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Optics

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Optics PDF Notes, Important Questions and Formulas

Geometrical Optics

     1. Properties of light

  1. Speed of light in vaccum, denoted by c, is equal to 3 x 108 m/s approximately
  2. Light is electromagnetic wave (proposed by Maxwell). It consists of varying electric field.
  3. Light carries energy and momentum.
  4. The formula v = fλ is applicable to light.

     

     2. Ray optics

Ray optics treats propagation of light in terms of rays and is valid only if the size of the obstacle is much greater than the wavelength of light. It concern with the image formation and deals with the study of the simply facts such as rectilinear propagation, laws of reflection and refraction by geometrical methods.

2.1 Ray
A ray can be defined as an imaginary line drawn in the direction in which light is travelling. Light behaves as a stream of energy propagated along the direction of rays. The rays are directed outward from the source of light in straight lines.

2.2 Beam of Light
A beam of light is a collection of these rays. There are mainly three types of beams.

  1. Parallel


    A search light and the headlight of a vehicle emit a parallel beam of light. The source of light at a very large distance like sun effectively gives a parallel beam

  2. Divergent beam of light


    The rays going out from a point source generally from a divergent beam.

  3. Convergent beam of light
    A beam of light that is going to meet (or converge) at a point is known as a convergent beam. A parallel beam of light after passing through a convex lens becomes a convergent beam.



     3. Reflection

When a ray of light is incident at a point on the surface, the surface throws partly or wholly the incident energy back into the medium of incidence. This phenomenon is called reflection. Surfaces that cause reflection are known as mirrors or reflectors. Mirrors can be plane or curved.

            



In the figure 1.5,
O is the point of incidence, AO is the incident ray,
OB is the reflected ray, ON is the normal at the incidence.

Angle of incidence
The angle which the incident ray makes with the normal at the point of incidence is called the angle of incidence. It is generally denoted by 'i'.

Angle of reflection
The angle which the reflected ray makes with the normal at the point of incidence is called the angle of reflection. It is generally denoted by 'r'.

Glancing angle
The angle which the incident ray makes with the plane reflecting surface is called glancing angle. It is generally denoted by 'g'.
g = 90° – i

3.1  Law of reflection

  1. The incident ray, the reflected ray and the normal to the reflecting surface at the point of incidence, all lie in the same plane.
  2. The angle of incidence is equal to the angle of reflection, i.e., ∠I = ∠r
    These laws hold good for all reflecting surfaces either plane or curved.

 

     Some important points

  1. If ∠I =0, ∠r=0, i.e., if a ray is incident normally on a boundary, after reflection it retraces its path.



                                         
  2. None of the frequency wavelength and speed changes due to reflection. However, intensity and hence amplitude (Iα A2) usually decreases.

  3. If the surface is irregular, the reflected rays on an incident beam of parallel light rays will be in random direction. Such an irregular reflection is called diffused reflection.

 

 

Wave Optics 


ELECTROMAGNETIC SPECTRUM

 


Visible light is that part of electromagnetic spectrum which is visible to us Light is studied under two sections.

1. Geometrical optics (If the dimensions of body is larger as compared to wavelength of light)

2.  Wave optics (If the dimensions of body is comparable to wavelength of light)

 

WAVE FRONT

    • Wave front is a locus of particles having same phase.
    • Direction of propagation of wave is perpendicular to wave front.
    • Every particle of a wave front act as a new source & is known as secondary wavelet.

 

Shape of wavefronts vary from source to source.

Point source → Spherical wave Fronts

Distant Parallel Rays →Planar wave front

Line source → cylindrical wave fronts



Coherent source

If the phase difference due to two source at a particular point remains constant with time, then the two sources are considered as coherent source


Note

Sources lying on same wavefront are coherent in nature because their phase difference=0



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PRINCIPLE OF SUPERPOSITION

When two or more waves simultaneously pass through a point, the disturbance of the point is given by the sum of the disturbances each wave would produce in absence of the other wave(s). In case of wave on string disturbance means displacement, in case of sound wave it means pressure change, in case of electromagnetic wave (E.M.W) it is electrified or magnetic field. Superposition of two light travelling in almost same direction results in  modification in the distribution of intensity of light in the region of superposition. This phenomenon is called interference.

 

Superposition of two sinusoidal waves

Consider superposition of two sinusoidal wave (having same frequency), at a particular point.

Let, x1(t) = a1 sin ωt 

  x2(t) = a2 sin (ωt + ϕ)

x=A sin (ωt + ϕ0)

Where A2=a12+a22+2a1.a2 cosϕ            …… (2.1)

{Refer topic: combination of SHM}

begin mathsize 12px style And      tan straight space straight ϕ subscript degree equals fraction numerator straight a subscript 2 space sin space straight ϕ over denominator straight a subscript 1 plus straight a subscript 2 space cos space straight ϕ end fraction                  . ....  ( 2.2 ) end style

INTERFERENCE

Interference implies super position of waves. Whenever two or more than two waves superimpose each other they give sum of their individual displacement.

Let the two waves coming from sources S1 and S2 be

y1= A1 sin (ωt +kx1)

y2= A2 sin (ωt + kx2)   respectively.

Due to superposition

ynet = y1 +y2

ynet =A1 sin (ωt +kx1)+ A2 sin (ωt +kx2)

Phase difference between y1 & y2=k(x2 –x2)

i.e., ∆ϕ=k(x1-x2)

begin mathsize 12px style As    space space Δϕ equals fraction numerator 2 straight pi over denominator straight lambda end fraction Δx end style

(Where ∆x= path difference & ∆ϕ=phase difference)

begin mathsize 12px style straight A subscript net equals square root of straight A subscript 1 superscript 2 plus straight A subscript 2 superscript 2 plus 2 straight A subscript 1 straight A subscript 2 cosϕ end root end style

⇒Anet2=A12+ A12+2A1A2cosϕ

begin mathsize 12px style therefore space straight I subscript net equals straight I subscript 1 plus straight I subscript 2 plus 2 square root of straight I subscript 1 end root straight I subscript 2 cosϕ space left parenthesis as space straight I proportional to space straight A squared right parenthesis end style

When the two displacement are in phase, then the resultant amplitude will be sum of the two amplitude & Inet will be maximum, this is known of constructive interference.

For Inet to be maximum

Cos ϕ =1

⇒ ϕ = 2nπ

Where n= {0, 1, 2, 3, 4, 5…..}

begin mathsize 12px style fraction numerator 2 straight pi over denominator straight lambda end fraction Δx equals 2 nπ end style

⇒ ∆x = nλ

For constructive interference

begin mathsize 12px style straight I subscript net equals left parenthesis square root of straight l subscript 1 end root plus square root of straight l subscript 2 end root right parenthesis squared end style

When I1=I2=I

           Inet= 4 I

     Anet = A1+ A2

When superposing waves are in opposite phase, the resultant amplitude is the difference of two amplitude & I­net is minimum; this is known as destructive interference.

For Inet to be minimum,

Cos ∆ϕ =-1

∆ϕ = (2n + 1) π

Where n = {o, 1, 2, 3, 4, 5…}

begin mathsize 12px style table attributes columnalign left end attributes row cell space space space space space fraction numerator 2 straight pi over denominator straight lambda end fraction Δx equals left parenthesis 2 straight n plus 1 right parenthesis straight pi end cell row cell rightwards double arrow Δx equals left parenthesis 2 straight n plus 1 right parenthesis straight lambda over 2 end cell end table end style

For destructive interference

begin mathsize 12px style straight I subscript net equals left parenthesis square root of straight I subscript 1 end root minus square root of straight I subscript 2 end root right parenthesis squared end style

If I1= I2

Inet = 0

Anet=A1 –A2

Generally,

If I1=I2=I

Inet=2I +2I cosϕ

begin mathsize 12px style table attributes columnalign left end attributes row cell straight I subscript net equals 2 straight I left parenthesis 1 plus cosϕ right parenthesis equals 4 Icos squared ΔQ over 2 end cell row cell Ratio space of space straight I subscript max  &  straight I subscript min equals left parenthesis square root of straight I subscript 1 end root plus square root of straight I subscript 2 end root right parenthesis squared over left parenthesis square root of straight I subscript 1 end root minus square root of straight I subscript 2 end root right parenthesis squared end cell end table end style



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