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Optics

1. Sign Convention

(a) All the distances are measured from pole.
(b) Distances measured in the direction of incident ray are taken to be positive while the distances measured in opposite direction to the incident ray are taken as negative.
(c) Heights above the principle axis are positive while the heights below the principe axis are negative.
(d) With or without optic centre (it being thin lens)as origin of coordinate system, the distances are heights (sizes) of objects and images are given positive, negative signs as per their location (quadrant) in the rectangular axes system.
 

2. Reflection And Refraction

(a) Laws of reflection
(i) The angle of incidence is equal to the angle of reflection.
(ii) The incident ray, the normal to the mirror at the point of incidence, and the reflected ray are in the same plane.

(b) Rotation of mirrors
When a mirror is turned through an angle θ, for the same incident ray, the reflected ray is turned through the angle 2θ.

(c) Plane mirror
The image formed by a plane mirror is as far behind the mirror as the object is in front of it. The image is erect, virtual, is of the same size as the object, and is laterally inverted.

(d) Deviation produced by a plane mirror
 δ = 180° - 2i

(e) Combination of two plane mirrors
If two plane mirrors are inclined at angle θ, then number of images formed
= Combination of two plane mirrors, ifCombination of two plane mirrors  is an even integer.
= Combination of two plane mirrors,Combination of two plane mirrors if is an odd integer, and object is placed unsymmetrical to the mirrors.
= Combination of two plane mirrors, Combination of two plane mirrorsif is an odd integer, and object is placed symmetrical to the mirrors.

(f) Spherical mirror
f = R/2.

(g) Mirror formula
1/v + 1/u = 1/f
Magnification produced by a spherical mirror

(h) Laws of refraction
(i) The incident ray, the normal to the refracting surface at the point of incidence and the refracted ray are in the same plane.
(ii) The ratio of the sine of the angle of incidence to the sine of the angle of refraction is constant for the two media.
1 sin θ1 = μ2 sin θ2


(i) Refractive index
When an object inside a medium of refractive index, μ is viewed normally from air,

(j) Refractive index in terms of velocity

(k)Shift due to a slab
When a slab of glass is placed between an object and observer, the object appears shifted through a distance.
Shift

(l)Critical angle
If C is the critical angle of a medium, the refractive index,
(m)Refraction through prism
(i)Angle of deviation,
    (ii)    
    (iii)In minimum deviation,
           
        Refraction through thin prism
    (iv)Angle of deviation,
    (v)Dispersive power,
    (vi)Angular dispersion
        
    (vii)For deviation without dispersion
    
    (viii)For dispersion without deviation

    (n)Refraction at a curved surface     

       (o)Lens maker’s formula
       
       (p)Thin lens formula
       
    Magnification (m) =

      (q)Silvering of lens
      If one of the surfaces of a lens is slivered as shown in figure. The rays are refracted at surface 1, reflected       at surface 2 and again refracted at surface 1. The focal length, F of the effective mirror is given by,
           
      where fn = focal length of lens or mirror repeated as many times as there are reflections and refractions.
      In the case shown in figure, there are two refractions and one reflection.
    Pmirror = Plens + Pmirror + Plens
    Hence,     
     (r)Power of lens
     The reciprocal of focal length (expressed in metres) of a lens is called its power (unit is dioptre)
     Equivalent focal length of two lenses in contact,  
     Power,
     If two lenses are separated by a distance d, then the focal length of the combination
               
               
    (s)Focal length f1 of lens immersed in liquid of μl
               
               
    where fa is the focal length of the lens in air



3. Optical Instruments


Magnifying power of an instrument is defined as
            

  (a)Simple microscope
           
  where D is the least distance of distinct vision
  (b)Compound microscope
    M
    If the final is formed at infinity,
  (c)Astronomical telescope
           
  For the normal adjustment,
  Maximum length of the telescope,
  For the Terrestrial telescope,
  For normal adjustment,
  Maximum length of the telescope, , where f is the focal length of the erecting lens.
  (d)Limit of resolution of telescope
   Limit


4. Interference


    (a)The wavelengths of visible light are between 4.3 × 10–7 m to 6.8 × 10–7 m.
    (b)Result of superposition of two wave trains with a phase difference, φ is
          
    (c)Relation between phase difference and path difference,
    where δ is the path difference and λ is the wavelength.
   (d)If two coherent waves with intensity I1 and I2 are superimposed with a phase difference of φ, the resulting wave intensity,
               
   The resulting amplitude, , where   
   (e)Young’s Double Slit Experiment
       (i)Condition for dark band , when n = 1, 2, 3…………
       (ii)Condition for bright band, δ = mλ, where m = 0,1,2 ………..
       (iii)Fringe width,
       where D is the distance between the double slits and the screen and a is the distance between the two slits
       (iv)Distance of mth dark fringe from centre , where m = 1, 2 ,3, etc
   (f)Thin films
    Path difference Δx = 2μt cos r for nth bright fringe
    Path difference Δx = 2μt cos r= mλ for mth dark fringe.
    Here, t is the thickness of film, μ is refractive index and r is the angle of refraction at first surface.


5. Diffraction


This is due to bending of waves from obstacles of size of the order of wavelength. Diffraction is of two types.
    (a)Fresnel’s diffraction
    Fresnel’s diffraction is near field diffraction. It may be assumed that rays from small imaginary strips to point P are not parallel.
    (b)Fraunhoffer diffraction
    Fraunhoffer diffraction is far field diffraction. The rays are parallel as the screen is at a very large distance. A converging lens will bring them at sharp focus on screen placed at the focus of lens.
               
   (c)Diffraction from a single slit
  and
   Where I0 is intensity at central maxima and d is width of slit and θ is angular position of a point with respect to central axis.


6. Polarization


   (a)If plane of vibration of electric and magnetic field is fixed then light will travel only in the direction perpendicular to the fixed plane of vibration and is called plane polarized light. In  the figure electric field is along y-axis and magnetic field along z-axis. Light travels along x-axis and plane of polarization is y-z.  
   
   Only transverse waves can be polarized. Plane polarized light can be achieved using
  (i)  Reflection  (ii)  refraction (iii) scattering  (iv) nicol prism (v) double refracting crystals
  (b)Brewster’s Law
  If light is incident on the interface of two media such that angle between reflected and refracted rays is 900 then reflected rays are plane polarized. Angle of incidence is called angle  of polarization (θP)
    Then μ = tan θP    
  (c)Malus’s Law
   When the plane of polarization is rotated by an angle θ then intensity of emergent light is given by , where I0 is intensity of incident polarized light


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