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UNES
SCO-NIGERIA TECHNICAL
L&
VO
OCATION
NAL EDU
UCATION
N
REVITA
ALISATIO
ON PROJJECT-PHA
ASE II
NA
ATIONAL DIPLO
OMA IN SCIENCE LLABORA
ATORY TTECHNO
OLOGY OPTTICS A
AND W
WAVEES
CO
OURSE C
CODE: STTP 122 YEAR I- SEMEST
TER II
TH
HEORY
V
Version
1:: Decembeer 2008
1` TABLE OF CONTENTS WEEK 1….Reflection and refraction of light at plane surfaces…………………………………………….3 WEEK 2…. Refraction of light at plane surfaces…………………………………………………………………..15 WEEK 3….Spherical mirrors………………………………………………………………………………………………..24 WEEK 4….Mirror formula……………………………………………………………………………………………………47 WEEK 5….Refraction at curved surfaces……………………………………………………………………………..55 WEEK 6….Lenses contd………………………………………………………………………………………………………69 WEEK 7….Optical instruments……………………………………………………………………………………………76 WEEK 8…. Optical instruments contd…………………………………………………………………………………83 WEEK 9…. Optical instruments contd…………………………………………………………………………………89 WEEK 10.. Optical instruments contd…………………………………………………………………………………92 WEEK 11.. Optical instruments contd…………………………………………………………………………………96 WEEK 12..Photometry………………………………………………………………………………………………………..99 WEEK 13.. Photometer……………………………………………………………………………………………………….105 WEEK 14..Waves and Motion……………………………………………………………………………………………..109 WEEK 15..Sound waves………………………………………………………………………………………………………126 2` WEEK 1: REFLECTION AND REFRACTION OF LIGHT AT PLANE SURFACES 1.0.0 Introduction Light is a form of energy and can be transformed into other forms of energy. You would have observed the path of 'a beam of light' inside a room. This beam is nothing but the scattered light produced by the dust particles and this beam of light becomes invisible if the room is dust free. Thus light makes things visible even though light by itself is invisible. Light does not require a material medium for its propagation. In a given medium light travels with a very high but finite velocity. The velocity of light in air or vacuum is 3 x 108 m/s Light is a form of energy. Energy can be transferred from one point to another point either by particle motion or by wave motion. Accordingly, different theories on the nature of light have been proposed. 1.1 Characteristics of Light •
Light is a form of energy produced by luminous objects. •
Light can travel through vacuum. •
Light can penetrate through transparent materials but cannot pass through opaque objects. •
Light travels in a straight line in an optically homogeneous medium. •
Light bounces back when made to fall on polished surfaces such as mirrors or metal surfaces. This bouncing back of light is described as reflection. •
The change in the velocity of light when it travels from one transparent medium to another is described as refraction. 3` •
Light takes the path of least time in passing from one point to the other. This is nothing but Fermat's principle. The shortest distance between any two given points is a straight line. Thus Fermat's principle proves the rectilinear propagation of light. •
Light appears to have a dual nature. During propagation, light exhibits wave characteristics but when it interacts with matter, it behaves like particles. •
In a homogenous transparent medium light travels in a straight line and this is known as rectilinear propagation of light. This can be demonstrated by the following experiment: The image part with relationship ID rId12 was not found in the file.
Take three cardboards A, B and C and make a pinhole at their centers. Place a burning candle on one side of A and arrange the cardboards in such a way that the three pinholes and the candle flame are in a straight line. The candle flame will be visible through the pinhole of the cardboard C. Now slightly displace any one of the cardboards and try to see the flame through the pinhole of the cardboard C. The flame will not be visible. From this it is clear that light travels in a straight line. 4` 1.2 Refle
ection of Ligh
ht When a rray of light falls on any ssurface, a paart of the ligh
ht is sent back to the sam
me medium. This phenomeenon where the incident light fallingg on a surfacce is sent back to the sam
me medium is known ass reflection. There are tw
wo types of reflection off light: Regular rreflection Irregular reflection Reflection Regular R
Regular Reflection on a Sm
mooth Surfacce hen a ray of light is incid
dent on a polished smoo
oth surface liike a Regular rreflection takes place wh
mirror. H
Here the refle
ected ray off light movess only in a fixxed direction
n. Irregularr Reflection or Diffused Reflection
5` Irregular reflection o
or diffused reeflection takkes place when a ray of light is incideent on a wall or wood, which is not smooth or po
olished. In th
his case, the different po
ortions of the surface reflect the incident light in d
different direections. In such cases no
o definite im
mage is formed, but the surface b
becomes visiible. It is com
mmonly know
wn as scatteering of lightt. Thus diffussed reflectio
on makes no
on‐luminouss objects visiible. Not all ligght, which hits an objectt, is reflected
d. Some of tthe incident light is abso
orbed. The brightnesss of an obje
ect depends on the inten
nsity of the iincident ligh
ht and also on the reflecttivity of the ob
bject. If a surface allows the entire inciident light to
o undergo reegular reflecction then it will becomee invisible. The figurre shows how
w a ray of ligght is reflectted by a plan
ne surface. LLet MM' reprresent a reflectingg surface. W
When a ray off light is incid
dent on MM
M' in the direection IO it gets reflected
d along thee direction O
OR. IO is the incident rayy; O is the po
oint of incideence and OR
R is the refleccted ray. 6` Reflectio
on of a Ray Liight by a Plaane Mirror he point of in
ncidence. Th
he Let ON be the normaal drawn perrpendicular tto the surfacce MM' at th
hich the incid
dent ray makkes with thee normal at tthe point of incidence is called the angle angle wh
of incidence and is denoted by th
he letter 'i'. The angle th
hat the reflected ray makes with thee normal aat the point o
of incidence is called thee angle of reeflection 'r'. M
Mirror is an example of a reflectingg surface. The Lawss of Reflection The refleection at anyy plane surface is found tto obey the laws of refleection. The laws of reflection are: •
The incident ray, the refleected ray an
nd the normaal at the point of inciden
nce lie in thee saame plane. •
The angle of iincidence is equal to thee angle of reeflection. 7` Nature of the Image Formed by a Plane Reflecting Surface An image can be real or virtual. A real image is formed when the rays of light actually intersect after reflection. A virtual image is formed when the light rays after reflection do not actually intersect but appear to diverge from it (these rays of light intersect when produced backwards). The image part with relationship ID rId15 was not found in the file.
Virtual Image Formed by a Plane Mirror In fig (i) we can see that the light beam from the point source O is a diverging beam. After reflection from the mirror it is still a diverging beam, which appears to come from I. The image formed by a plane mirror is virtual. Formation of Image by a Plane Mirror ‐ Ray Diagrams The following rays are usually considered while constructing ray diagrams. 8` The image part with relationship ID rId16 was not found in the file.
A Ray Diagram Showing the formation of an Image by a Plane Mirror A ray of light incident on a plane mirror at 90o gets reflected from the mirror along the same path. A ray of light falling on a plane mirror at any angle gets reflected from the mirror such that the angle of incidence is equal to the angle of reflection. Image Formation When an Object is Placed between Two Inclined Mirrors It has been found that if the mirrors are inclined at an angle q then the number of images is The image part with relationship ID rId17 was not found in the file.
The image part with relationship ID rId18 was not found in the file.
given by the relation If is not a whole number, then the number of images will be rounded off to the nearest integer. This can be verified by actual drawing. 9` If the mirrors are inclined at 120o the number of images formed by the mirrors is given by the The image part with relationship ID rId19 was not found in the file.
relation Case – I Let MM and MM' be two plane mirrors inclined at an angle 120o and O be the object placed in between these mirrors. In this case there will be only two images viz., O1 and O2. The image part with relationship ID rId20 was not found in the file.
10` Case II The image part with relationship ID rId21 was not found in the file.
The Angle of Inclination Between the Mirrors is 900. •
Place the mirrors MM' perpendicular to MM. •
An object O is kept in between these mirrors. •
OA and OB are the two rays, which are incident on the mirror MM. •
OA being normal to the surface retraces its path. •
OB makes an angle i with the normal N and gets reflected along BC according to the laws of reflection •
Extend the rays OA and BC backwards. •
They meet at O1, which is the virtual image of O. •
OD and OE represent the rays which are incident on the mirror MM'. •
OD is perpendicular to the mirror MM' and hence gets reflected along the same path. 11` •
OE is the incident ray and N2 is the normal at the point of incidence and OE gets reflected along the path EF. •
Extend OD and EF backwards. They meet at O2, which is the virtual image of O. •
The reflected ray BC gets internally reflected by the mirror MM' along CG. •
The ray DG appears to comes from O3, which is the image of O1, •
Similarly EF the reflected ray gets internally reflected by the mirror MM along FH. •
The ray FH appears to come from O4, which is the image of O2. •
The position of O1 and O2 coincide. •
Thus when the angle of inclination between the mirrors is 900 we get three images. Case III Let us now calculate the number of images formed if the two mirrors are placed parallel to each other i.e., the angle of inclination between them is 00. The image part with relationship ID rId22 was not found in the file.
•
Place the mirrors MM and MM' parallel to each other. 12` •
An object O is kept between these mirrors. •
OA and OO' represent the rays which are incident on the mirror MM. •
OO' being normal retraces its path. •
OA makes an angle i with the normal N1 and gets reflected along AB according to the laws of reflection. •
Extend the rays AB and OM backwards. •
They meet at I1, which is the virtual image of the object O. •
The reflected ray AB gets reflected by the mirror MM' and forms an image I2. •
Similarly I3, I4 etc. are formed. •
The light from I1, I2, I3, I4 etc. gets reflected and forms their images. In this manner, many images are formed but the intensity of the remote images goes on decreasing due to absorption of light energy at every successive reflection and thus we see only finite number of images even though infinite images will be formed. Uses of Plane Mirrors A plane mirror is used: as a looking glass to view ourselves by interior designers to create an illusion of depth to fold light as in a periscope and other optical instruments to make kaleidoscope, an interesting toy 13` Application of Plane Mirrors – Periscope It is an instrument in which plane mirrors are used to fold light so that the image of an object can be brought down to a lower level. It is used for observing enemy movements from trenches without any danger of being seen. Sailors on submarines use periscopes to see things above the water level. The image part with relationship ID rId23 was not found in the file.
It consists of two parallel mirrors A and B facing each other and each fixed at 450 to the frame work. Rays of light entering through the aperture, strike the mirror A at an angle of incidence equal to 450 and are reflected along the axis of the tube, striking the mirror B at 450. From the mirror B, these rays are reflected parallel to their original path reaching the eyes of the observer. 14` WEEK 2: REFRACTION OF LIGHT AT PLANE SURFACE 2.1 Introduction In the previous week we have seen how light gets reflected when it is incident on a surface. Now let us see what happens when a ray of light traveling from one medium to another medium of different density. It is a matter of common experience that a swimming pool appears to have less depth than its actual depth. Similarly a straight stick partly immersed in water appears to be bent at the surface of water. The above observations suggest that light changes its path as it passes from one medium to another. This change in the path of light is due to the fact that the velocity of light varies as it travels from one medium to another. Thus the deviation in the path of light when it passes from one medium to another medium of different density is called refraction. A part of the light gets reflected and rest of the light changes its direction as it enters the second medium. 15` Refraction and
d Reflection o
of Light dent ray IO = incid
ORl =reflected ray OR =refraacted ray The diagrram shows h
how the light gets refraccted when itt is traveling from one op
ptical mediu
um to another. Like refleection, refracction of lightt takes placee according tto certain laws. Before w
we state these laws let u
us get familiar with certaain terms wh
hich are com
mmonly used
d to explain the phenomeenon of refraction. Incident R
Ray 16` The ray of light striking the surface of separation of the mediums through which it is traveling is known as the incident ray. Point of Incidence The point at which the incident ray strikes the surface of separation of the two mediums is called the point of incidence. Normal The perpendicular drawn to the surface of separation at the point of incidence is called the normal. Refracted Ray The ray of light which travels into the second medium, when the incident ray strikes the surface of separation between the mediums 1 and 2, is called the refracted ray. Angle of Incidence (i) The angle which the incident ray makes with the normal at the point of incidence, is called angle of incidence. Angle of Refraction (r) The angle which the refracted ray makes with the normal at the point of incidence, is called angle of refraction. 17` Cause of Refraction A ray of light refracts or deviates from its original path as it passes from one optical medium to another because the speed of light changes. 2.2 Laws of Refraction •
The incident ray, the refracted ray and the normal to the surface at the point of incidence all lie in one plane. •
For any two given pair of mediums, the ratio of the sine of the angle of incidence to the sine of the angle of refraction is a constant. The above law is called Snell's law after the scientist Willebrod Snellius who first formulated it Where µ is the refractive index of the second medium with respect to the first medium. We know that the phenomenon of refraction is taking place because the speed of light changes when it is traveling from one optical medium to another. Thus we can define refractive index in terms of the speed of light in the two media. The refractive index of glass with respect to air is given by the relation. 18` Refraaction of Ligh
ht In generaal, if a ray off light is passsing from meedium 1 to m
medium 2, th
hen If the meedium 1 is air or vacuum
m, the refracttive index off medium 2 is referred to
o as the abso
olute refractivee index. The refraactive index of a medium
m depends o
on the follow
wing factors:
•
th
he nature off the medium
m •
th
he color or w
wave length of the incideent light 19` 2.3 Relattion betwee
en the Refracctive Indicess According to the prin
nciple of reversibility of light, the paath of a ray o
of light is revversible. Thee figure beelow shows h
how light gets refracted from mediu
um 1 to med
dium 2. According to Snell's law Suppose the refractio
on is taking place from m
medium 2 to
o medium 1,, then accord
ding to the OI will be the incident raay and the refracted rayy respectively. principle of reversibility RO and O
By Snell'ss law 20` From the above relation it is clear that the refractive index of the second medium with respect to the first medium is the reciprocal of the refractive index of the first medium with respect to the second. 2.4 Total Internal Reflection Let us consider a ray of light passing from glass to air, that is from a denser medium to a rarer medium. The figure shows that for a small angle of incidence, major part of the incident light is refracted and a small portion is reflected. The refracted ray bends away from the normal after the refraction. That is, the angle of refraction r is greater than the angle of incidence i. Now if we increase the angle of incidence, the angle of refraction also increases and for a certain angle of incidence (say ic), the refracted ray grazes over the surface of separation and the angle of refraction will be 90o. ic is referred to as the critical angle. If we further increase angle of incidence, the light ray instead of getting refracted bounces back into the same medium obeying the laws of reflection. This is known as total internal reflection. Critical angle is that angle of incidence for which a ray of light while moving from a denser to a rarer medium just grazes over the surface of separation of the two media (that is, angle of refraction = 90o). 21` If the anggle of incidence of a ray of light travveling from aa denser medium to a raarer medium
m is greater than the critiical angle for the two media, then th
he ray is refllected into the denser medium and this phe
enomenon iss described as total internal reflectio
on. 22` Conditions to be Satisfied for Total Internal Reflection to take Place The conditions to be satisfied for total internal reflection to take place are •
the ray of light must travel from a denser medium to a rarer medium. •
the angle of incidence must be greater than the critical angle for those two mediums. 23` WEEK 3 SPHERICAL MIRRORS 3.1 Introduction A mirror whose polished, reflecting surface is a part of a hollow sphere of glass or plastic is called a spherical mirror. In a spherical mirror, one of the two curved surfaces is coated with a thin layer of silver followed by a coating of red lead oxide paint. Thus, one side of the spherical mirror is opaque and the other side is a highly polished reflecting surface. Depending upon the nature of the reflecting surface of a mirror, the spherical mirror is classified as: •
Concave mirror •
Convex mirror Concave Mirror Concave mirror is a spherical mirror whose reflecting surface is towards the center of the sphere of which the mirror is a part. Convex Mirror Convex mirror is a spherical mirror whose reflecting surface is away from the center of the sphere of which the mirror is a part. 24` Concave aand Convex M
Mirror Let us no
ow define certain physical terms relaating to spheerical mirrorrs. Center off Curvature Center o
of Curvature
e is the centeer of the sph
here of which the spheriical mirror fo
orms a part. It is denoted
d by the letteer C. Center of Curvatu
ure Radius off Curvature 25` Radius off Curvature is the radiuss of the spheere of which the mirror is a part. It iss representeed by the letter R. Radius of Curvatu
ure perture Linear Ap
Linear ap
perture is the
e distance between the extreme po
oints (X and YY) on the periphery of th
he mirror. XY is the Aperturre Pole he midpoint of the apertture of the spherical mirrror. It is rep
presented byy the letter P
P. Pole is th
26` Mid
dpoint of xy
Principal Axis Principal axis is the straight line p
passing thro
ough the pole and the ceenter of curvvature of a sphericall mirror. Priincipal Axis
Secondarry Axis Secondarry axis is anyy other radiaal line passin
ng through th
he center off curvature o
other than th
he principal axis. 27` Secondary Axis
Normal The norm
mal at any po
oint of the spherical mirrror is the strraight line obtained by joining that p
point with the center of th
he mirror. Th
he normal att point A on tthe mirror iss the line AC
C obtained byy joining A
A to the cente
er of curvatu
ure of the m
mirror. Normal at any poiint on a spheerical mirrorr is equal to the radius o
of the spheree of which th
he mirror is aa part. Normal Principal Focus or Focus 28` The rays of light paraallel to the p
principal axiss of a mirror after reflecttion, either p
pass through
h a point (in case of a co
oncave mirro
or) or appearr to diverge from a point (in the case of a conveex mirror) o
on the principal axis and this point iss referred to as the princcipal focus o
or focal pointt of the mirro
or. 29` The princcipal focus o
of a spherical mirror mayy be defined as a point o
on its princip
pal axis wherre a beam of light paralle
el to the prin
ncipal axis co
onverges to o
or appears to diverge fro
om after reflection
n from the spherical mirrror. Focal Len
ngth Focal len
ngth is the distance betw
ween the pole and the fo
ocus of a mirrror. It is rep
presented byy the letter f. Characte
eristics of Fo
ocus of a Con
ncave and a Convex Mirrror 3.2 Relattion Betwee
en f and R To show that f = R/2 where f is th
he focal lenggth of a mirrror and R its radius of cu
urvature. oncave Mirro
or Case I Co
30` To Show that f = R/2 for a Concave Mirror Let a ray of light AB b
be incident, parallel to the principal axis, on a co
oncave mirro
or. After reflection
n, the ray AB
B passes alon
ng BD, throu
ugh the focu
us F. BC is no
ormal to the concave mirrror at B. We know
w that AB and PC are parrallel to each
h other. From equ
uations (1) aand (2) we geet Hence triangle BCF iss isosceles …. (3) ∴ BF = CF …………………………………
If the apeerture of the
e mirror is sm
mall then B w
will be very close to P. 31` BF = PFF …………………
…………………. (4) From equ
uations (3) aand (4) we co
onclude thatt But by deefinition PF =
= f (focal len
ngth) and PC = R (radius of curvaturee) Note: Wh
hile Derivingg the Relatio
on we have cconsidered o
only one of the incident rays Case II Co
onvex Mirror Let a ray of light AB b
be incident, parallel to the principal axis, on a co
onvex mirror. After reflection
n the ray AB
B appears to come from F. BC is the n
normal to th
he convex mirror at B. 32` From equ
uations (2) aand (3) Hence triangle BCF iss isosceles BF = CF ………………………………………. (4) If the apeerture of the
e mirror is sm
mall then B w
will be very close to P. BF = PF ……………………………………….(5) uations (4) aand (5) we co
onclude thatt From equ
By definition PF = f (ffocal length)) and PC = R (raadius of curvvature) 33` From thee above relattion we concclude that th
he radius of curvature of a mirror is twice its foccal length. 3.3 Sign C
Convention for Sphericaal Mirrors The following sign co
onvention is used for meeasuring various distances in the rayy diagrams o
of sphericall mirrors: •
All distances are measureed from the pole of the mirror. A
•
D
Distances me
easured in th
he direction of the incideent ray are p
positive and the distancees m
measured in t
the direction
n opposite to that of thee incident raays are negattive. •
D
Distances me
easured abovve the princiipal axis are positive and
d that measu
ured below tthe principal axis are negative. 34` Table Sho
owing the Siign Convention 3.4 Form
mation of Imaages by Spherical Mirro
ors Case I Co
oncave Mirro
or When an
n object is placed in front of a concave mirror, ligght rays from
m the objectt fall on the mirror an
nd get refleccted. The refflected rays produce an image at a p
point where they interseect or appear to
o intersect. Formation o
of an image b
by mirrors iss usually sho
own by consttructing ray diagramss. To constru
uct a ray diaggram, we neeed at least ttwo rays whose paths affter reflectio
on 35` from the mirror are kknown. Thesse rays mustt be chosen according to
o our conven
nience. Any ttwo of the rayys can be co
onsidered to obtain the iimage. A ray of light parallell to the princcipal axis aftter reflection
n from a con
ncave mirrorr passes thro
ough its focus.. A ray of light passingg through thee focus of a concave mirrror after refflection emeerges paralleel to the princcipal axis. 36` A ray of light passingg through thee center of ccurvature off a concave m
mirror retracces its path aafter reflection
n as the ray passing thro
ough the cen
nter of curvaature acts ass a normal to
o the sphericcal mirror. A ray of light which sstrikes the m
mirror at its p
pole gets refflected accorrding to the law of reflection
n. When thee Object is a
at Infinity 37` When an
n object is placed at infin
nity, the rayss coming from it are parallel to each
h other. Let u
us consider two rays, on
ne striking th
he mirror att its pole and
d the other p
passing throu
ugh the centter of curvatture. The rayy which is inccident at thee pole gets rreflected acccording to th
he law of reflection
n and the se
econd ray wh
hich passes tthrough the center of cu
urvature of tthe mirror retraces its path. The
ese rays afteer reflection form an imaage at the fo
ocus. The image formed is minished. real, inveerted and dim
The imagge is •
A
At F •
Real •
In
nverted •
D
Diminished When thee Object is P
Placed Beyon
nd C The two rays which aare considerred to obtain
n the image are: h the centerr of curvaturre a ray passsing through
38` a ray parrallel to the p
principal axiss. The ray paassing through the centeer of curvatu
ure retraces its path and
d the ray whiich is paralleel to the prin
ncipal axis paasses througgh the focus after reflecttion. These rayys after refle
ection meet at a point between C an
nd F. The image is invertted, real and
d diminisheed The imagge is •
Fo
ormed betw
ween C and FF •
Real •
In
nverted •
D
Diminished When thee Object is P
Placed at thee Center of C
Curvature Here we consider the
e two rays, o
one parallel to the principal axis and
d the other p
passing through o the princip
pal axis passees through tthe focus aftter the focuss. The ray off light which is parallel to
reflection
n. The otherr ray passingg through thee focus afterr reflection eemerges parrallel to the aaxis. 39` After refllection these
e rays meet at the centeer of curvatu
ure to form aan inverted iimage, which is real and of the same
e size as the o
object. The imagge is •
Fo
ormed at C •
Real •
nverted In
•
Saame size as the object When thee Object is B
Between C an
nd F Here we consider a rray of light w
which is paraallel to the principal axis and another ray passingg through tthe focus. Th
he ray which
h is parallel tto the principal axis passses through the principaal focus and
d the ray wh
hich passes tthrough the focus after rreflection em
merges parallel to the principal axis. The reflected rays meet at a p
point beyond
d C and the image is reall, inverted and magnified. 40` The imagge is •
Fo
ormed beyo
ond C •
Real •
In
nverted •
M
Magnified When thee Object is a
at the Focus Here, wee consider a ray of light w
which is paraallel to the p
principal axiss and anotheer ray passin
ng through tthe center o
of curvature.. The ray which is paralleel to the prin
ncipal axis paasses througgh the focuss and the rayy which passses through the center o
of curvature retraces its path. The reflected
d rays are parallel to each other, and
d would meeet only at inffinity i.e., thee image is formed aat infinity and it is a real,, inverted, enlarged image of the ob
bject. •
Im
mage is form
med at Infinitty •
Real •
In
nverted 41` •
M
Magnified When thee Object is B
Between the Pole and thee Focus Here we consider a rray of light w
which is paraallel to the in
ncident ray aand another ray which iss passing tthrough the center of cu
urvature. Thee ray which iis passing th
hrough the center of curvaturee retraces its path and the other rayy which is paarallel to thee principal axxis after reflection
n passes through the foccus. These raays appear tto meet behind the mirrror when thee reflected
d rays are exttended backkwards. The image is virttual, erect and magnifieed. 42` The following table ggives the possition, size and nature of the image formed in a concave mirror correspo
onding to diffferent positiions of the o
object and th
he use of concave mirror. 43` A convexx mirror alwaays gives a vvirtual imagee irrespectivee of the posiition of the o
object. 44` Case II Co
onvex Mirror The following rays arre considereed while consstructing rayy diagrams.
ng parallel to
o the princip
pal axis after reflection frrom a conveex mirror app
pear A ray of light travelin
us behind th
he mirror. to come from its focu
A ray of light travelin
ng towards the center off curvature b
behind the m
mirror hits th
he mirror at 90o and is refflected backk along its paath. 45` A convexx mirror alwaays gives a vvirtual imagee irrespectivee of the posiition of the o
object. 46` W
WEEK 4: Mirror Fo
ormula 4.0. Conccave Mirror Mirror fo
ormula is the
e relationship between o
object distan
nce (u), imagge distance ((v) and focall length. on Derivatio
The figurre shows an object AB att a distance u from the p
pole of a con
ncave mirrorr. The imagee A1B1 is fo
ormed at a d
distance v fro
om the mirro
or. The posittion of the im
mage is obtaained by drawing a ray diaggram. Considerr the ∆A1CB1 and ∆ACB 47` [when two angles of ∆A1CB1 and ∆ACB are equal then the third angle But ED = AB From equations (1) and (2) If D is very close to P then EF = PF 48` But PC = R, PB = u, PB1 = v, PF = f By sign convention PC = ‐R, PB = ‐u, PF = ‐f and PB1 = ‐v Equation (3) can be written as 49` Dividing equation (4)) throughoutt by uvf we gget n (5) gives the mirror formula Equation
4.1 Mirro
or Formula ((Convex Mirrror) Let AB bee an object p
placed on the principal aaxis of a convex mirror o
of focal lengtth f. u is the distance between the object and
d the mirror and v is the distance beetween the image and th
he mirror. 50` But DE = AB and when the aperture is very small EF = PF. Equation (2) becomes From equations (1) and (3) we get 51` [PF = f, PB1 = v, PB = u, PC = 2f] Dividing both sides of the equation (4) by uvf we get 4.2 Magnification The ratio of the height of the image to the height of the object is called the linear magnification. It is denoted by the letter m. While deriving the mirror formula it has been proved that ∆ ACB and ∆A1CB1 are similar and so also ∆FB1A1 and ∆FED are similar. 52` Where h1 is the height of the imaage and ho iss the height of the objecct We know
w that 53` For a real image u and v are negative and the magnification is negative. Negative magnification means the image is inverted. On the other hand for a virtual image u is negative and v is positive and hence the magnification is positive, i.e., the image is erect. 54` WEEK 5: RE
W
EFRACTIO
ON AT CU
URVED SU
URFACES 5.1 Lense
es A lens is a portion off a transpareent refractingg medium bounded by ttwo surfacess which are generallyy spherical o
or cylindrical or one curvved and one plane surfacce. Basically,, the lenses are classified
d as •
•
co
onvex lens o
or converging lens co
oncave or diiverging lenss Convex Lens A lens wh
hich is thicke
er in the mid
ddle and thin
nner at the eedges is calleed a convex lens. In a convvex lens at le
east one of itts surfaces iss bulging outt at the midd
dle. According to their shapes th
he convex le
enses are claassified as •
•
•
bi‐convex or double convvex onvexo – plaano co
co
onvexo ‐ con
ncave Concave Lens hich is thinner at the middle and thicker at the eedges is calleed a concavee lens. A lens wh
Like convvex lenses th
hese lenses aare also classified as •
•
•
bi‐concave co
oncave – plaano co
oncavo – convex 55` Optical C
Center It is the ccenter of a le
ens. It is den
noted by the letter O. A rray of light p
passing through the optiical center off a lens doess not suffer aany deviation. It is also rreferred to aas optic centter. Principal Axis nters of curvvatures of th
he two curveed surfaces o
of a lens. Is the straight line joining the cen
Principal Foci Rays of liight can passs through th
he lens in anyy direction aand hence th
here will be ttwo principaal foci on either side off the lens and they are reeferred to ass the first prrincipal focus and the seecond principal focus of a le
ens. First Prin
ncipal Focus (F1) 56` It is a poiint on the prrincipal axis of the lens ssuch that thee rays of ligh
ht starting frrom it (conveex lens) or aappearing to
o meet at thee point (concave lens) affter refractio
on from the two surfacees of the lens become parallel to the p
principal axiss of the lenss. First Prin
ncipal Focus of a Convex Lens First Prin
ncipal Focus o
of a Concavee Lens The distaance from th
he optic centter to the first focus is caalled the firsst focal lengtth (f1) of thee lens. Principal Focu
us (F2) Second P
It is a poiint on the prrincipal axis of the lens ssuch that thee rays of ligh
ht parallel to
o the principal axis of th
he lens after refraction from both th
he surfaces o
of the lens paass through this point (convex llens) or appe
ear to be coming from this point. 57` Seco
ond Principaal Focus of a convex lenss Seco
ond Principall Focus of a cconcave lens The distaance from th
he optic centter to the second princip
pal focus is ccalled the second focal length (f2) of the lenss. If the meedium on both sides of the lens is same then thee first and th
he second fo
ocal lengths w
will be equal. Focus of a convex lens is real wherreas that of tthe concavee lens is virtu
ual. mation of Im
mage by a Co
onvex Lens 5. 2 Form
When an
n object is placed in front of a lens, light rays com
ming from th
he object falll on the lenss and get refracted. The re
efracted rayss produce an
n image at a point wheree they interssect or appeaar to intersectt each other.. The formattion of imagees by lenses is usually sh
hown by a raay diagram. TTo constructt a ray diagrram we need
d atleast two
o rays whosee path after refraction th
hrough the lens is known. Any two off the following rays are u
usually conssidered for constructing ray diagram
ms. A ray of light passingg through thee optical cen
nter of the leens travels straight without suffering any deviaation. This holds good only in the caase of a thin lens. 58` An incideent ray paralllel to the prrincipal axis aafter refracttion passes tthrough the focus. An incideent ray passiing through the focus off a lens emerrge parallel tto the princiipal axis afteer refraction. ure of imagess formed by a convex lens depends upon the disstance of the object from
m The natu
the opticcal center of the lens. Let us now seee how the im
mage is form
med by a convex lens for various p
positions of tthe object. 59` When thee Object is P
Placed Betweeen F1 and O
O Formatio
on of Image by a Convexx Lens The imagge is – •
•
•
•
Fo
ormed on th
he same sidee of the lens
V
Virtual Erect M
Magnified When thee Object is P
Placed Betweeen the Optiical Center (O
O) and First Focus (F1) Here we consider tw
wo rays startiing from thee top of the o
object placed at F1 and o
optical centeer. The ray p
parallel to th
he principal aaxis after reffraction passses through the focus (FF2). The ray passing tthrough the optical center goes thro
ough the lens undeviated
d. These refrracted rays appear to
o meet only when produ
uced backwaards. Thus, w
when an object is placed
d between F1 and O of a co
onvex lens, a virtual, erecct and magn
nified image of the objecct is formed on the samee side of th
he lens as the object. When the Object is p
placed at 2F1 1
The image is –
– •
fo
ormed at 2F2 2
60` •
•
•
reeal In
nverted Saame size as the object Here onee of the rays starting from the top off the object placed at 2FF1 passes thrrough the op
ptic center w
without any d
deviation and
d the other ray which is parallel to tthe principall axis after refraction passes thrrough the focus. These two refracted
d rays meet at 2F2. Thuss, when an object is placed at 2FF1 of a conveex lens, inverted and reaal image of the same sizee as the objeect is formed aat 2F2 on the
e other side o
of the lens.
When the Object is P
Placed Betweeen F1 and FF2 The imagge is •
•
•
•
fo
ormed beyond 2F2 reeal in
nverted m
magnified Let us co
onsider two rrays coming from the ob
bject. The ray which is paarallel to thee principal axis after refrraction passe
es through tthe lens and passes thro
ough F2 on th
he other sidee of the lens. The ray passing through tthe optic center comes out of the leens without any deviatio
on. The two refracted
d rays interse
ect each oth
her at a point beyond 2FF2. So, when an object is placed betw
ween F1 and 2FF1 of a conve
ex lens the im
mage is form
med beyond 2F2. When thee Object is P
Placed at F1 61` The imagge is – •
•
•
•
Fo
ormed at inffinity Real In
nverted M
Magnified Here agaain we consid
der two rayss coming from the top off the object.. One of the rays which iis parallel tto the princip
pal axis after refraction passes throu
ugh F2 and the other rayy which passses through tthe optical ccenter comees out withou
ut any deviation. These two refracteed rays are parallel tto each othe
er and paralleel rays meett only at infin
nity. Thus, w
when an objeect is placed at F1 of a convvex lens, the image is forrmed at infin
nity and it is inverted, reeal and magn
nified. When thee Object is P
Placed Beyon
nd 2F1 The imagge is – •
•
•
•
fo
ormed between F2 and 2
2F2 reeal In
nverted D
Diminished 62` The ray p
parallel to th
he principal aaxis after reffraction passses through F2 and the rray which paasses through tthe optical ccenter comees out withou
ut any deviation. The refracted rays intersect att a point bettween F2 and
d 2F2. The im
mage is inverrted, real an
nd diminisheed. When thee Object is p
placed at Infiinity The imagge is – •
•
•
•
fo
ormed at F2 in
nverted reeal highly diminisshed When the object is at infinity, the rays comin
ng from it arre parallel to
o each otherr. Let one of the he other rayy pass througgh the opticaal center. Th
he parallel rrays pass thrrough the focus F1 and th
ray which
h passes through F1 beco
omes paralleel to the prin
ncipal axis after refractio
on and the rray which paasses through the optical center doees not suffer any deviatio
on. The tablee below give
es at a glancee the positio
on, size and n
nature of the image form
med by a convex lens corresponding to the differeent positionss of the objeect and also its applicatio
on. 63` 5.4 Form
mation of Imaage by a Con
ncave Lens
The following rays arre considereed while consstructing rayy diagrams ffor locating tthe images formed b
by a concave
e lens for thee various possition of thee object. An incideent ray of ligght coming frrom the objeect parallel tto the principal axis of a concave len
ns after refrraction appe
ears to comee from its foccus. An incideent ray of ligght passing through the o
optical centeer comes ou
ut of the lenss without an
ny deviation
n. 64` A concavve lens alwayys gives a virrtual, erect aand diminish
hed image w
whatever mayy be the possition of the ob
bject. Let us no
ow draw ray diagrams to
o show the p
position of th
he images w
when the objeect is placed
d •
•
•
att infinity and
d between O an
nd F1 and any position b
between inffinity and O.
When thee Object is a
at Infinity The imagge is – •
•
•
•
fo
ormed at F1 erect viirtual diminished placed betweeen O and F1 1
When thee Object is p
65` The imagge is – •
•
•
•
fo
ormed between O and FF1 erect viirtual diminished When thee Object is p
placed at anyy Position beetween O an
nd infinity The imagge is – •
•
•
•
fo
ormed between O and FF1 erect viirtual diminished Uses of cconcave lens It is used •
•
n spectacles for correctin
ng myopia.
in
along with co
onvex lens it is used to overcome deefects like chromatic abeerration and sp
pherical abe
erration (the failure of raays to converge at one fo
ocus becausse of a defecct in a leens or mirror). 5.5 Sign C
Convention for Lenses 66` Followingg sign conve
ention is used for measuring various distances during the formation of images b
by lenses: All distan
nces on the p
principal axis are measu
ured from the optical cen
nter. •
•
The distancess measured in the directtion of incideent rays are positive and
d all the he direction o
opposite to that of the incident rayss are negativve. distances measured in th
A
All distances measured above the priincipal axis aare positive. Thus, heigh
ht of an object and that of an
n erect imagge are positivve and all distances meaasured below
w the princip
pal axxis are negative. The following table ggives the sign
n convention
n for lenses:: 67` 68` W
WEEK 6: LENSES C
CONTD 6.1 Lens Formula The relattionship betw
ween distance of the ob
bject (u), disttance of the image (v) an
nd focal lenggth (f) of the lens is calle
ed lens formu
ula or lens eequation. This lens formula is aapplicable to
o both conveex and concaave lenses.
ns Formula ((Convex Len
ns) 6.2 Derivvation of Len
Let AB reepresent an object placeed at right an
ngles to the principal axiis at a distan
nce greater tthan 1 1
the focal length f of tthe convex lens. The imaage A B is formed beyo
ond 2F2 and is real and inverted.. OA = Objject distance
e = u OA1 = Image distance
e = v OF2 = Foccal length = f O
OAB and OA
O 1B1 are sim
milar 69` But we know that OC = AB the above equation can be written as From equation (1) and (2), we get Dividing equation (3) throughout by uvf 70` 6.3 Derivvation of Len
ns Formula ((Concave Le
ens) Let AB reepresent an object placeed at right an
ngles to the principal axiis at a distan
nce greater tthan 1 1
the focal length f of tthe convex lens. The imaage A B is formed betw
ween O and FF1 on the sam
me he object is kkept and thee image is errect and virtual. side as th
OF1 = Foccal length = f OA = Objject distance
e = u OA1 = Image distance
e = v m the ray diaggram we seee that OC = A
AB But from
71` From equation (1) and equation (2), we get Dividing throughout by uvf 6.4 Magnification Magnification is the ratio of the size of the image (hI) to the size of the object (ho) Magnification produced by a lens can be equal to one, greater than one or less than one depending upon the size and nature of the image. Case I When, height of the image (hI) = height of the object (ho) Thus, when the magnification is one, the size of the image is equal to the size of the object. Case II 72` When hI > ho Case III For both type of lensses, the heigght of the ob
bject is alwayys positive, w
while the height of the upon its natu
ure. image may be + or ‐ depending u
As per siggn conventio
on for lensess, the heightt of an inverted and reall image is neegative and hence the magnificattion of a lens is negativee when it pro
oduces an in
nverted and real image. For an erect and virtual iimage, the h
height of thee image is po
ositive. So, th
he magnificaation is posittive when an erect and virtual image is formed.
6.5 Thin Lenses Place
ed in Contacct Leet two thin lenses L1 and L2 of focall lengths f1 aand f2 be plaaced in conttact so as to have a commo
on principal axis. It is req
quired to find the effectiive focal length of this combination.. Let O be a po
oint object o
on the princiipal axis. Thee refractionss through the two lensess are consideered separately and the re
esults are co
ombined. Wh
hile dealing with the ind
dividual lensees, the distances are to bee measured ffrom the resspective optic centers; since the lensses are thin,, these distances can also be measured from the ccenter of thee lens system
m (point of ccontact in the case of tw
wo m. Assumingg that the len
ns L1 lenses). LLet u be the distance of O from the ccenter of thee lens system
alone pro
oduces the rrefraction. Leet the imagee be formed at I at a disttance v. Writting the lenss equation
n in this case
e, we get 73` The imagge I' due to tthe first lens acts on the virtual object for the seecond lens. LLet the final image bee formed at I, at a distan
nce v from th
he center of the lens system. Writingg the lens equation
n in this case
e, we get, Adding eequations (i) and (ii) we gget Let the tw
wo lenses be
e replaced b
by a single lens which can produce th
he same effeect as the tw
wo lenses pu
ut together p
produce, i.e.., for an objeect O placed
d at a distancce u from it, the image I must be formed at a distance v. Such a lens is called an eequivalent leens and its ffocal length iis called thee equivalentt focal length
h. Writing th
he lens equaation in this ccase, we gett
Comparin
ng equations (iii) and (ivv) we get Hence, w
when thin len
nses are com
mbined, the reciprocal of their effecttive focal len
ngth will be equal to the sum of tthe reciprocals of the individual focaal lengths. Since thee reciprocal o
of focal lenggth represen
nts the poweer the above equation, in
n terms of power, m
may be written as 74` P = P1 + P2 Therefore, the power of a combination of thin lenses is equal to the algebraic sum of the powers of the individual lenses. 75` WEEK 7: OPTIICAL INSTTRUMENTTS 7.1 The H
Human Eye Our eye iis the most iimportant naatural opticaal instrumen
nt. The eye iss nearly spheerical in shape with a slight bulge in
n the front paart. The Impo
ortant Parts of the Eye aand their Fun
nctions Cornea The frontt part of the eye is coverred by a tran
nsparent sph
herical mem
mbrane called
d the corneaa. Light entters the eye through cornea. The spaace behind tthe cornea iss filled with a liquid calleed aqueous humor. Iris Just behind the cornea is a dark coloured mu
uscular diaphragm which has a small circular opening in the middlle. Pupil Pupil is th
he small circcular opening of iris. Thee pupil appears black because no light is reflecteed from it. TThe iris regulates the am
mount of ligh
ht entering th
he eye by ad
djusting the size of the p
pupil. 76` Let us see how iris regulates the amount of light entering the eye. When the intensity of light is more or if it is a bright source of light then the iris makes the pupil to contract and as a result the amount of light entering the eye decreases. When the intensity of light is less or if the light is dim then the iris dilates the pupil so that more light can enter the eye. Eye Lens The eye lens is a convex lens made of a transparent jelly ‐ like proteinaceous material. The eye lens is hard at the middle and gradually becomes soft towards the outer edges. The eye lens is held in position by ciliary muscles. The ciliary muscles help in changing the curvature and focal length of the eye ‐ lens. Retina The inner back surface of the eye ball is called retina. It is a semi‐transparent membrane which is light sensitive and is equivalent to the screen of a camera. The light sensitive receptors of the retina are called rods and cones. When light falls on these receptors they send electrical signals to the brain through the optic nerve. The space between the retina and eye lens is filled with another fluid called vitreous humor. Blind Spot It is a spot at which the optic nerve enters the eye and is insensitive to light and hence the name. 7.2 Working of an Eye The light coming from an object enters the eye through cornea and pupil. The eye lens converges these light rays to form a real, inverted and diminished image on the retina. The light sensitive cells of the retina gets activated with the incidence of light and generate electric signals. These electric signals are sent to the brain by the optic nerves and the brain interprets the electrical signals in such away that we see an image which is erect and of the same size as the object. Before we go into the defects of vision let us be familiar with the terminology used by the ophthalmologists like least distance of distinct vision, far point and power of accommodation of the eye. Far point It is the farthest point up to which an eye can see clearly. For a normal eye, the far point is at infinity. Power of Accommodation 77` A normal eye can see
e both the distant and th
he nearby objects clearly. In the casse of the eyee, the image disstance (v) is fixed as thee distance beetween the eeye lens and
d retina remaains the sam
me but the o
object distan
nce (u) variess. The eye fo
ocuses the im
mages of all the objects,, distant or nearby, aat the same place on thee retina by changing the focal length
h of its lens. The eye lens changes its focal lenggth by changging its thickkness with th
he help of itss ciliary musscles. The ab
bility of the eyye lens to chaange its focaal length to ffocus the im
mages of all the objects, d
distant or neearby on the reetina is know
wn as the power of accommodation.. Whenever the eye is fo
ocused on a distant o
object, the ciliary musclees are relaxed and the ciliary muscles are tensed
d when the eeye is focused o
on a nearby object. Least Disstance of Disstinct Vision The minimum distance up to which an eye can see clearrly is called the least disttance of disttinct he least distaance of distinct vision is 25 cm for a normal eyee and for infaants it is 5 to
o 8 vision. Th
cm. Range off Vision The rangge of distance over which
h the eye can see clearlyy is called itss range of vission. The ran
nge of vision of a normal healthy eyee is from infinity to 25cm
m from the eeye. The term
ms like poweer of odation, focusing, least distance of distinct visio
on tell us thaat the eye is similar to a accommo
photograaphic cameraa. Similaritiies between the Human Eye and Cam
mera Dissimila
arities betweeen the Human Eye and C
Camera 78` 7.3 Defeccts of Vision
n A normal eye can see
e all objects over a widee range of disstances i.e., from 25 cm to infinity. B
But due to ceertain abnorrmalities the eye is not aable see objeects over succh a wide range of distan
nces and such
h an eye is saaid to be deffective. Some of the defeects of vision
n are •
•
•
Hypermetrop
H
pia or long sightedness M
Myopia or sh
ort sightedn
ness and A
Astigmatism Hypermeetropia You mustt have seen middle aged
d people holding a book away from their eyes to
o read propeerly. This is beecause they are not ablee to see the n
nearby objects clearly. W
We say that those people are sufferingg from hyperrmetropia (lo
ong sightedn
ness). Hyperrmetropia orr hyperopia iis the defectt of the eye d
due to which
h the eye is n
not able to ssee clearly th
he nearby ob
bjects thouggh it can see the distant o
objects clearlly. Causes off Hypermetrropia Hypermeetropia is cau
used due to the followin
ng reasons:
•
•
Shortening off the eyeball, that is, thee eyeball beccomes smalller. In
ncrease in fo
ocal length o
of the eye len
ns. 79` Focussingg of light rayys in normal eye and lon
ng sighted eyye Let us no
ow see how tthis defect iss rectified. A
A long sighteed eye formss image of a nearby objeect behind th
he retina. Th
hus, long sightedness is d
due to the d
decreased co
onverging po
ower of the llens. Therefore hypermetropia can bee rectified byy making thee eye lens more converggent. This is d
done ng a convex lens of suitab
ble focal len
ngth before tthe eye lens as shown in
n the figure.
by placin
Myopia You mustt have seen some people holding bo
ooks very clo
ose to their eeyes. This is because theey sufferingg from myopia (short sightedness). A
A myopic perrson cannot see distant objects cleaarly 80` because the far point of his eye iis less than infinity. Myo
opia is the deefect of the eye due to which the eye is not able to see tthe distant o
objects clearrly. Myopia is due to: •
•
th
he elongatio
on of the eyee ball, that iss, the distancce between the retina and eye lens is in
ncreased. decrease in fo
ocal length o
of the eye leens. Normal Eye ‐ The im
mage is formed on the reetina Eye B
Ball Too Long A myopicc eye forms tthe image of a far off ob
bject in frontt of the retin
na because o
of the increaase in converging power off the eye lens. Thereforee myopia can
n be rectified
d by using a suitable divergent or concave
e lens. 81` Correection of myopia using a concave len
ns Astigmattism At times the eye is not able to fo
ocus the light coming fro
om the horizontal and veertical planees. As will not be the same. Such a defect of the a result, the horizonttal and vertical views of an object w
eye is callled astigmatism. Astigm
matism occurrs when the eye‐lens or the cornea or both are not perfectlyy spherical in
n shape. In th
his case the image of a d
distant pointt object appears as a linee. Astigmattism is rectified using cyllindrical lensses. 82` WEEK 8: W
OPTICALL INSTRUM
MENT (CO
ONTD) 8.1Simplle Microscop
pe A microscope is an optical devicee which prod
duces a high
hly magnified
d image of very small ob
bject such as m
micro‐organiisms. Based on the desiggn, there aree two types o
of microscop
pes. They arre, simple m
microscope and compoun
nd microscopes. A simplle microscop
pe is nothingg but a singlee biconvexx lens. It is re
eferred to ass magnifyingg glass. Usually the focal lenggth of the co
onvex lens iss around 2.5 cm. The objject to be viewed througgh a simple m
microscope iss placed betw
ween the op
ptic center and the focuss and the im
mage is erect,, virtual an
nd magnified
d. Magnifyiing Power off a Simple M
Microscope The magnifying powe
er or angular magnificattion of a miccroscope may be defined
d as the ratio
o of the anglee subtended at the eye b
by the imagee formed at the distancee of the distiinct vision to
o the angle sub
btended by tthe object w
when placed at the distance of the distinct vision
n. The ray d
diagram show
ws that the image of thee object AB is formed att A1B1. A1B1 is formed at the least disttance of distinct vision. The figurre shows thaat the angle A
A1OB1 subteended at thee eye by the object in thee position A1B1 is greater than the anggle AOB subtended by it in the positiion AB. From
m this it is cleear that the eye estimates the angle ssubtended b
by an object on it and no
ot the linear size of the o
object. 83` But OB1 = Least distance of distinct vision from the lens or eye = D OB = u = distance between the lens and the object The distance between the image and the lens is negative as the image is virtual. The lens formula for a convex lens is Where f is the focal length of the lens Multiplying both sides of the equation (1) by v we get 84` D = 25 cm From equation (3) it is clear that a convex lens of short focal length has a large magnifying power. The highest magnification which can be obtained from a simple microscope is about 20. Uses of simple Microscope A simple microscope is used as a magnifying glass. It cannot be used to observe very tiny objects like bacteria and cells because of its low magnification. 8.2 Compound Microscope A compound microscope is an optical instrument which is used to magnify very small objects like blood cells, bacteria which otherwise cannot be seen with the naked eye. The essential parts of a compound microscope are two convex lenses of short focal length. These lenses are referred to as: •
•
the objective lens or objective the eye piece or lens 8.2.1 Working Principle of a Compound Microscope A compound microscope consists of the following parts: Objective Lens 85` The objective lens off a compoun
nd microscop
pe is a conveex lens of very short focaal length (fo) that )
is fo < 1cm
m. The objecct to be seen
n is kept very close to th
he objective lens. Eye piecee The eye p
piece of a co
ompound miicroscope is also a conveex lens of short focal len
ngth fe. But ffe > fo. Microsco
ope tube The objective lens an
nd the eyepiece are mou
unted coaxiaally (having aa common axis) at the ends which can be made to slid
de into each
h other so th
hat the distan
nce between
n the of two brrass tubes w
two lensees can be ad
djusted. The ray d
diagram give
en below gives the principle of a com
mpound microscope. The object is mounted
d on the stan
nd below thee microscope tube. The objective lens forms a real, inverted
d and magnified image (I1) of the objecct. The image I1 acts as aan object forr the eye pieece. The posiition of the eyyepiece is so adjusted that the imagee lies within the focus off the eyepiece (Fe). The eyepiecee acts like a m
magnifying gglass and forrms a virtual erect and m
magnified im
mage of the object. •
•
•
Image
e Formation in a Compound Microsccope The object (O
O) is placed just outside FFo, the princcipal focus off the objective lens. Fe is the princcipal focus o
of the eye len
ns. A
A real, inverte
ed magnified
d image I1 is formed. Thee magnified image I1 actts as an objeect fo
or the eye le
ens. 86` •
The final imagge I2 is virtuaal and is maggnified still ffurther. It is inverted com
mpared with
h the object. I2 mayy appear 100
00 times largger than the object. Magnifyiing Power off a Compoun
nd Microscop
pe The magnifying powe
er of a comp
pound micro
oscope is deffined as the ratio of the size of the ffinal image (I2) as seen through the m
microscope to
o the size of f the object aas seen with
h a naked eyee. Im
mage Formation in a Com
mpound Miccroscope = mobjectivve x meyepiece = mo x me mobjective (mo) and meyepiecce (me) are th
he magnificaation producced by the ob
bjective and Where m
eyepiecee respectively. m = mo xx me Eye piecee is nothing but a simplee microscopee 87` The lens formula is But distance between the object and the lens is ‐u. Multiply equation (2) by V m = mo x me 88` WEEK 9:: OPTICALL INSTRU
UMENT CO
ONTD 9.0 TELESSCOPE Telescop
pe is an opticcal instrument which is u
used for view
wing heaven
nly bodies an
nd distant objects. TThere are baasically two ttypes of teleescopes: •
•
reefracting telescope reeflecting tele
escope Refractin
ng telescope uses a combination of llenses to forrm the imagees of distantt objects. Astronom
mical telesco
ope devised by Kepler an
nd terrestriaal telescope devised by G
Galileo are exampless for refractiing telescopes. Reflecting telescope uses a comb
bination of aa lens and a concave mirrror to obtain the imagees of distant o
objects. 9.1 Astro
onomical Telescope This typee of telescop
pe is used to view heavenly bodies like stars, planets and sattellites. It consists o
of two conve
ex lenses called objectivve and eyepiiece. The objjective is of large focal length whereas the e
eyepiece is o
of short focaal length. Thee distance between the two lenses ccan be which hold
ds the lens.
be adjustted by adjussting the tub
The ray d
diagram show
wing the principle of thee astronomical telescope is given beelow. Astrono
omical Telesccope 89` The rays of light coming from a distant object (PQ) form a parallel beam of light. This parallel beam of light is focused by the objective in a plane passing through its focus and perpendicular to the axis and forms the image (PlQl). This plane is known as focal plane. The eyepiece is adjusted so that the image PlQl lies in its focal plane. The light beam after striking the eye lens emerges parallel and final image PllQll is formed at infinity. This adjustment of the telescope is known as normal adjustment. Magnifying Power of an Astronomical Telescope Magnifying power of an astronomical telescope may be defined as the ratio of the angle subtended at the eye by the image to the angle subtended at the eye by the object. From the ray diagram we know that: PlC is the focal length of the objective and PlD is the focal length of the eye piece. Distinction between a Compound Microscope and an Astronomical Telescope 90` 91` WEEK 10
0: OPTICA
AL INSTRU
UMENT C
CONTD 10.1 Terrrestrial Telescope From thee figure, it caan be seen tthat the top
p point of th
he distant ob
bject is abovve the axis o
of the lens, butt the top point of the fin
nal image iss below the axis. Thus th
he image in an astronomical telescopee is inverte
ed. This insstrument is suitable fo
or astronom
my becausee it makes little differencce if a star, ffor example,, is inverted,, but it is useeless for viewing objects on the earrth or sea, in which case an
n erect image is required
d. pe provides an erect im
mage. In add
dition to thee objective and a eye‐piece of A terresttrial telescop
the astro
onomical tele
escope, it haas a convergging lens L off focal length/between them, L is placed at a distaance 2/in fro
ont of the in
nverted real image Ij form
med by the objective, in
n which casee, the image I in L of I, (i) is inverted, real, and th
he same sizee as I,, (ii) is also at a diistance 2/fro
om L. me way up as the distant object. Iff I is at the ffocus of thee eye‐
Thus thee image 1 is now the sam
piece, the final image
e is formed aat infinity an
nd is also ereect. The lens L is often known as thee "erecting" lens of the telescope, aas its only fu
unction is th
hat of invertingg the image Ij. Since the image I pro
oduced by L is the samee size as Ij, the presencee of L does nott affect the magnitude of the anggular magniffication of the telescop
pe, which is thus f1
he erecting lens, however, reduces the intensityy of the light emerging through thee eye‐
f 2 . Th
piece, as light is refle
ected at the lens surfacees. Yet anoth
her disadvan
ntage is the increased leength of the teelescope wh
hen L is useed; the disttance from the objectivve to the eye‐piece e
is now e astronomical telescop
pe. f 1 + f 2 + 4 f . Compared with f 1 + f 2 in the
GALILEO'S TELESCOP
PE About 16
610, with ch
haracteristic genius, Galileo designeed a telesco
ope which provides p
an erect image off an object with the aid a of only two t
lenses. The Galileaan telescope consists of o an objectivee which is a converging lens of longg focal length, and an eyye‐piece which is a diveerging lens of short s
focal length. The distance beetween the lenses is eq
qual to the difference d
in
n the magnitud
des of their ffocal lengthss, i.e., QCj =
92` /, — /j, w
where /„ /£ aare the focal lengths of the objectivve and eye‐p
piece respectively. Fig. 2
23.13. The imagge of the disstant object in the objecctive 1^ wou
uld be formeed atlj, where CJj = f\, in the absence of the diverrging lens Ls;; but since LL., is at a disttance/,, from
m 11% the raays falling on the merge paralllel. It will no
ow be noted from eye‐piecee are refractted through this lens so that they em
Fig. 23.13 that an ob
bserver seess the top po
oint of the final image aabove the axxis of the leenses, and hencce the image
e is through tthe eye‐piecce L2. The lo
op point of th
he image formed at infinity is thus aa virtual imaage in L‐ of the virtual object P. But a raay C:P throu
ugh the midd
dle of Lj passses straight tthrough the lens, and this will also be a ray which passes thro
ough the top
p point of th
he image at infinity. Thu
us the hown emergging from the eye‐piece in Fig. 23.13
3 are paralleel to the linee PC,. three parallel rays sh
placed close to the diverrging lens, th
he angle a' ssubtended aat it by the im
mage Hence if the eye is p
at infinityy is angle I]C
C£P. The angle a subten
nded at thee eye by the distant object o
is praactically equ
ual to the angle a
subtendeed at the ob
bjective. Figg. 23.13. Now v = /: Cjlj = />,/;, wh
here/! is thee objective focal length an
nd h is the le
ength IjP; and o' = /j/CjIi = hf,. ulnr magnificcation, an ob
bjective of lo
ong focal length (/j) and
d an eye‐pieece of Thus for htrh Rn.sr.u
1. short foccal length (/j are required, as in the ccase of the aastronomical telescope ((see p. 5341
Advantagge and Disad
dvantage of Galilean Telescope. Opeera Glasses
The distaance CjCs be
etween the o
objective and
d the eye‐piece in the G
Galilean telesscope is (/, —
— /.); the distaance betwee
en the same lenses in th
he terrestrial telescope is (/j ‐ /2 + 4
4/1. p. 537. Thus the Galillean telesco
ope is a mu
uch shorter instrumentt than the terrestrial t
telescope, and is thereforee used for op
pera glassess. As alread
dy explained
d (p. 532), the eye‐ring is the imagge of the objjective in th
he eye‐piecee. But the eye‐p
piece is a diverging lens. Thus the eye‐ring is virtual, and correspond
ds to M. which is between
n L {and Lj (FFig. 23.13). SSince it is im
mpossible to place the eyye at M, thee best positio
on of 93` the eye iin the circum
mstances is as close as possible to the eye‐pieece L‐. and cconsequently the field of vview of the G
Galilean telescope is veryy limited compared with
h that of thee astronomiccal or terrestriaal telescope.. This is a dissadvantage o
of the Galilean telescopee. Final Imaage at Near P
Point The final image in a G
Galilean teleescope can aalso be vieweed at the near poin
nt of the eye
e, when the telescope is not in norm
mal adjustmeent. Fig. 23.14
4 illustrates the form;' o
on of the ereect image in this ni^c. Th
he ;nce C.l) is now more than focal lenggth/t of the eye‐piece; aand Optical in
nstruments Since C2I2 = D, the least distancee of distinct vvision, we haave v = D (th
he image in L21 is virtual) and 1 1 1
+ = , we obtain. f2 is negaative since v u f
1
1
1
+ =
− D u − f2
Assumingg f2 is the nu
umerical valu
ue of the divverging lens focal length, from which
h u = With thee usual notattion, the an guilar magnification M == − f2 D
D − f2
a'
notatio
on But a’ = h//u,a=h/f1. th
hus m a
= f/u. But ∴ M
M = u = − f2 D
D − f2
f1 ⎛ f 2
⎞
− ⎟ ⎜ −1
f2 ⎝ D
⎠
Measure
ement of maagnifying power of telesscope and m
microscope
Method I: The magn
nifying poweer of a telesccope can bee measured b
by placing a well‐illumin
nated large scaale S at one end of the laboratory, and viewing it through
h the telesco
ope at the other o
94` end. 1 f the telescope consists of converging lenses O. E acting as objective and eye‐piece respectively, the distance. 95` WEEK 11
1: OPTICA
AL INSTRU
UMENT C
CONTD 11.1 Spectrometer An opticaal instrumen
nt, which is u
used for observing pure spectra of sources of ligght in the laboratorry is called SSpectrometeer. Main partts of the speectrometer aare: 1) Collim
mator 2) Prism table 3) TTelescope Descriptiion The collim
mator rende
ers a parallell beam of ligght through tthe two coaxxial cylindriccal tubes. On
ne end of th
he collimatorr has a slit th
hrough whicch light enterrs the tube aand falls on lens L situated at the other end. Prism table is a circular plate fixed over aa vertical stand of adjusttable height.. The free end of stand con
nsists of a circular scale graduated in degrees frrom 0o to 360o along witth o read the po
osition of thee prism. Teleescope is meeant for observing the verniers to enable to
spectrum
m and is mou
unted horizo
ontally on a vvertical stand attached tto the circulaar scale. Thee telescopee can be rotated about tthe prism table. Adjustmeents The telescope
e is turned towards a disstant object and is focussed to see a clear image of object. Itt is then brou
ught in line w
with the collimator. A clear image o
of the slit is o
obtained by adjustingg the screws in the collim
mator. The p
prism is kept over the prism table. 11.2 Determination of Angle of a Prism m is placed o
over the tablle such that parallel rayss from collim
mator falls on
n the sides A
AB The prism
and AC. M
Move the telescope in th
he position TT1 to catch the brightestt image of th
he slit formeed by reflection
n of light at ffaces AB and
d AC. The cro
oss wire is m
made to coin
ncide with im
mage and reaading on the circular scale is noted. The telescope is turned to
o position T2 and the sam
me procedurre is 96` repeated
d. If θ is the d
difference between the two readinggs through w
which the tellescope is tu
urned then D
Determinatio
on of angle o
of prism 11.3 Determination of Angle of Minimum D
Deviation To determ
mine the angle of minim
mum deviatio
on the side A
AB of the priism is made to face the ray of light. O
On looking through the fface AC and rotating thee prism tablee, the imagee of slit also turns. Fo
or a particulaar position of the prism, the slit beco
omes station
nary. On furtther rotatingg the prism tab
ble, image of slit turns in
n the opposite direction. Fix the prissm when thee image of th
he slit is stattionary. Thiss is the posittion of minim
mum deviation. Coincidee the cross w
wires of the telescopee in this position and no
ote the readiing. Removee the prism aand catch thee direct ray and once agaain note the reading. Thee difference between the two readin
ngs gives thee angle of minimum
m position. 97` Knowing δm and A, tthe refractivve index (µ) o
of the materrial of the prrism can be ccalculated using the prism
m formula 98` WEEK 12: PHOTOMETRY 12.1 Introduction Photometry is the science of measuring visible light in units that are weighted according to the sensitivity of the human eye. It is a quantitative science based on a statistical model of the human visual response to light ‐‐ that is, our perception of light ‐‐ under carefully controlled conditions. The foundations of photometry were laid in 1729 by Pierre Bouguer. In his L’Essai d’Optique, Bouguer discussed photometric principles in terms of the convenient light source of his time: a wax candle. This became the basis of the point source concept in photometric theory. Wax candles were used as national light source standards in the 18th and 19th centuries. England, for example, used spermaceti (a wax derived from sperm whale oil). These were replaced in 1909 by an international standard based on a group of carbon filament vacuum lamps and again in 1948 by a crucible containing liquid platinum at its freezing point. Today the international standard is a theoretical point source that has a luminous intensity of one candela 12 (the Latin word for “candle”). It emits monochromatic radiation with a frequency of 540 x 10
Hertz (or approximately 555 nm, corresponding with the wavelength of maximum photopic luminous efficiency) and has a radiant intensity (in the direction of measurement) of 1/683 watts per steradian. Together with the CIE photometric curve, the candela provides the weighting factor needed to convert between radiometric and photometric measurements. Consider, for example, a monochromatic point source with a wavelength of 510 nm and a radiant intensity of 1/683 watts per steradian. The photopic luminous efficiency at 510 nm is 0.503. The source therefore has a luminous intensity of 0.503 candela. The human visual system is a marvelously complex and highly nonlinear detector of electromagnetic radiation with wavelengths ranging from 380 to 770 nanometers (nm). We see light of different wavelengths as a continuum of colors ranging through the visible spectrum: 650 nm is red, 540 nm is green, 450 nm is blue, and so on. The sensitivity of the human eye to light varies with wavelength. A light source with a radiance 2
of one watt/m ‐steradian of green light, for example, appears much brighter than the same 2
source with a radiance of one watt/m ‐steradian of red or blue light. In photometry, we do not measure watts of radiant energy. Rather, we attempt to measure the subjective impression produced by stimulating the human eye‐brain visual system with radiant energy. This task is complicated immensely by the eye’s nonlinear response to light. It varies not only with wavelength but also with the amount of radiant flux, whether the light is constant or flickering, the spatial complexity of the scene being perceived, the adaptation of the iris and retina, the psychological and physiological state of the observer, and a host of other variables. 99` Nevertheless, the subjective impression of seeing can be quantified for “normal” viewing conditions. In 1924, the Commission Internationale d’Eclairage (International Commission on Illumination, or CIE) asked over one hundred observers to visually match the “brightness” of monochromatic light sources with different wavelengths The sensitivity of the human eye to light varies with wavelength. A light source with a a given radiance of green light, for example, appears much brighter than the same source with the same radiance of red or blue light. To convert radiometric and photometric units, the eye's response is approximated by the CIE weighting function. Photometric theory does not address how we perceive colors. The light being measured can be monochromatic or a combination or continuum of wavelengths 12.2 Basic Illumination Concept (a) Luminous flux, F •
•
•
•
•
Luminous flux is the rate at which light energy flows from the source. It is measured in 'lumen', abbreviated as lm The units 'lumen' and 'watt' have the same dimension as they both represent energy per second. However, it is incorrect to convert photometric quantities directly into energy quantities or vice‐versa, as the luminous effect of radiant energy depends on its wavelength. A lumen is the amount of energy per second or power. Hence there must be a relation between lumen and watt, the mechanical unit of power. Experiment shows that 621 lumens of a green light of wavelength 5.540 X 10‐10m is equivalent to 1 watt. (b) Solid Angle A lamp radiates luminous flux in all directions round it. If we think of a particular small lamp and a certain direction from it, for example that of the corner of a table, we can see that the flux is radiated towards the centre of the table in a cone whose apex is the lamp. A thorough study of photometry must therefore include a discussion of the measurement of an angle in three dimension, such as that of a cone, which is known as a solid angle. An angle in two dimensions, i.e., in a plane, is given in radians by the ratio ⁄ , where s is the length of the arc cutoff by the bounding lines of the angle on a circle of a radius r. In analogous manner, the solid angle ω of a cone is defined by the relation Where s is the area of the surface of a sphere of radius r cut off by the bounding lines of the cone. 100` Luminou
us Intensity, I The conccept of a unit of brightneess of a sourrce to allow ffor spatial distribution o
of light energgy, which is measured in
n 'candela', aabbreviated as 'Cd'. Thee luminous In
ntensity of aa source is defined b
by the relation. where, ω
ω = solid anggle, in sterad
dian (st) 101` Luminous Intensity of Source (c) Illuminance, E Illuminance also called illumination of a surface is the result of illumination on a lighted surface and is defined as the luminous flux falling on unit area of the surface under consideration. where, E = illuminance of a surface, lm/m2 or lux F = luminous flux incident on the surface, lumen A = area of the surface, m2 (d) Luminance of a Light source, L 102` The luminance of a light source, L ,in a given direction is defined as the luminous intensity per unit projected surface area of the light source in that direction, i.e. The unit for luminance of a light source is Cd/m2. (e) Luminance of an Illuminated surface. B The luminance of an illuminated surface in a given direction is defined as the luminous intensity per unit area coming from the surface in the particular direction. where, B = luminance of a surface, Cd/m2 E = illuminance of that surface, lux r = luminous factor (sometimes called reflectance factor), of the surface. 12.3 Relationship between Luminous Intensity and Illumination illumination of the surface, but, so, This is known as the Inverse Square Law of illumination. In case the light falls on a surface at an incidence angle of q as in Figure 4, then the illumination on the surface is given by: 103` This is known as the Lambert's Cosine Rule Proof of Lambert's Cosine Rule 104` W
WEEK 13:
: PHOTOM
METER 13.0 Introduction meter is an instrument which can be used forr comparingg the lumino
ous intensities of A Photom
sources o
of light. One
e of the mosst accurate fforms of pho
otometer w
was designed
d by Lummer and Brodhun, and the esssential featu
ures of the in
nstrument are illustrated
d below. 13.1 The Lummer‐Brrodhun Phottmeter 1 and I2 are placed on o
opposite sides of a whitte opaque sccreen Lamps off luminous intensities I1
and som
me of the difffusely refleected light from the opposite surfaaces A, B is incident on
n two identical totally refle
ecting prisms P,Q as sho
own above. The light reeflected from
m the prism then passes to
owards the ““Lummer‐Brodhun cube”, which is the main feature of the p
photometer. This consist of o two right angled isco
osceles prism
ms in opticaal contact att their centrral portion C C but with edgges of one cu
ut away so that an air film exist at M
M, N all roun
nd C betweeen the prism
m. The rays leavving the prism
m P are those transmittted through the central portion C off the “cube””, but totally reeflected at the t edges. Similarly, S
the rays refleected from the t prism cu
ube towards the “cube” are a transmittted through the centrral portion but totally reflected at the edgess. An observerr of the cube
e thus sees aa central circcular patch B
B of light duee to initially to the light from the sourrce of intensity I1 and outer portion A due in
nitially to th
he light from
m the sourcce of intensity I2.(see fig2 above) 105` 13.2 Comparison of luminous intensities In general, the brightness of central and outer portion of the field of view in a Lummer‐Brodhun photometer is different, so that one appers darker than the other. By moving one of the sources, however, a portion is obtained when both portions appears equally bright in which case cannot be distinguished from each other and the field of view is uniformly bright. A “photometric balance” is then said to exist. Suppose that the distance of I1 and I from the screen are d1, d respectively as shown above. The intensity f illumination E due to the source I I cos θ
is generally given by E =
d2
But θ = 0 in this case, as the line joining the source to the screen is normal to the screen. I
o
E = 2 . Hence, since cos 0 = 1,
d
Similarly, the intensity of illumination, E1 , at the screen due to the source I 1 is given by E1 = I 1 2 . Now the luminance of the of the surface B = r1 E1 , where r1 is the reflection factor d1
of the surface and the luminance of the surface A = rE , where r is the reflection factor of this surface. Hence for a photometric balance, ∴
r1 E1 = rE r1 I 1 rI
= 2 .........................................(i ) d12
d
If the reflection factor r1 , r of A, B are equal, equation (i) becomes I1
I
= 2 2
d1
d
I1 d12
∴ = 2 I d
The ratio of the intensities are hence proportional to the square of the corresponding distances of the source from the screen. The reflection factor r1 , r are not likely to be exactly equal, however in which case another or auxiliary lamp is required to compare candle powers I1, I. The auxiliary lamp, I2 is placed on the right side, say, of the screen at a distance d2, and one of the other lamps is placed on the other side. A photometric balance is then obtained as shown below. 106` (i) A I1 (Auxiliary) B d1 d2 r1 r A B I2 (ii) I I2 d2 d1 r1 r In this case, if I1 is the intensity of the lamp and d1 is its distance from the screen I
I1
= r 22 .............................(ii ) 2
d2
d1
r1
The remaining lamp I is then used instead of the lamp I1 and a photometric balance is then obtained by moving this lamp keeping the position of the lamp I2 unaltered as shown below. Suppose the distance of the lamp I from the screen is d, then I
I
= r 22 .................................................(iii ) 2
d2
d
r1
From (ii) and (iii), it follows that r1
∴
I1
d1
2
= r1
I
d2
I
I
∴ 12 = 2
d1 d
I 1 d12
=
.....................................................(*) I d2
107` The intensities I1, I are hence proportional to the squares of the corresponding lamp distances from the screen. It should be noted that from equation (*) that the auxiliary lamp’s intensities is not required in the comparison of I1 and I nor is its constant distance d2 from the screen required. Luminous Energy Luminous energy is photometrically weighted radiant energy. It is measured in lumen seconds. Luminous Flux Density (Illuminance and Luminous Exitance) Luminous flux density is photometrically weighted radiant flux density. Illuminance is the photometric equivalent of irradiance, whereas luminous exitance is the photometric equivalent of radiant exitance. Luminous flux density is measured in lumens per square meter. (A footcandle is one lumen per square foot.) Luminance Luminance is photometrically weighted radiance. In terms of visual perception, we perceive luminance. It is an approximate measure of how “bright” a surface appears when we view it from a given direction. Luminance used to be called “photometric brightness.” This term is no longer used in illumination engineering because the subjective sensation of visual brightness is influenced by many other physical, physiological, and psychological factors. Luminance is measured in lumens per square meter per steradian. 108` WEEK 14: WAVES AND WAVE MOTION 14.1 Introduction The main characteristics of wave motion are: •
•
Particles of the medium vibrate about their mean position while the wave moves forward. Each particle of the medium vibrate, takes energy from its preceding particle and transmits it to the next particle. During a wave motion, the medium does not move as a whole. Only the disturbance travels through the medium. 14.2 Mechanical Waves and Electromagnetic Waves Depending upon their propagation in a medium, the waves are classified as follows: •
•
Mechanical waves or elastic waves and Electromagnetic waves. Mechanical Waves A mechanical wave is a periodic disturbance, which requires a material medium (solid, liquid or gas) for its propagation. These waves are also known as elastic waves because their propagation depends upon the elastic properties of the medium through which they pass. Examples for mechanical waves are sound waves and water waves. In these waves, the particles of the medium just vibrate to and fro about their mean position. Requisites of the medium to propagate mechanical waves The medium must possess the following properties for the propagation of the waves: The medium should be able to return to its original condition after being disturbed, i.e., the medium must possess elasticity. The medium must be capable of storing energy. The frictional resistance must be negligible so as not to damp the oscillatory movement. Electromagnetic waves 109` An electrromagnetic w
wave is a dissturbance, w
which does n
not require aany material medium forr its propagattion and can
n travel even
n through vaccuum. They are caused due to varying electric aand magneticc fields. Radio waaves and ligh
ht waves aree examples fo
or electromaagnetic waves. Let us no
ow tabulate tthe differences between
n the mechaanical waves and electro
omagnetic w
waves. 14.3 Longgitudinal an
nd Transversse Waves In generaal all waves m
mechanical and electrom
magnetic (no
on‐mechaniccal) waves aare classified
d into two typees. These are
e Longitudinal Wave Transverse Waves Longitud
dinal waves 110` Propagation of sound waves in air A wave motion in which the particles of the medium oscillate about their mean positions in the direction of propagation of the wave, is called longitudinal wave. Sound waves are classified as longitudinal waves. Let us now see how sound waves propagate. Take a tuning fork, vibrate it and concentrate on the motion of one of its prongs, say prong A. The normal position of the tuning fork and the initial condition of air particles is shown in the fig (a). As the prong A moves towards right, it compresses air particles near it, forming a compression as shown in fig (b). Due to vibrating air layers, this compression moves forward as a disturbance. As the prong A moves back to its original position, the pressure on its right decreases, thereby forming a rarefaction. This rarefaction moves forward like compression as a disturbance. As the tuning fork goes on vibrating, waves consisting of alternate compressions and rarefactions spread in air as shown in fig (d). The direction of motion of the sound waves is same as that of air particles, hence they are classified as longitudinal waves. The longitudinal waves travel in the form of compressions and rarefactions. Transverse waves A wave motion, in which the particles of the medium oscillate about their mean positions at right angles to the direction of propagation of the wave, is called transverse wave. These waves can propagate through solids and liquids but not through gases, because gases do not possess elastic properties. Examples of these waves are: vibrations in strings, ripples on water surface and electromagnetic waves. In a transverse wave the particles of the medium oscillate in a direction perpendicular to the direction of propagation as shown in the figure. 111` Particles of th
he medium o
oscillate in aa direction perpendicular to the direection of propagattion Thus, durring their oscillations, th
he particles m
may move upwards or d
downwards ffrom the plaane passing tthrough their mean positions. The uppermost po
oint of the w
wave, i.e., th
he position o
of maximum
m positive diisplacementt is crest and
d the lowest point, i.e. th
he position o
of maximum
m displacem
ment is calle
ed trough. Th
hus in a transverse wavee crests and troughs app
pear alternativvely. 14.4 Wavve Propertie
es On obserrving waves,, it may be o
observed thaat they traveel with a definite speed tthrough a uniform medium. If w
we watch a p
particular sp
pot, we find that the wavves pass thaat spot at reggular intervals of time. The
e following d
definitions help in describing wave m
motion. Period (TT) The perio
od of wave m
motion is thee time in wh
hich a particle of the med
dium completes one oscillatio
on i.e., a to and fro motio
on about its mean positiion. Frequenccy (f) It is the n
number of w
waves producced per seco
ond or the number of osscillations made by a parrticle of the meedium per se
econd. In T sseconds onee oscillation is completed
d. Thereforee in 1 second
d the number o
of oscillation
ns made willl be This is the definition of freequency. Wavelength (λ) It is the d
distance travvelled by thee wave in thee time in wh
hich a particle of the med
dium completes one oscillation. In the
e case of a transverse w
wave, the distance betweeen the centtres of two nearest trougghs gives thee wavelengtth. In the casse of nearest ccrests or bettween the ceentres two n
112` a longitudinal wave, the distance between the centres of two nearest compressions or between the centres of two nearest rarefactions gives the wavelength. Velocity (v) The distance travelled by the wave in one second gives its velocity. In a time equal to the period, the wave covers a distance equal to wavelength (λ). But according to equation 1.1, Amplitude (A) It is the maximum displacement of a particle of the medium from its mean position. Phase When a wave passes through a medium, the particles of the medium vibrate about their respective mean positions in the same manner, but reach the corresponding positions in their paths at different instants of time. These relative positions represent the phase of the motion. It is measured either in terms of the angle that the particle has described (denoted as a fraction of 2π) or the time that has elapsed (measured as a fraction of the time period T), since the particle last passed through its mean position in the positive direction. The phase difference between any two particles indicates the extent by which they are out of step with each other. For example, a particle on the crest and a particle on the adjacent trough of a wave differ in phase by 180o or π radians. Wavefront A surface on which all the particles of the medium are in identical state of motion at a given instant, is called a wavefront. It is the locus of all the points which are in the same phase. In a homogenous and isotropic medium, the wave front is always perpendicular to the direction of propagation of the wave. 113` Angular ffrequency (ω) t is defined as the ratte of changee of phase w
with time. On
ne full wave (i.e., a crest and a trouggh) is o
associateed with a phase change of 360 or 2 radians. Th
he time requ
uired for thiss change is tthe period T.. Thus, the angular frequ
uency 14. 5 Graaphical Reprresentation of Waves Wheneveer a wave paasses througgh a medium
m, there is a cchange in so
ome property of the medium.. Hence, the waves can b
be graphically representted by showing the chan
nges in the vvalue of any su
uch propertyy of the medium as the w
waves travel through it.
Graphica
al representa
ation of a lon
ngitudinal w
wave The longitudinal wavves travel in the form of compressions and rareffactions and always compression follow aa rarefaction
n or vice versa. Whenever there is compression the density of particles of the mediium is higher than the normal density, while in aa rarefaction
n, the densitty of the particles is lowerr than the no
ormal densitty. The figuree below givees the distan
nce‐density graph forr a longitudinal wave. 114` Distance‐density graph for a longitudinal wave In the above graph, the horizontal dotted line represents the normal density of the medium. All the points above the dotted line represents higher densities and the points below the dotted line represents lower densities than the normal density. The particle density at any particular point on a longitudinal wave alternatively increases and decreases also with time at regular intervals as shown in the figure. Particle density in a longitudinal wave Graphical representation of a transverse wave Transverse waves propagate in the form of crests and troughs, i.e., some particles get displaced upwards, while some others downwards from their mean positions. Therefore, a transverse wave can be described graphically by a displacement‐distance graph as shown below. 115` Displacement‐distance graph A transverse wave dampens after travelling some distance i.e. with time. Therefore, the displacement‐time graph for a transverse wave is as shown in the figure. Displacement‐time graph A periodic wave is a wave in which the particles of the medium oscillate continuously repeating their vibratory motion regularly at fixed intervals of time. 14.6 Stationary Waves Stationary or standing waves are formed in a medium when two waves having equal amplitude and frequency moving in opposite directions along the same line, interfere in a confined space. Generally, such waves are formed by the superposition of a forward wave and the reflected wave. Both longitudinal and transverse types of waves can form a stationary wave. When reflection occurs at a free end, there is no reversal of phase. i.e., a crest returns as a crest and a trough as a trough. Example, 116` 1) A rope held vertically in the hand with the lower end hanging free, made to vibrate briskly at the upper end 2) An open end of an organ pipe into which air is blown When reflection occurs at a fixed end, there is a reversal of phase but there is no change in amplitude, frequency and velocity. i.e., a crest returns as a trough and a trough as a crest. Example, 1) A guitar string plucked in the middle. 2) A closed end of an organ pipe, into which air is blown. Formation of stationary waves explained graphically Figure(a) shows two sinusoidal waves A and B having the same amplitude and frequency, traveling in opposite directions. At an instant of time t = 0, the resultant displacement graph is a straight line. All the particles of the medium affected by the two waves are at their equilibrium positions. Figure shows the situation after a time , where T is the period of oscillation of the particles of the medium. It is seen in the figure that the wave A has advanced through a distance λ/4 towards the right and the wave B has advanced through the same distance towards the left. The two crests and the two troughs will add up, giving rise to a bigger wave pattern as shown in the figure. The particles 1, 3, 5 and 7 are at their extreme positions while the particles 2, 4 and 6 are at their equilibrium positions. 117` Figure (c) shows the situation at the end of . The wave A has advanced through a distance λ/4 towards the right and the wave B through the same distance towards the left. All the particles are now in their equilibrium positions and the resultant wave pattern is a straight line. Figure (d) shows the situation at the end of . The wave A has advanced through a distance towards the right and the wave B through the same distance towards the left. The resultant wave pattern is bigger than either wave as the amplitudes add up. It may be noted that the particles 1, 3, 5 and 7 are again at their equilibrium positions. Figure (e) shows the dynamic condition of the particles at the end of t = T. The wave A has advanced through a distance λ towards the right and the wave B through the same distance, towards the left. The resultant displacement pattern is a straight line. All the particles are in their equilibrium position. From the above discussion, it follows that some particles like 2,4 and 6 always remain at rest while particles like 1, 3, 5 and 7 vibrate (simple harmonic) about their mean positions. Such particles have the maximum amplitude, equal to twice that of the individual waves. 118` The resultant displaccements at times are shown in figure (ff). The positiions of particlles like 2, 4 aand 6, which
h always rem
main at their mean positiions are called 'nodes'. TThe positionss of particless like 1, 3, 5 aand 7 at whiich the resulltant amplitu
ude is maxim
mum, are called 'antinodees'. The distaance betweeen any two cconsecutive nodes or an
ntinodes is equal to λ/2. Between
n a node and
d an antinod
de, the amplitude of the particles varies from zeero to 2 A. Characteeristics of Sta
ationary Waves •
•
•
•
•
•
•
•
•
In
n stationary waves, there are certain
n points calleed nodes wh
here the parrticles are permanently at rest and certain otheer points callled antinodees where thee particles viibrate with m
maximum am
mplitude. Th
he nodes and
d antinodes are formed alternately. A
All the particl
les of the meedium excep
pt those at the nodes, viibrate simplee harmonicaally w
with a time p
period equal to that of th
he componeent waves. The amplitud
de of vibratio
on increases gradually frrom zero to maximum frrom a node tto an antinode. The medium is split up in
nto segmentss. The particcles in a segm
ment vibratee in phase. TThe particles in on
ne segment are out of p
phase with th
he particles in the neighbouring o
seegment by 1
180 . V
Velocity and a
acceleration
n of all the particles sepaarated by a d
distance aare the same at a given instan
nt. In
n a given seggment, the p
particles attaain their maxximum or minimum velo
ocity and accceleration aat the same instant. There is no ne
et transportt of energy in
n the medium. Compressions and rarefactions do no
ot travel forw
ward as in progressive w
waves. They appear and disappear altternately, at the same pllace. D
During each v
vibration, alll the particlees pass simultaneously through their mean positions tw
wice, with m
maximum vellocity which is different for differentt particles. Comparisson between
n Progressivee Waves and
d Stationary y Waves 119` 120` 14.7 Vibrration of Strretched Strin
ngs In physics, the word 'string' is used in a moree general sense than wh
hat it normally denotes. In olden days, musical iinstruments employed sstrings of twisted intestines of animaals, such as ccat‐
gut. Now
wadays, the sstrings of mu
usical instrum
ments like th
he veena, violin and guittar are madee of metal wires. An ideal string is an infinitely th
hin, perfectlyy flexible corrd of uniform
m area of cro
oss‐
section. IIt should offfer no resistaance to bend
ding and theere should bee no changee in its length
h during vibration. In p
practice, a th
hin, long mettal wire of uniform crosss‐ section, sttretched between
n two fixed supports can be considerred as a strin
ng. Anyone w
who has playyed a musicaal instrumeent knows th
hat •
•
•
•
a thick heavy string, when made to vvibrate, has aa lower natu
ural pitch thaan a thin onee a short stringg has a higheer pitch than a long one and th
he tighter a string is streetched, higheer is its pitch
h a stretched sttring can be excited by sstriking with light felt maallets as in aa piano or byy bowing with resined horssehair as in aa violin or byy plucking w
with finger naails or picks aas in a veena or gu
uitar. n all the above casess, transversee vibrations aare produced in the strin
ng. The velocity of the transversse wave thatt travels alon
ng the length of the string, dependss on the natu
ure of the string and its sttate of tensio
on. Modes off vibration o
of a stretched
d string A string ccan be made
e to vibrate iin different m
modes. Wheen it vibratess as a whole with two no
odes at the exxtremities an
nd an antinode in the middle, it is the simplest o
or the fundam
mental mod
de of vibration
n. The freque
ency is then called fundaamental freq
quency or th
he first harm
monic and it is given by the equation If a stringg vibrating in
n the fundam
mental modee is gently to
ouched at th
he center, a n
node is form
med at that po
oint and the
e frequency o
of vibration becomes tw
wice that of tthe fundameental mode. If the stringg is touched lightly at a p
point one th
hird the distaance from th
he end, it will vibrate in three seggments and have a frequ
uency three times that o
of the fundamental mod
de. The different modes off vibration aare shown in figure. 121` A string can be set into vibrations with its fundamental and several of its higher modes at the same time. This is accomplished by plucking or bowing the string vigorously. The figure shows a string vibrating with two modes at the same time. Velocity of a transverse wave along a stretched string Let LAM represent a portion of a stretched string in which a transverse wave is travelling towards the right, with a velocity V. If the string is drawn towards the left with the same velocity, the wave becomes stationary. Let PQ represent a small element of this portion of the string. It is in the form of an arc with its centre of curvature at O. Let For the sake of clarity in the diagram has been shown to be large, but it will be quite small in practice. Let the tension in the string be T at P or Q. The tensions will be along the tangents, meeting at A. Join AO. AO represents the radius of the circular arc PQ, which is represented by r. If m is the mass per unit length of the string, then length of the arc PQ = m . PQ
The components of the tensions at P and Q along the radius will add up, while those perpendicular to it will cancel out. Therefore, the resultant tension in the element PQ is 2T sin acting along AO and this provides the necessary centripetal force, making the particles of the string trace a circular path. 122` Frequenccy of vibratio
on of a stretcched string
The fund
damental mo
ode of vibrattion of a streetched stringg is shown in
n the figure. It has two n
nodes at the en
nds and an antinode in th
he middle. Iff L is the length of the viibrating segm
ment betweeen the two n
nodes, then Substitutting for v fro
om equation 1.47 we gett 123` Laws of transverse vibrations of stretched strings Law of length "For a given string under constant tension, the frequency of vibration is inversely proportional to the length of the string”. Law of tension For a given string of constant length, the frequency of vibration is directly proportional to the square root of the tension”. Law of mass "For a string of constant length and under a constant tension, the frequency of vibration is inversely proportional to the square root of its mass per unit length”. If M is the mass and L is the length of the string, then If d is the diameter of the wire, then 124` Substituting in equation (1.48), we get The law of mass may be put into two additional laws, for strings of circular cross‐section, as given below. Law of diameter "For a string of a given material and length and under a constant tension, the frequency is inversely proportional to its diameter”. Law of density "For a string of a given length and diameter and under constant tension, the frequency is inversely proportional to the square root of the density of the material of the string”. 125` WEEK 15: SOUND WAVES 15.1 Introduction Sound is always produced due to the vibration of a body. In some cases the vibrations of the source may be very small or very large that it may not be possible to detect them. This type of vibrations is produced by tuning fork, drum, bell, the string of a guitar etc. Human voice originates from the vibrations of the vocal chords and the sound from the musical instruments is due to the vibrations of the air columns. We have already discussed that the sound travels in the form of longitudinal wave motion. Sound waves require a material medium for its propagation. Applications of sound waves (1) Submarines can be detected by the under‐water sound waves produced by the propellers. (2) Some bones conduct sound and this type of bones are used to design hearing aids. Speed of Sound The flash of lightning due to collision of clouds is seen much before the thunder, although both occur simultaneously. This happens because the velocity of light is greater than the velocity of sound. The speed of sound depends on (1) elasticity and (2) density of the medium through which it propagates. Sound waves are emitted from a vibrating source and transmitted through air. The human ear can hear sound waves in the range 20 Hz and 20 kHz. This range is known as audible range. The sound waves having frequencies above the audible range are known as ultrasonic waves and it is usually referred as ultrasound. The sound waves having frequencies less than the audible range are called infrasonic waves. Reflection of Sound Like light, sound waves also obey the laws of reflection and refraction. For sound waves to reflect, we need extended surface or obstacle of large size. For example, the rolling of thunder is due to successive reflections from clouds and land surfaces. Practical Applications of Reflection of Sound Some applications of the principle of Reflection of Sound are: (1) Megaphone, (2) Hearing Board, (3) Sound Boards. 126` (1)
(2)
(3)
Megaphone: Megaphone is a horn‐shaped tube. The sound waves are prevented from spreading out by successive reflections and are confined to the air in the tube. Hearing aid: It is a device used by the people who are hard of hearing. Here the sound waves, which are received by the hearing aid are reflected into a narrower area leading to the ear. Sound Boards: Curved surfaces can reflect sound waves. This reflection of sound waves is used in auditorium to spread the waves uniformly throughout the hall. Reflection of sound waves is done by using Sound Boards. The speaker is located at the focus of the sound board. 15.2 Musical Sound and Noise A musical sound can be defined as a pleasant continuous and uniform sound produced by regular and periodic vibrations. Example: The pleasant sound produced by a guitar, piano, tuning fork etc. Noise can be defined as an irregular succession of disturbances, which are discordant and unpleasant to the ear. Echo The sound heard after reflection from a rigid obstacle (such as cliff, wall) is called echo. The sensation of sound lasts or persists in our brain for 0.1 sec., even after the sources of sound has stopped vibrating. If d is the distance between the observer and the obstacle and V is the speed of sound, then the time taken by the sound to reach the obstacle and then to come back is calculated using the formula: Or If velocity of sound is taken to be equal to 334 m/s and t = 0.1 sec, Thus, in order to hear the echo of a sound, the reflecting surface should be at a minimum distance of 17.2 m from the observer. Bats and dolphins can detect the presence of an obstacle by hearing the echo of the sound produced by them. This process is called sound ranging. One of the most important applications of the reflection of sound is oceanographic studies. For this purpose, we use a system called the SONAR. The SONAR is abbreviated form of Sound 127` Navigation and Ranging. The SONAR system is used for detecting the presence of unseen under water object, such as submerged submarine, a sunken ship, iceberg and locating them. In Sonar ultrasonic waves are sent in all directions from the ship and are then received on their return after reflection. Applications of ultrasound (1)
(2)
(3)
(4)
(5)
It is used for medical diagnosis and therapy and also as a surgical tool. Bats and porpoises use ultrasound for navigation and to locate food in darkness. It is used to detect defective foetus. It is used as a tool in the treatment of muscular pain. Sonogram (is a technique of 3‐dimensional photographs with the help of ultrasonic waves) is used to locate the exact position of an eye tumor. 15.3 Superposition of Waves When a number of similar waves pass through a medium simultaneously, each wave travels through the medium as though the others were not present. However, at any point in the medium, the net effect of the waves reaching that point at any instant, is produced. The principle of superposition of waves states that 'if two or more waves of the same nature travel past a point of the medium, then the resultant displacement of the medium at that point is given by the vector sum of the individual displacements due to the waves. If are the individual displacements caused by two waves reaching a point, then the resultant displacement is given by This principle is applicable when the equations of the waves under consideration are linear. It holds good in the case of waves in an elastic medium and electromagnetic waves, but not in the case of shock waves produced by large explosions. Superposition of waves can be studied in three different cases •
•
•
Superposition of two waves of the same frequency traveling in the same direction, giving rise to a sustained interference pattern. Superposition of two waves of slightly different frequencies traveling in the same direction, giving rise to the phenomenon of beats. Superposition of two waves of same frequency and same amplitude moving in opposite directions, giving rise to the formation of standing waves or stationary waves. 128` 15.4 Modes of Vibrations in Pipes Organ pipe An organ pipe is the simplest form of a wind instrument. Figure (a) shows the longitudinal section of an organ pipe whose one end is closed and figure (b) shows an organ pipe, both ends of which are open. It consists of a hollow tube BD in which air can be blown through a pipe A (also called the mouthpiece). The air moves through a narrow slit B and strikes against the sharp edge C, called the lip. This lip vibrates and sets up vibration in the air column enclosed in the pipe. These vibrations travel to the other end of the pipe and get reflected. Due to superposition of the incident wave and the reflected wave, longitudinal stationary waves are formed. The frequency of the note produced, depends mainly on the length of the pipe and the type of the pipe, i.e., whether it is closed or open. In a closed pipe, the end D is always a seat of node and in an open pipe, the end D is always a seat of antinode. In both the pipes, the end B is always the seat of an antinode. Modes of vibration in a closed pipe 129` In the simplest or the fundamental mode of vibration, the air column vibrates with an antinode A at the open end and a node N at the closed end as shown in figure (a). Since the distance between a node and an antinode is to , the length of the tube l in this case will be equal . If v is the velocity of sound, then The frequency f1 of the fundamental note is called the ‘first harmonic’ Figure (b) shows the first overtone in a closed pipe. Two nodes and two antinodes are formed. The wavelength and the frequency of the sound corresponding to this mode of vibration will be different from those corresponding to the fundamental mode. Let the wavelength be 2 and the corresponding frequency be f2. Then, as seen in the figure, i.e l =
3λ2
4
∴ λ2 =
f2 =
4
l..........................................(iii )
3
v
3 v
= . ..................................(iv )
λ2 4 l
Figure (c) shows the formation of the second overtone with 3 nodes and 3 antinodes. If λ3 and f3 are the corresponding wavelength and frequency, then 5λ
l= 3
4 130` Or 4
5
λ3 = l...............................................(v)
f3 =
v
λ3
=
5v
…………………………(vi) 4l
Similarly, it can be shown that f4 =
7v
,
4l
f5 =
9v
....
4l ∴ f1 : f 2 : f 3 : f 4 .................. =
v 3v 5v 7v
: : :
4l 4l 4l 4l = 1: 3 : 5 : 7 Thus, in a closed pipe, only harmonics proportional to the odd natural numbers are present. Therefore, the quality of the note given out by a closed pipe lacks in fullness. Modes of vibration in an open pipe In an open pipe, when a compressed wave reaches the far end, the air at that point is, for an instant, at a pressure greater than the atmospheric pressure. Being an open end, the air there can vibrate with maximum freedom and so, it suddenly expands into the surrounding air. Thus, the pressure diminishes so quickly that it becomes lesser than the pressure of the surrounding air, which causes a sudden rarefaction at the end of the pipe. This sets up a rarefied wave which passes back along the pipe. Within the tube, the reflected pulses meet the direct ones and the result is the formation of the standing waves. Figure (a) shows the fundamental mode of vibration with two antinodes and one node. If λ1 and f1 are the wavelength and frequency of the sound producing this mode of vibration, then f1 gives the frequency of the first harmonic. 131` The first overtone formation is shown in figure (b). Three antinodes and two nodes are formed. From the figure, Figure(c) shows the formation of the second overtone. Four antinodes and three nodes are formed. Similarly, it can be shown that The frequencies of the harmonics present in an open pipe are proportional to the natural numbers. Owing to the presence of all harmonics, the quality of the note given out by an open pipe is richer and sweeter than that given out by a closed pipe. The fundamental frequency of vibration in a closed pipe is given by and for the same length of the tube, the fundamental frequency in an open pipe is given by . Therefore, the fundamental frequency of an open pipe is said to be an octave higher than that of the closed pipe. 15.5 Resonance A mechanical system which is free to vibrate like a hacksaw blade clamped at one end, a diving spring board or the air in pipes has a natural frequency of vibration f0, which depends on its dimensions. When a periodic force of a frequency different from f0 is applied to the system, it 132` vibrates with a small amplitude and undergoes forced vibrations. When the periodic force has a frequency equal to the natural frequency f0 of the system, the amplitude of the vibration becomes a maximum and the system is then set into resonance. When the diver on the edge of a springboard begins to jump up and down repeatedly, the board is forced to vibrate at the frequency equal to the frequency of the jump. Initially, the amplitude is small and the board is said to be undergoing forced vibrations. As the diver jumps up and down to gain increasing height for his dive, the frequency of the periodic downward force reaches a stage where it is almost the same as the natural frequency of the board. The amplitude of the board then becomes very large and the periodic force is said to have set the board in resonance. If the prongs of a vibrating tuning fork are held on the top of a pipe, the air inside it is set into vibration by the periodic force exerted on it, by the prongs. In general, however, the vibrations are feeble, as they are forced vibrations. So, the intensity of the sound heard is correspondingly small. But when a tuning fork of the same frequency of the pipe is held over the pipe, the air inside is set into resonance by the periodic force and the amplitude of the vibrations is large. So, a loud note, which has the same frequency as the fork is heard, coming from the pipe and a stationary wave is set up. By achieving resonance in a pipe, the frequency of the fork or the velocity of sound can be determined. 15.6 Beats Whenever two wave motions pass through a single region of a medium simultaneously, the motion of the particles in the medium will be the result of the combined disturbance due to the two waves. This effect of superposition of waves, is also known as interference. The interference of two waves with respect to space of two waves traveling in the same direction, has been described in previous section. The interference can also occur with respect to time (temporal interference) due to two waves of slightly different frequencies, travelling in the same direction. An observer will note a regular swelling and fading (or waxing and waning) of the sound resulting in a throbbing effect of sound called 'beats'. 133` Number of beats heaard per seco
ond Qualitatiive treatmen
nt Suppose two tuning forks havingg frequencies 256 and 25
57 per secon
nd respectively, are soun
nded eginning of aa given secon
nd, they vibrrate in the saame phase sso that the together. If at the be
compressions (or rarrefactions) o
of the corresponding wavves reach th
he ear togeth
her, the soun
nd will be reeinforced (sttrengthened). Half a seco
ond later, w
when one maakes 128 and
d the other vibratio
ons, they aree in oppositee phase, i.e.,, the compreession of onee wave o produce silence. At thee end of one combines with the raarefaction off the other aand tends to
und is reinfo
orced. By this time, one fork second, tthey are agaain be in the same phasee and the sou
is ahead of the otherr by one vibrration. Thus,, in the resultant sound, the observeer hears m sound at tthe interval o
of one secon
nd. Similarlyy, a minimum
m loudness iss heard at an
n maximum
interval o
of one secon
nd. As we maay consider a single beat to occupy the interval between tw
wo consecuttive maxima or minima, the beat pro
oduced in on
ne second in
n this case, iss one in each
h second. IIf the two tu
uning forks h
had frequenccies 256 and 258, a simillar analysis w
would show that the numb
ber of beats will be two per second.. Thus, in general, the nu
umber of beeats heard peer second w
will be equal to the difference in the frequenciess of the two sound wavees. Demonsttration of beeats Let two ttuning forks of the samee frequency b
be fitted on suitable ressonance boxes on a tablee, with the open ends o
of the boxes facing each
h other. Let tthe two tuning forks be sstruck with aa wooden hammer. A continuous loud sound is heard. It d
does not risee or fall. Let a small quan
ntity 134` of wax bee attached to a prong off one of the tuning forkss.. This reducces the frequ
uency of thaat tuning fo
ork. When th
he two forks are sounded again beatts will be heard. Uses of b
beats •
•
The phenome
enon of beatts is used for tuning a no
ote to any particular frequency. Thee note of the desired frequ
uency is soun
nded together with the note to be tuned. If therre is quencies, theen beats are produced. W
When they aare exactly in
n a slight differrence in freq
unison, i.e., h
have the sam
me frequencyy, they do no
ot produce aany beats wh
hen sounded
d to
ogether, butt produce the same num
mber of beatss with a third note of slightly differeent frrequency. Sttringed musiical instrumeents are tuned this way. The centrall note of a piano iss tuned to a standard value using thiis method.
The phenome
enon of beatts can be useed to determ
mine the frequency of a tuning fork.. Let A
A and B be tw
wo tuning forks of frequeencies fA (kn
nown) and fB (unknown). On soundin
ng A and B, let the
e number of beats produ
uced be n. Th
hen one of tthe followingg equations must be true. fA ‐ fB = n ……………. (i) or fB ‐ fA == n ……………
…. (ii) To find th
he correct equation, B iss loaded with a little waxx so that its frequency d
decreases. If the number o
of beats incrreases, then equation (i)) is to be useed. If the num
mber of beaats decreases, then equ
uation (ii) is tto be used. TThus, knowin
ng the valuee of fA and th
he number o
of beats, fB caan be calculated. •
•
ometimes, b
beats are deliberately caaused in mussical instrum
ments in a section of the So
orchestra to ccreate sound
d of a speciaal tonal quality. The phenome
enon of beatts is used in detecting daangerous gaases in miness. The apparratus used for this purpose con
nsists of two
o small and eexactly similaar pipes blow
wn togetherr, one by pure air from a reservoir and the o
other by thee air in the m
mine. If the air in the min
ne co
ontains methane, its density will be less than th
hat of pure air. The two n
notes produ
uced 135` •
by the pipes will then differ in the pitch and produce beats. Thus, the presence of the dangerous gas can be detected. The super heterodyne type of radio receiver makes use of the principle of beats. The incoming radio frequency signal is mixed with an internally generated signal from a local oscillator in the receiver. The output of the mixer has a carrier frequency equal to the difference between the transmitted carrier frequency and the locally generated frequency and is called the intermediate frequency. It is amplified and passed through a detector. This system enables the intermediate frequency signal to be amplified with less distortion, greater gain and easier elimination of noise. 15.7 Doppler Effect Let a stationary observer on a platform listen to the sound emitted by the whistle of an incoming train. As the train approaches the platform, an increase in the pitch of the sound will be observed. As the engine moves away from him, there will be a decrease in the pitch of the sound. The same effect is observed in the case of approaching or receding automobiles and jet aircrafts flying at a low altitude. This effect is also felt when the observer approaches or moves away from a stationary source. In these cases, it must be noted that the source is emitting sound of a particular frequency, but it is the observed frequency that changes. This phenomenon in which there is an apparent change in the frequency of sound as a result of relative motion between the source and the observer is called Doppler effect. The Doppler effect or Doppler shift was first discussed by Christian Johann Doppler (1803 ‐ 1853) in 1842, in connection with similar shifts in the frequency of light emitted by the stars revolving about each other in double‐star systems. It was tested experimentally, in connection with sound, in 1845, by Berys Ballot in Holland. The Doppler effect can be demonstrated in the following ways Let a tuning fork be fixed vertically near the rim of a circular turntable. On sounding the fork and spinning the turn table, a periodic variation in the pitch of the sound will be observed. It increases when the fork is moving towards the observer and decreases when it is moving away from him. Let a small whistle be placed inside one end of a long flexible rubber tube. Let the other end of the tube be held in the mouth and the whistle be blown. On holding the middle of the rubber 136` tube with
h the hand aand whirling the tube in a horizontal circle over the head, peeriodic variation
ns in the pitch of the sound will be observed. 15.8 Expression for tthe Apparen
nt Frequencyy Case (i) G
General casee Source, o
observer and
d the medium, all in mottion Let S and
d O denote the initial positions of a ssource of sound and an observer. Fo
or the sake o
of simplicityy, we shall assume that the source, the observer and the medium are all moving alo
ong the posittive direction
n. Let the velocity of so
ound in still aair be = v The veloccity of the so
ource = a The veloccity of the observer = b and the vvelocity of th
he medium ((wind blowin
ng) = w Let Sl and
d Ol represent the positiions of the source and th
he observer after 1 seco
ond. Distancee travelled
d in 1 second
d is nothing b
but the veloccity. The wavees produced
d by the sourrce travel a d
distance SA iin 1 second, but as the w
wind is blow
wing l
l
with a veelocity w, it ccarries the w
wavefront fro
om A to A w
where A A = w
w. The distan
nce travelled
d by the wavees relative to
o the source in 1 second. If f is the frequency o
of the wavess produced b
by the source, then f waves are acco
ommodated in a l l
distance S A . 137` Since the observer recedes by a distance b in 1 second, the relative velocity of the waves with respect to the observer is (v + w ‐ b). Therefore, the apparent frequency is given by the number of waves of wavelength λl contained within the above distance. Substituting for λl from equation (i), we get This is the general expression for the apparent frequency of the sound when the source of sound, observer and the medium are in motion, in the same direction. Discussion of equation (1‐38) for particular cases Case (i) Source moving towards a stationary observer Let us assume that the wind velocity is zero. Then w = 0 and b = 0. Equation (1‐38) becomes The apparent frequency will be greater than the actual frequency. Case (ii) Source moving away from a stationary observer Assuming the wind velocity to be zero, putting ‐a in the place of a and b = 0 in equation (1‐38), we get The apparent frequency will be lesser than the actual frequency. Case (iii) Observer moving away from a stationary source 138` Putting w = 0 and a = 0 in equation (1‐38), we get The apparent frequency will be lesser than the actual frequency. Case (iv) Observer moving towards a stationary source Putting w = 0, a = 0 and ‐b in the place of b in equation (1‐38), we get The apparent frequency will be greater than the actual frequency. Case (v) Observer and source moving in the same direction as sound in a stationary medium Putting w = 0, in equation (1‐38), we get •
•
•
If b < a, then (v ‐ b) > (v ‐ a) and fl > f. If b > a, then (v ‐ b) < (v ‐ a) and fl < f. If b = a, then fl = f. Thus, for a passenger sitting in a train the frequency of the whistle of the train appears to be the same, when the train moves, as when it was at rest. Case (vi) Observer and source moving towards each other in a stationary medium Now the velocity of the observer is opposite to the velocity of sound, while that of the source is the same as that of sound. 'b' is to be replaced by ‐b in equation (1‐38). Case (vii) 139` Observer and source moving away from each other in a stationary medium Considering the direction of the velocity of sound reaching the observer as positive, a is negative and b is positive. Then The apparent frequency is less than the actual frequency. Case (viii) Wind blowing opposite to the direction of the velocity of sound In this case, w is to be replaced by ‐w in equation (1‐38) Some Important Aspects of Doppler Effect •
When a source moves towards a stationary observer the apparent frequency, according to •
When the observer moves towards a stationary source, the apparent frequency, according to In both the cases, the apparent frequency is greater than the actual frequency. For comparing the apparent frequencies, let us assume that a = b, in magnitude. Let the apparent frequencies in the two cases be denoted by f1 and f2. Then And 140` Since thee denominattor in the RH
HS of the abo
ove equation
n is less than
n 1, f1 > f2. Th
hus, the incrrease in the apparent frequ
uency is greaater when th
he source ap
pproaches a stationary o
observer thaan, when thee observer approaches aa stationary source. In other words, Doppler effect in sound
d is 'asymmeetric'. This assymmetry is not observeed in Doppleer effect in caase of light. This is due tto the fact tthat the velo
ocity of light is very high when comp
pared to the velocity of tthe source o
or the observer. es away from
m the sourcee with a speeed greater th
han the speeed of sound, If the obsserver move
then the wave can ne
ever 'catch u
up' with the observer an
nd the formu
ula should no
ot be applied. Another problem arises when th
he source mo
oves with a vvelocity exceeeding the vvelocity of so
ound. Equation
n (1‐39) then
n, predicts a negative freequency. Since this does not make any physical sense, a different intterpretation has to be giiven. Consider the sourcce moving fro
om the poin
nt A to B in a time t with aa velocity grreater than vv, the velocitty of sound. By the timee the source reaches B
B, the sound
d wave from A will have covered a distance vt. TThe wavefront is shown by the largeest circle in the figure. Th
hese wavefronts become smaller an
nd smaller in
n size until th
he zero radius is reached at B. The eenvelope of these waveffronts is a co
one (called tthe Mach Co
one) whose su
urface makes an angle θ with the dirrection of m
motion of thee source. Fro
om the figuree 141` The ratio
number. i.e., the ratio of the velocity of a body to the velocity of sound is called the Mach The conical envelope represents a very narrow region of high pressure called the 'shockwave'. Some examples of the formation of the shockwaves are when (i)
(ii)
a supersonic aircraft flies across the sky a bullet is shot from a gun. A shockwave is a highly concentrated sound wave and is therefore, a pressure wave. It is this wave of increased pressure that produces a sound like a thunderclap or an explosion. The Doppler effect can be observed in all kinds of waves so long as the speed of the source is small when compared to the speed of the wave. The observed shift towards the red (lower frequencies) in the spectra of some stars indicate that the stars are moving away from the observers on the Earth and gives support to the 'expanding universe' concept. Applications of Doppler Effect •
•
•
The Doppler effect provides a convenient means of tracking a satellite that is emitting a radio signal of constant frequency. The frequency of the signal received on the Earth changes as the satellite is passing. If the received signal is combined with a constant signal generated in the receiver to give rise to beats, then the beat can have a frequency that produces an audible note, whose pitch decreases as the satellite passes overhead The traffic police use a technique based on Doppler principle to detect overspeeding of vehicles on highways. An electromagnetic wave is emitted by a source at the side of the road attached to a police car. The wave is reflected by a moving vehicle, which thus, acts as a moving source. The reflected wave will have a Doppler shift in frequency. Measurement of the frequency shift using the phenomenon of beats, permits the measurement of the speed of the vehicle. The Doppler effect for light is important in astronomy. Two stars which revolve around one another are called double stars or spectroscopic binaries. When one is approaching the Earth, the other will be receding and this causes a split in the spectral lines, due to change in the frequency of the light emitted. The phenomenon of red shift in the light from stars helps in understanding the theory of expanding universe. 142`