A cubical block of glass, refractive index $$1.5$$, has a spherical cavity of radius $$r = 9\ cm$$ inside it as shown in Fig. A luminous point object $$O$$ is at a distance of $$18\ cm$$ from the cube (see figure). What is the apparent position of $$O$$ as seen from $$A$$?
We have to consider four refractions at $$S_{1}, S_{2}, S_{3}$$, and $$S_{4}$$, respectively. At each refraction, we will apply single surface refraction equation.
For refractive at first surface $$S_{1}$$:
$$\dfrac {3/2}{v_{1}} - \dfrac {1}{(-18)} = 0$$
$$v_{1} = -27\ cm$$
First image lies to the left of $$S_{1}$$
For refractive at second surface $$S_{2}$$:
$$\dfrac {1}{v_{2}} - \dfrac {3/2}{-(27 + 9)} = \dfrac {(1 - 3/2)}{+9}$$
$$v_{2} = -\dfrac {72}{7} cm$$
Note that origin of Cartesian coordinate system lies at vertex of surface $$S_{2}$$. The object distance is $$(27 + 9) cm$$. The second image lies to left of $$S_{2}$$.
For refractive at third surface $$S_{3}$$:
$$u_{3} = -\left (\dfrac {72}{7} + 8\right ) = -\dfrac {198}{7}$$
$$\dfrac {1.5}{v_{3}} - \dfrac {1}{(-198/7)} = \dfrac {(1.5 - 1)}{(-9)}$$
$$v_{3} = -16.5\ cm$$
For refractive at fourth surface $$S_{4}$$:
$$u_{4} = -(16.5 + 9) = -25.5\ cm$$
$$\dfrac {1}{v_{4}} - \dfrac {3/2}{(-25.5)} = \dfrac {(1 - 3/2)}{\infty} = 0$$
$$v_{4} = -17\ cm$$
The final image lies at $$17\ cm$$ to the left of surface $$S_{4}$$.
An object $$2.4\ m$$ in front of a lens forms a sharp image on a film $$12\ cm$$ behind the lens. A glass plate $$1\ cm$$ thick of refractive index $$1.50$$ is interposed between lens and film with its plane faces parallel to film. At what distance (from lens) should object be shifted to be in sharp focus on film?
An object is placed at a distance of $$2f $$ from a concave mirror. Light reflected from the mirror falls on a plane mirror. The distance of the plane mirror from the concave mirror equals $$f$$. The distance of the final image from the concave mirror (due to reflection at both concave and plane mirror) is :
A small object is placed $$50\ cm$$ to the left of a thin convex lens of focal length $$30\ cm$$ . A convex spherical mirror of radius of curvature $$100\ cm$$ is placed to the right of the lens at a distance of $$50\ cm$$. The mirror is titled such that the axis of the mirror is at a angle $$\displaystyle \theta ={ 30 }^{ \circ }$$ to the axis of the lens, as shown in the figure. If the origin of the coordinate system is taken to be at the centre of the lens, the coordinates $$(in\ cm)$$ of the point $$(x, y)$$ at which the image is formed are:
A biconvex lens of focal length 15 cm is in front of a plane mirror. The distance between the lens and the mirror is 10 cm. A small object is kept at a distance of 30 cm from the lens. The final image is :
A diverting lens of focal length $$20 cm$$ and converging mirror of focal length $$10 cm$$ are placed coaxially at a separation of $$5 cm$$. Where should an object be placed so that a real image is found at the object itself?
shows a transparent hemisphere of radius $$3.0cm$$ made of a material of refractive index $$2.O$$. (a) A narrow beam of parallel rays is incident on the hemisphere as shown in the figure. Are the rays totally reflected at the plane surface? (b) Find the image formed by the refraction at the first surface. (c) Find the image formed by the reflection or by the refraction at the plane surface. (d) Trace qualitatively the final rays as they come out of the hemisphere.
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. 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) $$\dfrac{R}{v}$$ b) $$\dfrac{3R}{v}$$ c) $$\dfrac{5R}{v}$$
A diverging lens of focal length $$20 cm$$ and a converging lens of focal length $$30 cm$$ are placed $$15 cm$$ apart with their principal axes coinciding. Where should an object be placed on the principal axis so that its image is formed at infinity ?
Two convex lenses, each of focal length $$10 cm$$, are placed at a separation of $$15 cm$$ with their principal axes coinciding, (a) Show that a light beam coming parallel to the principal axis diverges as it comes out of the lens system. (b) Find the location of the virtual image formed by the lens system of an object placed far away. (c) Find the focal length of the equivalent lens. (Note that the sign of the focal length is positive although the lens system actually diverges a parallel beam incident on it).
The combination of convex lens and concave lens each of focal length $$10\ cm$$ when combines, behave as :