IIT JEE / CET – Main / Advanced Physics 2013 & 2014 Free Notes : Geometrical / Ray Optics

The beautiful LED lit toys being hawked by a kid - a night scene at a beach in Trivandrum.

O P T I C S

Def : The branch of physics dealing with the study of light and the broad phenomenon associated with the generation , transmission and detection of light. Here ‘ light ‘ refers to electromagnetic radiation having wavelengths greater than x-rays and shorter than microwaves i.e , visible light ( to human eye ).The energy basis for which is the excited atoms / molecules. Can be broadely divided into –

  • Geometrical / Ray optics : In this study , light is considered to travel in a straight line as a ray , and deals with the laws governing the reflection and refraction of light. The study , hence also involves the formation of images or an array of images.
  • Physical / Wave optics : In this branch of the study of light, the properties of light due to it’s wave nature like diffraction, interference and polarization are dealt with.

Light classification

  1. Ray –     The path of light travel , indicated by a straight line. The arrows indicate the direction of light propagation.
  2. Beam –   A group of rays. There are parallel , divergent and convergent beams.
  3. Pencil –  A very narrow beam of light , like that of a pointed laser.

Object – refers to the the source of light rays. It could be a pointed object or an extended object ( considered as a collection of points ).Objects are either 1. Real or 2. Virtual. Real objectAn object is considered a real object, if two or more incident rays actually emanate or seem to emanate from a point.  Virtual objectIn a virtual object, the rays appear to converge on a virtual point.

virtual object

Image : Refers to the representation of a physical object formed by a lens,mirror  or other optical devices. It is the point of convergence or apparent point of divergence of rays after interacting with an optical element. The optical element reflects or refracts the incident light rays , thereafter meeting at a point to form an image. Hence images could be  real or virtual.

  1. Real imageImages which can be caught on a screen.They are formed when the reflected or refracted rays actually meet or converge on a point.Note that both real and virtual objects can form real images.
  2. Virtual imageImages which cannot be caught on a screen. Here,rays do not meet at a point ,but appear to emanate from a point.Note that, here too, both real and virtual objects can form virtual images.

Images can be upright or inverted and magnified or dimnished.

real image
virtual image
  1. Law of rectilinear propagation of light –  light propagates in straight lines in homogenous medium.
  2. Law of independence of light rays –  rays do not disturb each other on intersection.
  3. Law of reversibility of light rays –  on reversing their direction , rays retrace their paths.

Reflection of light

The change in direction of light at an interface between two different media so that , (a part of  ) the light returns into the medium from which it originated.

Types of Reflection

  1. Regular ( Specular ) - When the reflection is from a perfect plane surface like that of a mirror.  Here,the reflected light has larger intensity in one direction than the other.
  2. Diffuse reflection -      When light strikes the surface of a (non-metallic) material it bounces off in all directions due to multiple reflections by the microscopic irregularities inside the material and by its surface, if it is rough. Thus, an ‘image’ is not formed. The exact form of the reflection depends on the structure of the material. This is the process which enables us to seen an object like a magazine or an apple.Interestingly , it’s the same reason why such objects don’t behave like mirrors – we cannot see our face on their surface.   
  3. Retro reflection - reflection from surfaces where , the light is returned in the direction from which it came.When flying over clouds illuminated by sunlight the region seen around the aircraft’s shadow will appear brighter, and a similar effect may be seen from dew on grass. This partial retro-reflection is created by the refractive properties of the curved droplet’s surface and reflective properties at the backside of the droplet.  A corner reflector is a retro reflector and it’s made by  placing three ordinary mirrors mutually perpendicular to one another. The image produced is the inverse of one produced by a single mirror. A surface can be made partially retroreflective by depositing a layer of tiny refractive spheres on it or by creating small pyramid like structures. In both cases internal reflection causes the light to be reflected back to where it originated. This is used to make traffic signs and automobile license plates reflect light mostly back in the direction from which it came. In this application perfect retroreflection is not desired, since the light would then be directed back into the headlights of an oncoming car rather than to the driver’s eyes.
  4. Complex conjugate reflection –  Light bounces exactly back in the direction from which it came due to a nonlinear optical process. Here , not only the direction of the light is reversed, but the actual wavefronts are reversed as well. A conjugate reflector can be used to remove aberrations from a beam by reflecting it and then passing the reflection through the aberrating optics a second time.
    corner reflector

diffuse reflection
regular reflection

Laws of Reflection -

  1. The incident ray , the reflected ray and the normal at the point of incidence lie in the same plane , called the plane of incidence ( or plane of reflection ).
  2. The angle of incidence ( the angle between the normal and the incident ray ) and the angle of reflection ( the angle between the reflected ray and the normal ) are equal .
laws of reflection

Note –

  • Normal incidencewhen light is incident normally , here  angle of incidence (i ) = angle of reflection(r) = 0 and δ = 180°.
  • Grazing incidencewhen light strikes a reflecting surface at a tangent,here i = r =90° and δ = 0°  or 360° .

Plane Mirror – Formed by polishing / silvering one surface of a plane thin glass plate. Beam of parallel rays of light , incident on a plane mirror,gets reflected , in the same mannerIt is the plane mirror , we use to see ourselves .They can be used to created the feeling of depth to a room and also to make instruments like periscope and  kaleidoscopeFeatures of reflected images of a plane mirror are -

  1. Distance of the object from the mirror is the same as that of the distance of the image from the mirror.Image is the single point where all the incident rays from a point object after reflection from a plane mirror meet.
  2. The line joining  a point object and its image is normal to the reflecting surface.
  3. The size of the image is the same as that of the object.
  4. The image is virtual for a real object and real for a virtual object.
  5. A ray of light incident on a plane mirror at 90o gets reflected from the mirror along the same path.
  6. 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.
Reflection by a plane mirror

Plane mirror and Image of Extended Object

If an extended object exists from points A to B , one could consider the whole object as a combination of infinite number of point objects extending from A to B. In such a case , every point object will form its own image and thus , all point images together form an extended image.The properties of such an extended image are -

  1. The size of the extended object equals that of the extended image.
  2. An extended object placed partallel to the plane mirror gives an upright image.
  3. An extended object placed perpendicular to the plane mirror gives an inverted image.

Left right reversal of Plane mirror images

An interesting characteristic of plane mirror images has to do with the orientation of the image. If you view an image of yourself in a plane mirror (like a bathroom mirror), you will quickly notice that there is an apparent left-right reversal of the image. That is, if you raise your left hand, you will notice that the image raises what would seem to be it’s right hand. If you raise your right hand, the image raises what would seem to be its left hand.While there is an apparent left-right reversal of the orientation of the image, there is no top-bottom vertical reversal. If you stand on your feet in front of a plane mirror, the image does not stand on its head. Similarly, the ceiling does not become the floor. The image is said to be upright, as opposed to inverted.( Of course , if you do a sirshasana,[ head on floor ] , the case would be different )

Velocity of object and image

For plane mirrors , the distance of the object from the mirror is equal to the equal to the distance of the image from the mirror. Considering the x ,y and z co-ordinates, let

  • Xim  – x co-ordinate of the image with respect to the mirror ,
  • Yim – y co-ordinate of the image with respect to the mirror,
  • Zim  – z co-ordinate of the image with respect to the mirror,
  • Xom – x co-ordinate of the object with respect to the mirror,
  • Yom – y co-ordinate of the object with respect to the mirror and
  • Zom – z co-ordinate of the object with respect to the mirror…    then,

Xim = – Xom  ,  Yim = – Yom    ,  Zim = – Zom.   Now, differentiating with respect to time ,

If  Vom – velocity of object with respect to mirror   and Vim – velocity of image with respect to mirror , then -

V(im)x = – V(om)x  ,    V(im) y  =  V(om)y  ,  and  V(im)z  =   V(om)z

Therefore for x axis   =>       Vi  – Vm  =   ( Vo – Vm ) , considering the direction normal to the mirror , the relative velocity of the image w.r.t mirror = relative velocity of the object w.r.t mirror. If  Vi = velocity of the image w.r.t ground and Vo = velocity of the object w.r.t ground , then….                      Vi – Vo = ( Vo – Vm )   i.e ,    Vi   = Vo  ( for y and z axis )

Hence , the velocity of object is equal to velocity of image when parallel to the mirror surface.

Image formation by two Plane mirrors

Consider an object placed between two plane mirrors , which are inclined at an angle θ , then the number of images is given by the relation  If is not a whole number, then the number of images ought to be rounded off to the nearest integer. If the mirrors are inclined at 120o the number of images formed by the mirrors is given by the relation 

plane mirrors inclined at 120 degrees.

Here , O is the object placed between the two mirrors , and two images – O1 and O2 are produced.

When the angle is  90° -

plane mirrors at 90 degrees.

Here , the number of images formed by the mirror is Points to note –

  • OA and OB are are the  incident rays , where , OA being normal to the surface retraces its path.
  • OB makes an angle i with the normal N and gets reflected along BC.
  • OD and OE are the rays which are incident on the mirror MM’ , of which the perpendicular ray ( OD ) gets reflected along the same path.
  • O1 and O2 are virtual images of O , and their positions coincide.
  • Rays BC and EF gets internally reflected.
  • O3 and O4 are the images of O1 and O2 respectively.

When the angle is 0°

plane mirrors at 0 degrees

This is a very interesting case resulting in infinite image formation . But we see only finite images , as the  intensity of the distant images goes on decreasing due to absorption of light energy at every successive reflection.

Spherical Mirror –  refers to a mirror, either convex or concave, whose surface forms part of a sphere .  Important sections are

  1. Pole ( P ) – refers to the geometrical centre of the spherical reflecting surface.
  2. Radius of curvature (R) – it’s the radius of the sphere , of which the mirror belongs to.A plane mirror could be considered a special case of a spherical mirror , whose radius of curvature is infinite.
  3. Principal axis – refer to the straight line joining the centre of curvature to the pole.
  4. Principal focus ( F) – refers to the point of intersection of all the reflected rays of the incident rays ,which are striking the mirror , parallel to the principal axis. ( real for concave and virtual for convex mirrors ). For spherical mirrors , F = R/2.
  5. Focal length ( f) – refers to the distance between the pole and the principal focus.
  6. Aperture - diameter of the mirror.
  7. Focus ( F ) – When a narrow beam of light rays , parallel to the principal axis and close to it, is incident on the surface of a mirror , the reflected beam is found to converge to or appear to diverge from a point on the principal axis called the focus.

Sign ( Cartesian ) Conventions used -

  1. All distances are to be measured from the pole.
  2. Distances which are measured in the direction of incident rays are taken to be positive.
  3. Distances which are measured in the direction opposite to that of incident rays are taken to be negative.
  4. Distances above the principal axis are considered as positive.
  5. Distances below the principal axis are considered as negative.
  6. Angles measured from the normal in anti-clockwise manner are taken as positive.
  7. Angles measured from the normal in clockwise manner are taken as negative.

Mirror Formula of Spherical Mirrors –            consider the figures given below –

Here , I is the image formed by a point object O. According to the laws of reflection –

angle at OBC = angle at PCB = θ ,  α + θ = β    ,  β + θ = γ   ,  α + γ = 2 β     and

=>    α = tan θ   ,   β  = tan β   , tan γ

=>    tan α + tan γ  = 2 tan β    ,

=>    BP ‘/OP ‘ +  BP ‘/IP ‘  = 2 BP ‘/ CP ‘     , now using sign conventions –

u = – OP  ,  v = – IP  ,  R = – CP  ,

=>      – 1 /u  +  [ – 1/v ] = – 2/R  ,

If u is at infinity , 1/v = 2 /R  , but when u = ∞ , v = f  ,

therefore , f = R/2      AND   1 / u  +  1 / v   =   1 / f (  same is the case with convex mirrors )  see dig. below –

mirror formula - convex

Based on the polished part , they are  either  –

  • 1.  Concave ( Converging ) mirror –    it has a reflecting surface that bulges inward (away from the incident light). They  refocus parallel incoming rays toward a point , thus reflecting  light inward to one focal point, and are hence used to focus light.  ( Reason – the light is reflected at different angles, since the normal to the surface differs with each spot on the mirror) .Unlike convex mirrors, concave mirrors show different image types depending on the distance between the object and the mirror.
concave mirror
Effect on image of object’s position relative to mirror focal point (Concave)
Object’s position (S),
focal point (F)
Image Diagram
S < F
(Object between focal point and mirror)
  • Virtual
  • Upright
  • Magnified (larger)
Concavemirror raydiagram F.svg
S = F
(Object at focal point)
  • the image is formed at infinity.

(Note that the reflected light rays are parallel and do not meet the others. In this way, no image is formed or more properly the image is formed at infinity.)

Concavemirror raydiagram FE.svg
F < S < 2F
(Object between focus and centre of curvature)
  • Real
  • Inverted (vertically)
  • Magnified (larger)
Concavemirror raydiagram 2FE.svg
S = 2F
(Object at centre of curvature)
  • Real
  • Inverted (vertically)
  • Same size
  • Image formed at centre of curvature
Image-Concavemirror raydiagram 2F F.svg
S > 2F
(Object beyond centre of curvature)
  • Real
  • Inverted (vertically)
  • Reduced (diminished/smaller)
Concavemirror raydiagram 2F.svg

Locating the image -

By taking into account any of the two ray paths from the paths given below , the image of an extended object can be located using a graphical drawing-

  1. A ray incident at the pole is reflected symmetrically.
  2. A ray moving through the centre of curvature is reflected back along itself.
  3. A ray initially parallel to the principal axis is reflected through the focus.
  4. A ray initially passing through the focus is reflected parallel to the principal axis.

( to continue……….. )

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