Derive the Mathematical Relation Between Refractive Indices N1 and N2 of Two Radii and Radius of Curvature R for Refraction at a Convex Spherical Surface. - Physics

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(i) Derive the mathematical relation between refractive indices n1 and n2 of two radii and radius of curvature R for refraction at a convex spherical surface. Consider the object to be a point since lying on the principle axis in rarer medium of refractive index n1 and a real image formed in the denser medium of refractive index n2. Hence, derive lens maker's formula.

(ii) Light from a point source in air falls on a convex spherical glass surface of refractive index 1.5 and radius of curvature 20 cm. The distance of light source from the glass surface is 100 cm. At what position is the image formed?

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Solution

 
 

(i)

The above figure shows the geometry of formation of real image I of an object O and the principal axis of a spherical surface with centre of curvature C and radius of curvature R

Assumptions:

(i) The aperture of the surface is small compared to other distance involved.

(ii) NM will be taken to be nearly equal to the length of the perpendicular from the point N on the principal axis.

`tan /_NOM=(MN)/(OM)`

 `tan/_NCM=(MN)/(MC)`

`tan /_NIM=(MN)/(MI)`

 For ΔNOC, i is the exterior angle.
Assuming the incident ray is very close to the principal axis, all the angles are very small. Hence, for very small angles

tan x = x = sin x

`:.i=/_NOM+/_NCM`

`i=(MN)/(OM)+(MN)/(MC)" .....(i)"`

Similarly, r = ∠NCM − ∠NIM

i.e,`r=(MN)/(MC)-(MN)/(MI)" ...(ii) "`

By Snell’s law,

n1sini = n2sinr

For small angles

n1in2 r

Substituting the values of i and r from equations (i) and (ii), we obtain

`n_1((MN)/(OM)+(MN)/(MC))=n_2((MN)/(MC)-(MN)/(MI))`

Or,`n_1/(OM)+n_2/(MI)=(n_2-n_1)/(MC)" ....(iii)"`

Applying new Cartesian sign conventions,

OM = − u, MI = + v, MC = + R

Substituting these in equation (iii), we obtain

`n_2/v-n_1/u=(n_2-n_1)/R`

...(iV)

 

(ii)

`n_2/v-n_1/u=(n_2-n_1)/R`

`=>1.5/V-1/((-100))=(1.5-1)/20=1/40`

`=>1.5/V=1/40-1/100=(5-2)/200=3/200`

`=>v=(1.5xx200)/3=100 cm`

Hence, the image is formed 100 cm in the denser medium
 

 


 
 
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2015-2016 (March) All India Set 2 C

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RELATED QUESTIONS

A point object 'O' is kept in a medium of refractive index n1 in front of a convex spherical surface of radius of curvature R which separates the second medium of refractive index n2 from the first one, as shown in the figure

Draw the ray diagram showing the image formation and deduce the relationship between the object distance and the image distance in terms of n1, n2 and R.

When the image formed above acts as a virtual object for a concave spherical surface separatig the medium n2 from n1 (n2 > n1), draw this ray diagram and write the similar (similar to (a)) relation. Hence obtain the expression for the lens maker's formula.

 


Mrs. Rashmi Singh broke her reading glasses. When she went to the shopkeeper to order new spects, he suggested that she should get spectacles with plastic lenses instead of glass lenses. On getting the new spectacles, she found that the new ones were thicker than the earlier ones. She asked this question to the shopkeeper but he could not offer satisfactory explanation for this. At home, Mrs. Singh raised the same question to her daughter Anuja who explained why plastic lenses were thicker. 

(a) Write two qualities displayed each by Anuja and her mother.

(b) How do you explain this fact using lens maker's formula?


Starting with an expression for refraction at a single spherical surface, obtain Lens Maker's Formula.


Lens maker’s formula is _________.


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A plano-convex lens is made of glass with a refractive index of 1.5. The radius of curvature of the convex surface is 25 cm.

  1. Calculate the focal length of the lens.
  2. If an object is placed 50 cm in front of the lens, find the nature and position of the image formed.

The power of a thin lens is +5 D. When it is immersed in a liquid, it behaves like a concave lens with a focal length of 100 cm. Calculate the refractive index of the liquid. Given the refractive index of glass = 1.5.


Read the following paragraph and answer the questions.

A lens is a transparent optical medium bounded by two surfaces; at least one of which should be spherical. Considering image formation by a single spherical surface successively at the two surfaces of a lens, the lens maker's formula is obtained. It is useful to design lenses of desired focal length using surfaces of suitable radii of curvature. This formula helps us obtain a relation between u, v and f for a lens. Lenses form images of objects and they are used in a number of optical devices, for example, microscopes and telescopes.
  1. An object AB is kept in front of a composite convex lens, as shown in the figure. Will the lens produce one image? If not, explain.
  2. A real image of an object formed by a convex lens is observed on a screen. If the screen is removed, will the image still be formed? Explain.
  3. A double convex lens is made of glass with a refractive index of 1.55 with both faces of the same radius of curvature. Find the radius of curvature required if the focal length is 20 cm.
    OR
    Two convex lenses A and B of focal lengths 15 cm and 10 cm respectively are placed coaxially 'd' distance apart. A point object is kept at a distance of 30 cm in front of lens A. Find the value of 'd' so that the rays emerging from lens B are parallel to its principal axis.

Draw a ray diagram for the formation of the image of a point object by a thin double convex lens having radii of curvature R1 and R2. Hence derive lens maker’s formula.


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