Electromagnetic radiation belonging to the wave length range between 400 nm to 800 nm is called light . It travels with an enormous speed in straight line in vacuum . It's speed in vacuum is 3 x 10^8 m/s . It's speed in vacuum is the highest speed attainable in nature . A light wave can be considered to travel from one point to another in a straight line joining them . Such a path of light is called ray of light . A bundle of light rays is called a beam of light .We will discuss the phenomena of reflection refraction and dispersion of light using ray picture. Next we understand some optical instruments such as microscope , telescope and human eye using the ray property of light .
Reflection on spherical surface :
We know the laws of reflection : 1) Incident ray , reflected ray and normal on the reflecting surface at the point of incident all lie in the same plane .2) Angle of incident ( angle between incident ray and normal ) and angle of reflection ( angle between reflected ray and normal ) are equal to each other . These laws are valid in all reflecting surfaces whether it is plane or curved .In curved surface normal is drawn on the tangent at the incident point and it passes through the centre of curvature of the curved surface. Centre of curvature is the centre point of the sphere forming the curved mirror .
The geometric centre of a spherical mirror is called its pole . In case of lens it is called optical centre.The line joining between pole and centre of curvature is called principal axis . In case of lens principal axis is the line joining between optical centre and principal focus .
Sign convention:
In Cartesian sign convention here pole is taken as origin and the distances measured in the direction of light rays are taken as positive and the direction measured against the light rays is taken negative . The distance upward to principal axis is taken positive and the downward to the principal axis is assumed negative .
Focul length of spherical mirror:
Light rays travelling parallel to the principal axis after reflection from spherical mirror converge to a point on the principal axis for concave mirror and for convex mirror light rays appears to diverge from a point on the principal axis . This point on the principal axis is called Principal focus . The distance between the pole and focal point is called focal length of the mirror.
The mirror equation :
If rays emanating from a point actually meet at another point after reflection or refraction then that point is called the image of the first point .The image is real if the rays converge to the point and image is virtual if rays appear to diverge from the point .
In principle any two rays emanating from a point find their point of intersection and thus get the image of the point due to reflection from a spherical mirror . To draw the image it is convenient to follow any of the two procedures from the following.
i) The ray from the point coming para to the principal axis after reflection goes through the focus of the mirror.
ii) The ray passing through the centre of curvature of the concave mirror retraces back to its path after reflection.
iii) Any ray passing through the focus of the mirror becomes parallel to the principal axis after reflection.
iv) The ray incident at any angle at the pole is reflected following the law of reflection.
Figure shows the rays diagram for the above three rays . Here AB is image and A'B' is real image .The two right angled triangles A'B'F and MPF are similar .Therefore,
A'B'/PM = B'F/PF ( MP is considered perpendicular to principal axis for small mirror)
Or A'B'/AB = B'F/PF ( PM = AB)
Since angle APB' = angle A'PB so triangle ABP and A'PB' are similar .
Hence
AB/A'B' = PB/PB'
From the above two relations we get ,
B'F/PF = PB'/PB
Or, (B'P - FP)/PF = PB'/PB
Or ,( - v + f )/-f = - v/-u
When simplified we get
1/v + 1/u = 1/f
This is known as mirror equation.
Magnification :
It is defined as the ratio of image height and the object height= A'B' / AB = h'/h From the above relation we get A'B'/ AB = PB'/PB = v/u
So magnification, m = -h'/h = -v/u ( image is inverted so the negative sign )
REFRACTION :
When a light rays travelling from one medium incidents obliquely on the interface of another medium a part of the light ray transmit with direction changed at the interface of two medium . It is called refraction of light . Snell experimentally obtained the following two laws of refraction .
I) The incident ray reflected rays and the normal to the interface at the point of incidence all lie in the same plane .
II) The ratio of sine of angle of incidence to the sine of angle of refraction is constant for a given pair of media and for a given colour of light .
We have sini/son r = n21, where n21 is a constant and called the refractive index of medium 2 with respect to medium 1 .It is Snall's law of refraction. If n21 is greater than 1 then medium 2 is optically deser medium .If n21 is less than unity theefium two is rarer medium . If n21 > 1 then i > r reverse is the case if n21<1 . Optical density bshould not be confused with mass density. Optical density may be greater where mass density is lower . In case of water and terpenpine oil .mass density of water is greater than terpentine oil. But optical density of terpentine oil is greater than water .
Since the path of light is reversible, n21 = n12 , where n12 is refractive index ofedium 1 wth respect to medium 2 .
It can also be easily proved ,
n23 = n21 x n13
From the laws of refraction it can be proved that when a ray of light pass through a rectangular glass slab the incident ray and emergent ray are parallel. Another familiar example is : the bottom of tank filled with water appears to be raised . It can shown that apparent depth h1 is real depth h divided by refractive index of the medium . Proof : n = sin I /sin r = tan i/ tanr ( i, r are small )= h2/h1 ( h2 real depth , h1 apparent depth )
When light travel from an optically denser medium to rarer medium at the interface light is partly reflected back to the 1st medium .it is called internal reflection . But the refractive angle is greater than incident angle i.e retracted ray bends away from the normal .If we increase the incident angle at a particular value of it refractive angle becomes 90 degree. Any further increase of refractive angle is not possible. This angle of incidence for which refractive angle is 90 degree is called critical angle of the two media .If incident angle is increased greater than critical angle light will get reflected totally from the interface between the two medium .This reflection is Total Internal Reflection.
So ,Sinic = n21
Hence n12 = 1/Sinic
Refractive index and critical angle of some media is given below :
Water = refractive index 1.33 Critical angle 48.75 degree
Crown glass = 1.54 , 41.14
Denser flint glass = 1.62 , 37.31
Diamond = 2.42 , 24.41
A demonstration for total Internal Reflection:
Take a glass beaker with clear water in it . Add a few drops of milk to make it a little turbid .Take a laser pointer and shine it's beam through the turbid water .The path of the last light rays can be seen inside the water .Shine the beam from below the beaker and it strikes the upper layer of water . We can see the beam is divided into two parts at the surface, one is transmitted to the air and the other is reflected back into the water. If we increase the incident angle gradually we find a value of incident angle there is no rays transmitted into the air ,the total rays is reflected into the water . This is total Internal Reflection.
i)
Prism : prism designed to bend light by 90 degree or 180 degree make use of total internal reflection.Such a prism is used to invert images without changing their shapes .In this case critical angle of the glass making the prism must be less than 45 degree .
ii) Optical fibres : Optical fibres are extensively used to transmit audio and vedio signal for long distance using the total internal reflection. Each fibre consists of two parts one is core and the external part is called cladding .The refractive index of the core material is greater than that of cladding . When a signal in the form of light is directed at one end of the fibre at a suitable angle it undergoes repeated total internal reflections along the length of the fibre and finally comes out of the other end . Since the rays undergoes total internal reflection there is no appreciabke loss of intensity of the light signal . Optical fibres are also used as 'light pipes' to facilitate visual examination of internal organs like stomach and intestines . In decorative lamp optical fibres are used.One end of the fibre is held at the light source and from the other end we get a dot of light .
Refraction at Spherical Surfacesand by Lenses :
So far we considered only refraction at plane interface . We shall now consider refraction at Spherical interface seprating between two transparent media . An infinitesimal part of a spherical surface can be considered as plane surface and the law of reflection hold good . The normal at the point of incidence is perpendicular to the tangent plane drawn at the point of incidence. First we consider refraction by a single spherical surface and then follow it by thin lenses .A thin lens is a transparent medium bounded by two surfaces , at least one of which should be spherical .Applying the formula of image formation by a single surface to two surfaces of a lens we will get lens maker formula .
Refraction at Spherical surface :
In the figure O is a point on the principal axis and I is it's image and R is radius of curvature.The rays are incident from a medium of refractive index n1 to another of refractive index n2 .We take the aparture of the surface is small compared to other distances involved. NM will be taken perpendicular to principal axis .For small angles ,
tanNOM = MN/OM
and tan NCM = MN/MC
tan NIM= MN/MI
For triangle NOC, i is exterior angle
Hence , i = angle NOM + angle NCM
i = MN/OM + MN/MC
Similarly , r = angle NCM - angle NIM
or, r = MN/MC - MN/MI
By Snell's law
n1 sini = n2 sin r
For small angles , n1 i = n2 r
n1( MN/OM + MN/MC) = n2(MN/MC - MN/MI)
Or, n1/OM + n2/MI = (n2 - n1)/MC
Now taking sign convention ,
OM = - u , MI = v , MC = R
So we have ,
n2/v - n1/u = ( n2 - n1)/R
u is object distance , v is image distance R is radius of curvature.n1 and n2 are refractive indices of 1 and 2 medium respectively.
Refraction at Convex lens :
The image formation can be seen in terms of two steps .
The first refracting surface forms the image I1 of the object O .The image I1 acts as a virtual object for the second surface that forms the image I .Applying the formula for refraction at Spherical surface to the first surface ABC we get ,
n1/OB + n2/BI1 = (n2 -n1)/BC1
A similar procedure applied to the second interface ADC we get ,
-n2/DI1 + n1/DI = (n2 - n1)/DC2
For a thin lens BI1 = DI1 and adding the above two equations ,
n1/OB + n1/DI = (n2 -n1)(1/BC1 + 1/DC2)
If the object is taken to infinity then OB tends to infinity and DI = f the above equation gives ,
n1/f = (n2 - n1)( 1/ BC1 + 1/DC2)
Now BC1 = + R1 and DC2 = -R2
So the equation becomes
1/f =( n21 - 1)( 1/R1 - 1/R2) as n21 = n2/n1
The above equation is known as lens maker's formula .
The formula is equally valid for concave lens also .In concave lens R1 is nagative , R2 positive giving f negative .
Derivation of thin lens formula :
Considering the refraction at two spherical surfaces , we have the equation :
n1/OB + n1/DI = ( n2 - n1)( 1/ BC1 + 1/BC2)
Or 1/OB + 1 /DI = (n21 - 1)(1/R1 - 1/R2 ) ( OB is object distance = -u, DI is image distance = v and using lens maker formula )
Or , 1/-u + 1/v = 1/f
Or, 1/v - 1/u = 1/f
It is known as lens formula for thin lens .
Image formation by a lens :
To locate the image of a point of an object kept on th principal axis we have to follow the path of any two rays emerging from the point and undergoes refraction at the lens and then find the point where the refracted rays meet .This meeting point is called the image of the point .
We may consider any two of the following three rays undergoing refraction.
i) A ray emanating from the point and parallel to the principal axis after refraction passes through the second focus F'
ii) A ray passing through the optical centre passes undeviated after refraction.
iii) A ray emerging from the object and passing through the 1st focus emerges parallel to the principal axis
In case of concave lens the rays will not meet after refraction but they seem to come out from a point .
The power of a lens is a measure of convergence or divergence of a lens when light falling on it . The power P is the tangent of the angle by which it is converged or diverged when light falling parallel to principal axis falling at a unit distance from optical centre .
tan (delta) = h/f ; if h = 1, then tan(delta)= 1/f
Delta= 1/f
Delta = 1/f for small values of delta
P = 1/f
The SI unit of power 1D = 1 m^-1
The power of 1 metre focal length lens is 1 doipotre . It is positive for converging lens and negative for divergence lens .
Combination of thin lenses in contact :
1/v1 - 1/u = 1/f1
For the image formed by 2nd lens B we have ,
1/v -1/v1 = 1/f2
From the above two relations we get ,
1/v - 1/u = 1/f1 + 1/f2 ,
From the above relation we can say that if the focal length of the combination is f then
1/f = 1/f1 + 1/f2
Here two lenses can be regarded as a single lens with focal length f
If there are more than two lenses the formula can be written as
1/f = 1/f1 +1/f2 + 1/f3 + ........
So the power of the equivalent lens is P = P1 +P2 +P3 +.........
If the individual magnification is m1 ,m2,m3 etc then total magnification m is given by
m = m1 m2 m3 .......
Such a combination is used to design lens for cameras, microscope and telescope etc
Refraction through a prism :
In the quadrilateral AQNR , the angles at Q and R are right angles .So the sum of other angles is 180 degree
Angle A + angle QNR = 180 degree
From the triangle QNR , r1 + r2 + QNR = 180 degree
Comparing the above two equations
A = r1 +r2
The total deviation delta is the sum of deviations at two surfaces
∆ = ( i - r1 ) + ( e - r2)
Thus ∆ = i + e - A
Thus angle of deviation depends on the angle of incidence i . For any value of delta we have two values i and e since path of light is reversible.
At the minimum deviation i = e and hence r1 = r2 .So we have at minimum deviation
2r = A or r= A/2
So Dm = 2i - A
Or i = ( A + Dm)/2
So the refractive index of the prism
n21
= n2/n1
= sin[(A + Dm)/2]/sin[A/2]
= (A + Dm)/2÷ A/2 ( for angles ,thin prism )
Dm = (n21 - 1) / A
It means prism do not deviate light much .
Optical Instruments :
A number of optical devices are designed utilising the property of refraction and reflection. Microscope , telescope and binocular are formed using the above properties of light .
The Microscope :
m = v/u = v( 1/v - 1/f) = ( 1 - v/f) .
If v= D , for clear vision and using the sign convention,
m = ( 1 + D/f) , here D is about 25 cm . magnification can also be written as m = h'/h where h' is image size ,and h is object size
If ,theta is the angle subtanded by object kept at D then
tan 'theta' = h/D = theta, for small angle .
In discussion of microscope and telescope we assume image is at infinity.
A simple microscope has a limited maximum magnification (< 9 ) for realistic focal length . For much larger magnification one uses two lenses compounding the effect .This is known as compound microscope.
The lens nearest to the object called objective form real inverted magnified image .This serves as object for the second lens called eye piece that functions as simple microscope producing enlarge virtual image .
β
magnification produced by objective m zero = h'/h = L/fo .magnification produced by eyepiece me = 1 + D/fe
For infinity focusing
me = D/fe
The final magnification,
m = mo me =( L/fo) (D/fe)
Telescope:
The telescope is used to see distant objects and it provides angular magnification. The angular magnification can be written as the ratio of angle subtainded by image and the angle that makes by the object at the objective .
So , m = mo .me
Or , m = tan β/ tan α = h/fe ÷h/fo
Hence ,m = fo/fe
Terrestrial telescope has in addition a pair of investing lens to make the final image erect. Refracting telescope can be used as both astronomical and terrestrial observation.
The main considerations with astronomical telescope are its light gathering power and its resolving power .The former is clearly depends on area of objective .With larger diameter of objective fainter objects can be seen .The resolving power is the ability to observe two distinct separated by small angular separation . it also depends on diameter of objective . But it is very difficult and expensive to make very large size lens that are free from also chromatic aberration and distorsion .For these reason modern telescope use concave mirror objective. Such telescope is called reflecting telescope .
One problem of the reflecting telescope is that it focuses light inside the telescope tube . One must have an observer and eye piece there obstructing some light .The solution to the problem is to deflect light bring focused by another convex mirror and the reflecting light passes through a hole in the objective.
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