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A paperweight in the form of a hemisphere of radius 3.0 cm is used to hold down a printed page. An observer looks at the page vertically through the paperweight. At what height above the page will the printed letters near the center appear to the observer?
Radius of the paper weight = R= 3.0 cm
Refractive index of paper weight =
Refractive index of air=
• Refraction at point A
Object distance = u= 0
By lens equation
Therefore, image of the printed letters will be formed at the point.
• Refraction at point B
Object distance u=-3 cm
Radius R= -3 cm
By lens equation
Hence, there will be no shift in the final image.
A particle goes in a circle of radius 2.0 cm. A concave mirror of focal length 20 cm is place with its principal axis passing through the center of the circle and perpendicular to its plane. The distance between the pole of the mirror and the center of the circle is 30 cm. Calculate the radius of the circle formed by the image.
A converging mirror M1, a point source S and a diverging mirror M2 are arranged as shown in figure (18-E4). The source is placed at a distance of 30 cm for M1. The focal length of each of the mirrors is 20 cm. Consider only the images formed by a maximum of two reflections. It is found that one image is formed on the source itself. (a) Find the distance between the two mirrors. (b) Find the location of the image formed by the single reflection from M2.
k transparent slabs are arranged one over another. The refractive indices of the slabs are μ1, μ2, μ3, … μk and the thickness are t1, t2, t3, … tk. An object is seen through this combination with nearly perpendicular light. Find the equivalent refractive index of the system which will allow the image to be formed at the same place.
A cylindrical vessel of diameter 12 cm contains 800π cm3 of water. A cylindrical glass piece of diameter 8.0 cm and height 8.0 cm is placed in the vessel. If the bottom of the vessel under the glass piece is seen by the paraxial rays (see figure 18-E6), locate its image. The index of refraction of glass is 1.50 and that of water is 1.33.
Consider the situation in figure (18-E7). The bottom of the pot is a reflecting plane mirror, S is a small fish and T is a human eye. Refractive index of water is μ.
(a) At what distance(s) from itself will the fish see the image(s) of the eye?
(b) At what distance(s) from itself will the eye see the image(s) of the fish?
A small object is placed at the center of the bottom of a cylindrical vessel of radius 3 cm and height 4 cm filled completely with water. Consider the ray leaving the vessel through corner. Suppose this ray and the ray along the axis of the vessel is used to trace the image. Find the apparent depth and the ratio of real depth to apparent depth under the assumptions taken. Refractive index of water = 1.33.
A cylindrical vessel, whose diameter and height both are equal to 30 cm, is placed on a horizontal surface and a small particle P is placed in it at a distance of 5.0 cm from the center. An eye is placed at a position such that the edge of the bottom is just visible (see figure 18-E8). The particle P is in the plane of drawing. Up to what minimum height should water be poured in the vessel to make the particle P visible?
A container contains water up to a height of 20 cm and there is a point source at the center of the bottom of the container. A rubber ring of radius r floats centrally on the water surface. The ceiling of the room is 2.0 m above the water surface.
(a) Find the radius of the shadow of the ring formed on the ceiling if r = 15 cm.
(b) Find the maximum value of r for which the shadow of the ring is formed on the ceiling. Refractive index of water = 4/3.
A spherical surface of radius 30 cm separates two transparent media A and B with refractive indices 1.33 and 1.48 respectively. The medium A is on the convex side of the surface. Where should a point object be placed in medium A so that the paraxial rays become parallel after refraction at the surface?
Figure (18-E12) shows a transparent hemisphere of radius 3.0 cm made of a material of refractive index 2.0.
(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.
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 center 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.
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 (18-E13). (a) Where should the 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 thin lens made of a material of refractive index μ2 has a medium of refractive index μ1 on one side and a medium of refractive index μ3 on the other side. The lens is biconvex and the two radii of curvature have equal magnitude. R. A beam of light travelling parallel to the principal axis is incident on the lens. Where will the image be formed if the beam is incident from (a) the medium μ1 and (b) from the medium μ3?
An extended object is placed at a distance of 5.0 cm from a convex lens of focal length 8.0 cm. (a) Draw the ray diagram (to the scale) to locate the image and from this, measure the distance of the image from the lens. (b) Find the position of the image from the lens formula and see how close the drawing is the correct result.
A converging lens and a diverging mirror are placed at a separation of 15 cm. The focal length of the lens is 25 cm and that of the mirror is 4o cm. Where should a point source be placed between the lens and the mirror so that the light, after getting reflected by the mirror and then getting transmitted by the lens, comes out parallel to the principal axis?
A convex lens of focal length 20 cm and a concave lens of focal length 10 cm apart with their principal axes coinciding. A beam of light travelling parallel to the principal axis and having a beam diameter 5.0 mm, is incident on the combination. Show that the emergent beam is parallel to the incident one. Find the beam diameter of the emergent beam.
Two convex lenses, each of focal length 10 cm, are places 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.)
A ball is kept at a height h above the surface of a heavy transparent sphere made of a material of refractive index μ. The radius of the sphere is R. At t = 0, the ball is dropped to fall normally on the sphere. Find the speed of the image formed as a function of time for t<. Consider only the image by a single refraction.
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 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 (18-E14). 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 (figure 18-E15).
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 2R and also the separation between B and the mirrors is 2R. 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 =
Consider the situation shown in figure (18-E16). The elevator is going up with an acceleration of 2.00 ms-2 and the focal length of the mirror is 12.) cm. All the surfaces are smooth and the pulley is light. The mass-pulley system is released from rest (with respect to the elevator) at t=0 when the distance of B from the mirror is 42.) cm. Find the distance between the image of the block B and the mirror at t=0.200 s. Take g= 10 ms-2