Karnataka Board PUCPUC Science 2nd PUC Class 12

A Hemispherical Portion of the Surface of a Solid Glass Sphere (μ = 1.5) of Radius R is Silvered to Make the Inner Side Reflecting. an Object is Placed on the Axis of the Hemisphere at - Physics


A hemispherical portion of the surface of a solid glass sphere (μ = 1.5) of radius r is silvered to make the inner side reflecting. An object is placed on the axis of the hemisphere at a distance 3r from the centre of the sphere. The light from the object is refracted at the unsilvered part, then reflected from the silvered part and again refracted at the unsilvered part. Locate the final image formed.



As shown in the figure, OQ = 3r and OP = r
Thus, PQ = 2r
For refraction at APB,
we know that,

\[\frac{\mu_2}{v} - \frac{\mu_1}{u} = \frac{\mu_2 - \mu_1}{R}\] 

\[ \Rightarrow \frac{1 . 5}{v} - \left( \frac{1}{- 2r} \right) = \frac{0 . 5}{r} = \frac{1}{2r}      \left( \because u = - 2r \right)\] 

\[ \Rightarrow v =  \infty \] 
For the reflection in the concave mirror,
u = ∞
Thus, = focal length of mirror = r/2
For the refraction of APB of the reflected image,
u = −3r/2

\[\Rightarrow \frac{1}{v} - \frac{1 . 5}{3r/2} = \frac{- 0 . 5}{- r}    \left( Here, \mu_1 = 1 . 5  , \mu_2 = 1  and  R = - r \right)\] 

\[ \Rightarrow v =  - 2r\] 
As negative sign indicates images formed inside APB, so image should be at C.
Therefore, the final image is formed on the reflecting surface of the sphere.

Concept: Concave Mirror
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Chapter 18: Geometrical Optics - Exercise [Page 415]


HC Verma Class 11, Class 12 Concepts of Physics Vol. 1
Chapter 18 Geometrical Optics
Exercise | Q 46 | Page 415

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A convex lens of focal length f1 is kept in contact with a concave lens of focal length f2. Find the focal length of the combination. 

Find the diameter of the image of the moon formed by a spherical concave mirror of focal length 7.6 m. The diameter of the moon is 3450 km and the distance between the earth and the moon is 3.8 × 105 km.

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A concave mirror of radius R is kept on a horizontal table (See figure). Water (refractive index = μ) is poured into it up to a height h. Where should an object be placed so that its image is formed on itself?

The convex surface of a thin concavo-convex lens of glass of refractive index 1.5 has a radius of curvature 20 cm. The concave surface has a radius of curvature 60 cm. The convex side is silvered and placed on a horizontal surface as shown in figure. (a) Where should a pin be placed on the axis so that its image is formed at the same place? (b) If the concave part is filled with water (μ = 4/3), find the distance through which the pin should be moved so that the image of the pin again coincides with the pin.

A particle is moving at a constant speed V from a large distance towards a concave mirror of radius R along its principal axis. Find the speed of the image formed by the mirror as a function of the distance x of the particle from the mirror.

A small block of mass m and a concave mirror of radius R fitted with a stand lie on a smooth horizontal table with a separation d between them. The mirror together with its stand has a mass m. The block is pushed at t = 0 towards the mirror so that it starts moving towards the mirror at a constant speed V and collides with it. The collision is perfectly elastic. Find the velocity of the image (a) at a time t < d/V, (b) at a time t > d/V.

A gun of mass M fires a bullet of mass m with a horizontal speed V. The gun is fitted with a concave mirror of focal length f facing towards the receding bullet. Find the speed of separation of the bullet and the image just after the gun was fired.

A mass m = 50 g is dropped on a vertical spring of spring constant 500 N m−1 from a height h = 10 cm as shown in figure. The mass sticks to the spring and executes simple harmonic oscillations after that. A concave mirror of focal length 12 cm facing the mass is fixed with its principal axis coinciding with the line of motion of the mass, its pole being at a distance of 30 cm from the free end of the spring. Find the length in which the image of the mass oscillates.

Two concave mirrors of equal radii of curvature R are fixed on a stand facing opposite directions. The whole system has a mass m and is kept on a frictionless horizontal table following figure. Two blocks A and B, each of mass m, are placed on the two sides of the stand. At t = 0, the separation between A and the mirrors is 2 R and also the separation between B and the mirrors is 2 R. The block B moves towards the mirror at a speed v. All collisions which take place are elastic. Taking the original position of the mirrors-stand system to be x = 0 and X-axis along AB, find the position of the images of A and B at t = (a) `R/v`  (b)  `3R/v` (c) `5R/v`.


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