Using lens maker’s formula, derive the thin lens formula 1 f = 1 v - 1 u for a biconvex lens.

The lens maker's formula is 1 v - 1 u = n - 1 1 R 1 - 1 R 2 ....................... 1 Here, n = n 2 n 1 that is the ratio of refractive index of lens to other medium. For a thin biconvex lens, assuming the object is at infinity (i.e. u = ∞ ) and the image is forming at focal point (i.e. v = f ). Substituting the above values in lens maker's formula we get, 1 f = n - 1 1 R 1 - 1 R 2 ..................... 2 From 1 and 2 , we get 1 v - 1 u = 1 f . Hence, thin lens formula 1 f = 1 v - 1 u has been derived for a biconvex lens.

A symmetric biconvex lens of the radius of curvature R and made of glass of refractive index 1 . 5 , is placed on a layer of liquid placed on top of a plane mirror as shown in the figure. An optical needle with its tip on the principal axis of the lens is moved along the axis until its real, inverted image coincides with the needle itself. The distance of the needle from the lens is measured to be x . On removing the liquid layer and repeating the experiment, the distance is found to be y . Obtain the expression for the refractive index of the liquid in terms of x and y .

We are given, the refractive index of the lens, μ g = 1 . 5 . The distance of the needle from the lens in the first case is equal to the focal length of the combination of a convex lens and a planoconcave lens formed by the liquid. Thus, f = x The distance measured in the second case is equal to the focal length of the convex lens. Thus, f 1 = y Let f 2 be the focal length of the planoconcave lens formed by the liquid. Then, 1 f = 1 f 1 + 1 f 2 1 f 2 = 1 x - 1 y = y - x x y f 2 = x y y - x Let the radius of curvature of the first surface be R The radius of curvature of the second surface will be - R Thus, for the biconvex lens, 1 f 1 = μ g - 1 1 R - 1 - R 1 y = 1 . 5 - 1 2 R ⇒ y = R Let the refractive index of liquid be μ . For plano-convex lens, R 2 = ∞ R 1 = - R Thus, 1 f 2 = μ - 1 1 - R - 1 ∞ Substituting values from above, y - x x y = μ - 1 1 - y μ - 1 = - ( y - x ) x μ = x - y x + 1 μ = 2 x - y x

The figure shows a ray of light falling normally on the face AB of an equilateral glass prism having a refractive index 3 / 2 , placed in water of refractive index 4 / 3 . Will this ray suffer total internal reflection on striking the face AC? Justify your answer.

It is given that the glass prism is equilateral. μ p = 3 / 2 μ w = 4 / 3 Thus, ∠ A = 60 ° . Let us draw a normal to the face AC. From the geometry, we can see, the angle of incidence on the face AC is 60 ° . Thus, i = 60 ° . Let this angle be the critical angle i.e. i = i c . Then the refractive index of the prism in the air, μ = 1 sin i c = 1 sin 60 ° μ = 2 3 = 1 . 54 Now if the prism is placed in water then the refractive index of a prism with water, μ A = μ p μ w = 3 2 4 3 = 9 8 = 1 . 125 The total internal reflection at face AC, is possible only when μ < μ A . But from the results obtained above, we can see that μ > μ A ∴ 1 . 54 > 1 . 125 No total internal reflection takes place and the light ray gets refracted from the prism placed in water. Thus, we can conclude that for total internal reflection, in this case, the Critical angle should be greater than 60 ° .

Using lens maker’s formula, derive the thin lens formula 1 f = 1 v - 1 u for a biconvex lens.

The lens maker's formula is 1 v - 1 u = n - 1 1 R 1 - 1 R 2 ....................... 1 Here, n = n 2 n 1 that is the ratio of refractive index of lens to other medium. For a thin biconvex lens, assuming the object is at infinity (i.e. u = ∞ ) and the image is forming at focal point (i.e. v = f ). Substituting the above values in lens maker's formula we get, 1 f = n - 1 1 R 1 - 1 R 2 ..................... 2 From 1 and 2 , we get 1 v - 1 u = 1 f . Hence, thin lens formula 1 f = 1 v - 1 u has been derived for a biconvex lens.

An object is placed 30 cm in front of a plano-convex lens with its spherical surface of radius of curvature 20 cm . If the refractive index of the material of the lens is 1 . 5 , find the position and nature of the image formed.

The radius of curvature of the spherical surface R 1 = 20 cm The radius of curvature of the plane surface R 2 = infinite Using Lens makers' formula to find the focal length f of the plano-convex lens: 1 f = μ - 1 1 R 1 - 1 R 2 1 f = 1 . 5 - 1 1 20 cm = 1 40 cm ⇒ f = 40 cm Using the lens equation: 1 f = 1 v - 1 u 1 40 = 1 v + 1 30 1 v = 1 40 - 1 30 = - 1 120 ⇒ v = - 120 cm The image will be formed at a distance of 120 cm in front of the lens. Image will be virtual in nature.

An object is placed in front of a concave mirror. It is observed that a virtual image is formed. Draw the ray diagram to show the image formation and hence derive the mirror equation 1 f = 1 u + 1 v .

Ray diagram to show the formation of a virtual image by concave mirror: Derivation of mirror equation using the above ray diagram: ∆ C A ' B ' and ∆ C A B are similar triangles. Hence, we can write: A ' B ' A B = C B ' C B = C P + P B ' C P - B P .......(i) D N is equal to A B and is a normal on the principal axis. ∆ F D N and ∆ F A ' B ' are similar triangles. The incident rays are considered to be paraxial rays hence, D and P are very close. We can write: A ' B ' A B = F B ' F D = F P + P B ' F D = F P + P B ' F P .....(ii) Using the above equations and substituting the values according to the sign convention we get: C P + P B ' C P - B P = F P + P B ' F P - 2 f + v - 2 f + u = - f + v - f ⇒ - 2 f 2 - f v = 2 f 2 - f u - 2 v f = u v ⇒ f u + f v = u v f On dividing both sides by u v f we get: 1 u + 1 v = 1 f The above equation is the required mirror equation.

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An equilateral glass prism has a refractive index $1.6$ in the air. Calculate the angle of minimum deviation of the prism, when kept in a medium of refractive index $\frac{4 \sqrt{2}}{5}$ .

Important Questions on Ray Optics and Optical Instruments

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EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR PHYSICS>Chapter 9 - Ray Optics and Optical Instruments>cbse-2019>Q 8

Draw a labelled ray diagram of an astronomical telescope in the near point adjustment position.

A giant refracting telescope at an observatory has an objective lens of focal length $15 m$ and an eyepiece of focal length $1.0 cm$ . If this telescope is used to view the Moon, find the diameter of the image of the Moon formed by the objective lens. The diameter of the Moon is $3.48 \times 10^{6} m$ , and the radius of the lunar orbit is $3.8 \times 10^{8} m$ .

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EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR PHYSICS>Chapter 9 - Ray Optics and Optical Instruments>cbse-2019>Q 9

Under what conditions is the phenomenon of total internal reflection of light observed? Obtain the relation between the critical angle of incidence and the refractive index of the medium.

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EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR PHYSICS>Chapter 9 - Ray Optics and Optical Instruments>cbse-2019>Q 10

Three lenses of focal length $+ 10 c m, - 10 c m, + 30 c m$ are arranged coaxially as in the figure given below. Find the position of the final image formed by the combination.

Question Image

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EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR PHYSICS>Chapter 9 - Ray Optics and Optical Instruments>cbse-2020>Q 11

A biconvex lens of glass having refractive index

1.47

is immersed in a liquid. It becomes invisible and behaves as a plane glass plate. The refractive index of the liquid is

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12th CBSE

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EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR PHYSICS>Chapter 9 - Ray Optics and Optical Instruments>cbse-2020>Q 12

For a glass prism, the angle of minimum deviation will be smallest for the light of

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12th CBSE

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EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR PHYSICS>Chapter 9 - Ray Optics and Optical Instruments>cbse-2020>Q 13

A ray of light on passing through an equilateral glass prism, suffers a minimum deviation equal to the angle of the prism. The value of refractive index of the material of the prism is _____.

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EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR PHYSICS>Chapter 9 - Ray Optics and Optical Instruments>cbse-2020>Q 14

Using lens maker’s formula, derive the thin lens formula

\frac{1}{f} = \frac{1}{v} - \frac{1}{u}

for a biconvex lens.

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12th CBSE

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EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR PHYSICS>Chapter 9 - Ray Optics and Optical Instruments>cbse-2020>Q 15

Draw the ray diagram of an astronomical telescope when the final image is formed at infinity. Write the expression for the resolving power of the telescope.

An equilateral glass prism has a refractive index 1.6 in the air. Calculate the angle of minimum deviation of the prism, when kept in a medium of refractive index 425.

Important Questions on Ray Optics and Optical Instruments

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An equilateral glass prism has a refractive index $1.6$ in the air. Calculate the angle of minimum deviation of the prism, when kept in a medium of refractive index $\frac{4 \sqrt{2}}{5}$ .