
6. Using the law of reflection for sound and light, we can measure the distances accurately
to objects.
7. Reflection is the reason why we hear the echo of sound.

Refraction Of Light : Law Of Refraction

What is Refraction?
Refraction of Light is a phenomenon wherein light bends and travels from one transparent
substance to another. Lets us understand this concept in-depth with an illustration. So, have you
ever observed the bottom of a thick glass slab when a printed paper is kept below it? You will
find that the printed matter seems to be raised. Similarly, when a pencil is partly immersed in
water in a glass tumbler, it appears to be displaced from its original position at the surface. Did
you ever think why does it happen? Why can’t it appear to be at the normal position?
Let us understand this with the help of the case of a partially immersed pencil in water. The
pencil seems to be displaced from its original position because light reaching our eyes from the
portion of the pencil inside water comes from a different direction. Due to this reason, the pencil
and printed matter seem to be displaced from the original position.
What if the water in the tumbler is replaced by kerosene or turpentine? Will the pencil appears to
be displaced to the same extent? No, now the pencil will be displaced to some other extent. The
extent of displacement is different for a different medium. This shows that light does not travel
in the same direction in all medium. The direction of propagation of light while travelling
obliquely changes from one medium to another. This phenomenon is named as refraction of
light. We can define refraction as the phenomenon of bending of light when it passes from one
substance to another. Rainbow, mirage a few real-life examples of refraction. Sunrise and sunset
is a result of atmospheric refraction. Let us understand it more clearly by the concept of
refraction through a rectangular glass slab.

Refraction through a rectangular glass slab:

Let us understand the phenomenon of refraction of light with the help of an activity.

1. Take a white sheet and put a glass slab over the sheet.
2. Draw the outline of the slab and mark it as ABCD as shown in the fig.
3. Take two pins, say E and F and put it at the edge of A and B.
4. Now, look at the images of the pins E and F through the opposite sides. Place two more
pins G and H such that G, F and the images of E and F are in the same straight line.
5. Remove the pins and the slab carefully.
6. Join the points E and F and extend the line up to AB. EF meets at O. In the same manner,
join G and H and extend it to the edge of CD. Mark O’ the point where HG meets CD.
7. Now, join O and O’. Also, extend EF till P as shown in the figure.
Refraction of Light in air and glass medium

In the above activity, we have observed that light ray changes its direction at O and O’ points at
the surfaces of two separating transparent media. Make a perpendicular line NN’ to AB at O and
another perpendicular line MM’ to CD at O’. At O the light ray enters from a rarer medium to a
denser medium i.e. from air to glass. Hence, the light travels towards the normal. While at O’,
the ray of light moves from denser to rarer medium. Hence, the light moves away from the
normal.

In the given figure EO is the incident ray, O’H is the emergent ray and the OO’ is the refracted
ray. In this, we can observe that emergent ray is parallel to the incident ray. From the above
activity, we can say that refraction happens due to change in the speed of light when light travels
from one medium to another.

Laws of Refraction
The laws of refraction govern the behaviour of light as they pass through the interface between
two media. It is generally known as Snell’s Law. From the above-depicted activity, we can say
that refraction of light follows two laws:
  The refracted ray, incident ray and the normal at the interface of two transparent media at
the point of incidence, all lie in the same plane.
 For the given pair of media, the ratio of the sine of the angle of incidence to the sine of
angle refraction is always constant.

Refractive Index:
We know that when light passes obliquely from one medium to another, it changes direction in
the second medium. The extent to which change in direction takes place in the given set of a
medium is termed as refractive index.

Difference Between Reflection and Refraction

There is a unique difference between Reflection and Refraction and it is important to analyze
both these terms and understand the definitions of both these terms. Reflection is simply the
property of a light that rebounds after hitting a surface. When the light that passes through a
surface undergoes some changes in the appearance, whenever it usually passes through a
medium, this phenomenon is usually referred to as Refraction. The two different types of lights
that are typically involved in this are incident ray and the reflected ray. Light energy is incredible
and has many uses to it.

Difference between Reflection and Refraction

Reflection Refraction

This phenomenon usually occurs in mirrors. This phenomenon usually occurs in Lenses.

Reflection can simply be defined as the Refraction can be defined as the process of the shift of
reflection of light when it strikes the light when it passes through a medium leading to the
medium on a plane. bending of light.

The light entering the medium returns to the The light entering the medium travels from one medium
same direction. to another.
Considering the light waves, they bounce The light waves pass through the surface while
from the plane and change direction. simultaneously change from medium to medium.

The angle of incidence of the light is equal The angle of incidence is not equal to the angle of
to the angle of reflection. reflection.

Examples

The speed of light in water is (3/4)c. what is the effect on the frequency and the wavelength
of the light in passing from vacuum (or the air ,to good approximation into water ?
compute the refraction index in the water.

Solution:

The same number of wave peaks leave the air each seconds as enter into the water. Hence
the frequency is the same in the two materials.

Because wave length =speed/frequency, the wave length in the water is ¾ that in air. To
find the refraction index,

n= speed of the vacuum/ speed of the water

n=

n= 4/3

A glass plate is 0.6cm thick and has a refractive index of 1.55. how long does it takes for a pulse
of light to pass through the plates?

Solution:

t = x/v

t= = 3.1

SUMMARY

 Refraction is the change in direction of a wave caused by a change in wave speed.

o An interface is the boundary between two different media…
 or two regions of a medium with different characteristics such as…
 density (which is often related to temperature)
 concentration of solute (salinity, for example)
 mechanical stress
o In geometric optics, when an incident ray meets an interface it will be partially
 reflected
 Reflected rays obey the law of reflection described in a previous section of this
book.
 Materials that reflect a significant portion of incident light
appear shiny or lustrous.
 transmitted
 Transmitted rays obey Snell's law of refraction, which is described in this
section of this book.
 Materials that transmit a significant portion of incident light
appear clear or transparent.
 Materials that do not transmit any incident light are said to be opaque.
 absorbed
 The energy of absorbed rays is not destroyed, but changes form.
 Materials that absorb a significant portion of incident light appear dark.
o Angles are measured with respect to the line normal to the surface.
 The angle of incidence is the angle between the incident ray and the normal.
 The angle of reflection is the angle between the reflected ray and the normal.
 The angle of refraction is the angle between the transmitted ray and the normal.

 Refraction is described mathematically by Snell's law.

n1 sin θ1 = n2 sin θ2

where

n1 n2 = indexes of refraction of the two media

θ1 θ2 = angles between the ray and the line normal to the surface in the two media

o Snell's law describes the path of least action between two points in different media.
 The index of refraction…
o is a property of a medium
o is a measure of the "slowness" of a wave
o is defined mathematically by the formula
c
n=
v

o where
n = index of refraction
 c = speed of light in a vacuum

v = speed of light in a medium

o is always greater than 1 (since the speed of light in a medium is always slower than the
speed of light in a vacuum)
o has no units (since it is the ratio of two speeds)
o generally increases with the density of the medium
 If a ray of light travels from…
o a medium with a low index to a medium with a high index (n1 < n2)…
 it slows down
 it refracts toward the normal
o a medium with a high index to a medium with low index (n1 > n2)…
 it speeds up
 it refracts away from the normal
 Total internal reflection occurs when…
o Snell's law has no real solution
o light travels from a "slow medium" to a "fast medium" (n1 > n2)
o the incident angle is greater than the critical angle
n2
sin θc =
n1

 The critical angle is the incident angle that corresponds to a refracted angle of 90°;
that is, the transmitted ray travels parallel to the interface.
 Dispersion
o occurs when the speed of light in a medium (and thus the index of refraction) is a
function of frequency and medium
 Informally, this can be summarized as different colors travel at different speeds in
some media.
o can be used to produce a spectrum
 violet refracts the most (n is "large", v is "slow" for violet light)
 red refracts the least (n is "small", v is "fast" for red light)
o is the cause of the colors seen in…
 rainbows (sunlight passing through raindrops)
 halos (sunlight passing through ice crystals)