optics iit material
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iit, reflection, optics notes, problems...
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REFLECTION OF LIGHT INTRODUCTION: Geometrical optics, or ray optics, describes light propagation in terms of "rays". The "ray" in geometrical optics is an abstraction, or "instrument", which can be used to approximately model how light will propagate simply put, a ray of light gives the direction of propagation of light.Light rays are defined to propagate in a rectilinear path as far as they travel in a homogeneous medium. Rays bend (and may split in two) at the interface between two dissimilar media, may curve in a medium where the refractive index changes, and may be absorbed and reflected. Geometrical optics provides rules, which may depend on the color (wavelength) of the ray, for propagation of these rays through an optical system.
SECTION I SOME DEFINITIONS: (i)
Ray: The path of light, as determined within the approximations of geometric optics,
is a ray. In a homogeneous medium, it is a straight line. (ii)
Beam: A collection of rays, usually referred to as a bundle of rays, forms a beam. One
may have a convergent, divergent or, a parallel beam. A convergent beam may converge to a point or, a line, a divergent beam may diverge from a point or, a line. A parallel beam consists of parallel rays. (iii)
Collimation: A process whereby a divergent (or, convergent) beam is rendered
parallel usually through the use of lenses and/or mirrors. (iv)
Object & Image: The term object is used to refer to any object (being photographed
or observed) that is a source of light, and its likeness (usually two dimensional) formed or observed by an optical system is the image. (v)
Optical System: An optical system consists of elements like lenses, mirrors, prisms, etc.
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Axis: The axis of an optical system is frequently an axis of symmetry such that a ray
directed along the axis continues in the same direction or, returns backwards (if reflected within the system)
(vii)
Centre of Curvature: Most lenses and curved mirrors being manufactured spherical
(i.e., their surfaces are spherical), the center of curvature of the curved mirror or, the curved surface of a lens is important and is frequently denoted by the letter C. The radius of curvature is also equally important in the analysis. (viii) Pole: The pole of a spherical surface (refracting or, reflecting) is the central point of the surface involved in the formation of the image. It is denoted by O or, P. The axis, for a single spherical surface, is the join of the pole with the center of curvature; it is known as the principal axis. (ix)
Optical Centre: The optical center of a thin lens is a point on the axis of the lens such
that a ray directed towards that point emerges parallel to itself after passing through the lens. (x)
Paraxial Rays: It is observed that the formation of clear images by spherical surfaces
takes place only with rays which are close to the principal axis and make very small angles with it. These rays are called paraxial rays. (xi)
Wave speed: It is the distance travelled by the wave disturbance in a unit time. It is
denoted by the letter (xii)
Frequency: It is the number of vibrations made by the medium particle in 1 s. In
other words, it is number of waves passing through a point of the medium in 1 second. It is generally represented by the letter n or . Its unit is hertz and it is represented as Hz or s–1. (xiii) Wavelength: It is the distance travelled by the wave in one complete period of a medium particle. In other words, distance between two consecutive crest or trough. It is generally denoted by the letter
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Relationship between the wavelength, wave speed and frequency: Wave speed Frequency X Wavelength.
1. REFLECTION OF LIGHT BY PLANE MIRROR When light rays strike the boundary of two media such as air and glass, a part of light is turned back into the same medium. This is called Reflection of Light. The wavelength and the velocity of the light wave remains the same. Incident Ray
N
Reflected Ray
N N
i r i r
i r
1.1 Laws of reflection: (a)
The incident ray (AB), the reflected ray (BC) and normal
(BN) to the surface (SS') of reflection at the point of incidence (B) lie in the same plane. This plane is called the plane of incidence (or plane of reflection). (b)
The angle of incidence (the angle between normal and the
incident ray) and the angle of reflection are equal. ( i r ) Example: A plane mirror makes an angle of 30o with horizontal. If a vertical ray strikes the mirror, find the angle between mirror and reflected ray (a)30o
(b) 45o
(c) 60o
(d) 90o
Solution: Since angle between mirror and normal is 90o and reflected ray (RR) makes an angle of 30o with the normal so required angle will be
IR
60 o .
30o
30o
RR = 60o
1.2 Reflection at plane surfaces
30o
The point I is called the image of the object A The image formed by reflection at plane surfaces has the following characteristics: EDUDIGM 1B Panditya Road, Kolkata 29
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(ii)
The line joining the object point with its image is normal to the reflecting surface.
(iii)
The image is laterally inverted (left right inversion).
The ray AB gets reflected at B and goes along BC. The ray AD that falls normally on the mirror is reflected back along DA. Let us produce BC and AD behind the mirror where they meet at I. it is simple to prove that triangles ABD and BDI are congruent and AD=DI. Thus, all reflected rays meet at I when produced behind the mirror. An eye receiving the reflected rays are diverging from the point I. (iv)
The size of the image is the same as that of the object.
(v)
For a real object the image is virtual and for a virtual object the image is real.
(vi)
The laws of reflection holds good for all kinds of reflection. Be it at plane or curved
surfaces (vii)
Image of an object is the point at which rays after reflection (or reflection) actually
converge or appear to diverge from that point Example: You are standing in front of a mirror 3m from you. There is a painting 2m behind you on the wall. How far from you is the image of the painting. Solution: Since image distance= object distance, the image distance of painting from the mirror
(
)
The distance of the image from me
(
)
1.3 Real and virtual images If light rays, after reflection or refraction, actually meets at a point then real image is formed and if they appears to meet, virtual image is formed. (Real image) Real image
I (Real image)
I
O
(Virtual object)
(Virtual object)
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(Real object)
I
O
(Virtual image)
(Virtual image)
(Virtual image)
(Real object)
1.4 Deviation by reflection Deviation is the angle between the original direction of the ray and the direction it goes along after reflection or refraction. Deviation produced by a plane mirror and by two inclined plane mirrors.
Final path
i
2
r
Original path
= (360 – 2)
= (180 – 2i)
Example: Two vertical plane mirrors are inclined at an angle of 60o with each other. A ray of
light travelling horizontally is reflected first from one mirror and then from the other. The resultant deviation is (a)60o
(b) 120o
(c) 180o
Solution : (d)By using (360 2 )
(d) 240o
360 2 60 240o
Note : If two plane mirrors are inclined to each other at
, the emergent ray is anti-parallel to incident ray, if it suffers one reflection from each. Whatever be the angle to incidence.
(1) Rotation : If a plane mirror is rotated in the plane of incidence through angle , by keeping the incident ray fixed, the reflected ray turned through an angle 2.
IR IR
RR RR
Example: What should be the height of transmitting antenna if the T.V. telecast is to cover a
radius of 128 Km EDUDIGM 1B Panditya Road, Kolkata 29
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Page 6 (c) 1050 m
Solution : (b) Height of transmitting antenna
h
(d) 79 m d (128 10 3 ) 2 1280 m 2 Re 2 6 .4 10 6 2
(2) Images by two inclined plane mirrors : When two plane mirrors are inclined to each other at an angle , then number of images (n) formed of an object which is kept between them. 360 360 n 1 ; If even integer
(i) (ii) If
360
odd integer then there are two possibilities
(a) Object is placed symmetrically
(b)
Object
is
placed
asymmetrically 360 1
(b) n
(a) n
Object
360
Object
/2 /2
Note : If θ = 0o i.e. mirrors are parallel to each other so n i.e. infinite images will be formed.
360 1 3 90 360 1 4 (If nothing is said object is supposed to be If θ = 72o, n 72 symmetrically placed). If θ = 90o, n
Example: A ray reflected successively from two plane mirrors inclined at a certain angle undergoes a deviation of 300o. The number of images observable are (a)10
(b) 11
(c) 12
(d) 13
Solution : (b)By using (360 2 ) 300 360 2
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30 o . Hence number of images
360 1 11 30
1.5 Additional Information 1. When two plane mirror, inclined to each other at an angle , and 360 If is a fraction, then the number of image formed will be equal to its integral
part. 2.
(i) When the object moves with speed u towards (or away) from the plane mirror then
image also moves toward (or away) with speed u. But relative speed of image w.r.t. object is 2u. (ii) When mirror moves towards the stationary object with speed u, the image will move
with
speed 2u.
I
O u
I
O Rest
u
2u
u Mirror at rest
Mirror is moving
Example:A cube falls down from the back of a truck moving at a speed of 10 km/hr. there is a mirror in the truck facing backward. Considering the cube to be at rest on the road, what will be the speed of its image in the mirror Solution: as the mirror is moving at 10 km/hr the image will move at double the speed as distance of image from is twice the distance from mirror. Speed of image =20 km/hr. 3.
A man of height h requires a mirror of length at least equal to h/2, to see his own complete
image.
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SECTION II 2. REFLECTION OF LIGHT BY CURVED SUFACES A spherical mirror is a reflecting surface which forms a part of a sphere (as shown in following a and b diagram). When the reflection takes place from the inner surface and outer surface is polished or silvered the mirror is known as concave mirror. Vice- versa, it is convex.
(a) concave mirror
C
F
(b) convex mirror
P
P
F
C
Convex Mirror
Concave Mirror
2.1 Key terms (i)
Pole (P) is generally taken as the midpoint of reflecting surface.
(ii)
Centre of curvature (C) is the center of the sphere of which the mirror is a part.
(iii)
Radius of curvature is the radius of the sphere of which the mirror is a part. Distance
between P and C. (iv)
Principal Axis is the straight line connecting pole P and center of curvature C.
(v)
Principal focus (F) is the point of intersection of all the reflected rays which strike
the mirror (with small aperture) parallel to the principal axis. In concave mirror it is real and in the convex mirror it is virtual. (vi)
Focal length (f) is the distance from pole to focus.
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Carbohydrates Page 9 (vii) Focal plane is the plane through the focus and perpendicular to the principal axis. If a parallel beam of light not parallel to the principal axis falls on a concave mirror. It gets reflected to converge at a point on the focal plane. (viii) Aperture is the diameter of the mirror. 2.2. Image Tracing When a point object is placed before a spherical mirror of small aperture, a point image is formed. To locate the position of the image, we draw two rays from the point object, make them incident on the mirror and trace the reflected rays. The line joining the point of incidence and the center of curvature is the normal. A reflected ray is traced by applying the laws of reflection. If the reflected rays intersect, the point of intersection is the real image. If the rays diverge after reflection, a virtual image is formed at the point from where the rays seem to diverge. Figure shows some examples
If the incident rays diverge from a point object, the object is called a real object. Sometimes the rays incident on the mirror do not diverge from a point rather they converge towards the mirror. In this case, the point where these rays would meet if there was no mirror is treated as the object. Such a point is called a virtual object Thus the point of intersection of the incident rays is called the object and the point of intersection of the corresponding reflected rays is called its image. 2.3. Sign convention (i) All distances are measured from the pole. (ii) Distances measured in the direction of incident rays are taken as positive while in the direction opposite of incident rays are taken negative. (iii) Distances above the principle axis are taken positive and below the principle axis are taken negative.
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Note
:
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Same sign convention are also valid for lenses.
Incident ray
+ +
–
Principle Mirror or Lens
–
axis
Use following sign while solving the problem : Concave mirror Real image (u ≥ f)
Convex mirror
Virtual image (u< f)
Distance of object
u –
u –
u –
Distance of image
v –
v +
v +
Focal length
f
–
f –
f
Height of object
O +
O+
O +
Height of image
I
–
I +
I
Radius of curvature
R –
R –
R +
Magnification
m –
m+
m +
+ +
2.4 Mirror formula and Magnification
If u = Distance of object from pole, v = distance of image from pole, f = Focal length, R = Radius of curvature, m = magnification (or linear magnification)
Mirror formula :
1 1 1 ; (use sign convention while solving the problems). f v u
Example: A lighted candle is placed at a distance 40 cm from the vertical wall. Where a concave mirror of radius of curvature 30cm should be placed so that an image of the flame is obtained on the wall Solution:
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Page 11 from the candle
The distance of the wall from the mirror
(
)
As the object and the image are both real, (
)
( (
) )
The relevant answer is So the mirror should be placed at 20 cm from the candle. Magnification : m = Size of image/ Size of object = -v/u Question to upload More Solved Examples Example: A convex lens in air produces a real image having the same size as object. When
the object and the lens immersed in a liquid, the real image formed is enlarged two times the object size. Find the refractive image of the liquid. (
)
Solution:Combining the lens equation and the lens maker’s formula we have (
)(
)
In air:
(
)(
)
()
In liquid:
(
)(
)
( )
Dividing (i) and (ii), we get:
(
)
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Hence the refractive index of liquid is Example:Photographs of the ground are taken, from an altitude of 2000 m, by a camera with a lens of focal length 50 cm. the size of the film in the camera is
what
area of the ground can be photographed by this camera at any one time? Solution:We will take ground to be photographed as object and the image will be formed on the camera film.
Thus u = 2000 cm. f= 50 cm
Let the area to photographed be x meters long x meters wide.
(size of real inverted image is taken negative)
The area photographed = Note: Area magnification = Example: An object is imaged by a lens on a screen placed 12 cm from the lens. When the lens is moved 20 cm away from the object, the screen must be moved 20cm closer to the object to reform it. Find the focal length of the lens.
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Let x = magnitude of initial object distance,
() After shifting the lens and the screen (
)
(
( )
)
Solving (i) and (ii), we get
(
(
)(
((
)
(
)
)
Hence, the focal length of the lens is 4 cm. Example: One surface of biconvex lens having focal length 40cm is silvered. The radius of the curvature of the other surface is 60 cm. at what distance from the silvered lens should an object be placed to obtain a real image magnified three times? Solution:Let x= magnitude of radius of the curvature of the silvered surface. EDUDIGM 1B Panditya Road, Kolkata 29
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)(
(
)
)( (
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)
)
The silvered lens will behave like a converging mirror with focal length F because it is a combination of a converging lens and a converging (concave) mirror.
Focal length of mirror
For the concave mirror: ⁄
Example-When the flat surface of the piano convex lens is silvered, an object coincides with its image at a distance of 15 cm from the lens. If the curved surface is silvered, the object coincides with its image at a distance of 5 cm from the lens. Calculate the refractive index of glass. Solution:In both the situations, the silvered lens will act like a concave mirror. Let
be
the equivalent focal lengths of the silvered lens when the flat and the curved surfaces are silvered respectively. Let
be the focal length of the unsilvered lens.
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As the object coincides with image, it must be at the center of curvature.
Is curved surface is silvered:
The radius of curvature of the curved side Now applying Lens maker’s formula we can calculate the refractive index
(
)(
)
(
)(
)
(
)(
)
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Example-A converging beam of light falls on one surface of a biconvex lens whose other surface is silvered. After reflection from the silvered lens, the beam converges to a point 24 cm in front of the lens. The focal length of the lens is 30 cm and the silvered surface has a radius of the curvature equal to 50 cm. where wills the beam of light coverage if the lens is removed from its path? Solution:The silvered lens is a combination of diverging lens and a diverging mirror. Hence it will behave like a convex mirror of focal length F. For equivalent convex mirror:
For equivalent convex mirror: ⁄ ( (
) ⁄ )
Hence the beam will coverage at a point +6.74 cm behind the lens if it is removed.
Example: A person is in a room whose ceiling and two adjacent walls are mirrors. How many images are formed
]
(a)5
(b) 6
(c) 7
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(d) 8
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images of the person. Now these images with object (Person) will act as objects
for the ceiling mirror and so ceiling will form 4 images as shown. Therefore total number of images formed = 3 + 4 = 7 O
I 1 I1
O I 2
I2
Note :
I 3
Four images by ceiling
I3
Three images by walls
The person will see only six images of himself (I1 , I 2 , I 3 , I1' , I 2' , I 3' )
Example: A ray of light incident on the first mirror parallel to the second and is reflected from the second mirror parallel to first mirror. The angle between two mirrors is (a) Solution : (b)
30o (b) 60o (c) From geometry of figure
75o (d)
90o
180 o 60
o
Example: A point object is placed mid-way between two plane mirrors distance 'a' apart. The plane mirror forms an infinite number of images due to multiple reflection. The distance between the nth order image formed in the two mirrors is (a)na solution (b)
(b) 2na
(c) na/2
(d) n2 a M
M' III order
II order
I order
image
image
image
I3'
I2'
I1' 3a/2 5a/2
a/2
I order
II order
III order
image
image
image
I1
I2
O a/2
a
a/2
a/2
I3
3a/2 5a/2
From above figure it can be proved that separation between nth order image formed in the two mirrors = 2na Example: Two plane mirrors P and Q are aligned parallel to each other, as shown in the figure. A light ray is incident at an angle of at a point just inside one end of A. The plane of incidence coincides with the plane of the figure. The maximum number of times the ray undergoes reflections (including the first one) before it emerges out is EDUDIGM 1B Panditya Road, Kolkata 29
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l d tan
(c) ld tan
(b)
Page 18 d l tan
l
(d)None of these
d
solution (a)Suppose n = Total number of reflection light ray undergoes before exist out.x = l Horizontal distance travelled by light ray in one reflection. x So nx = l
x d
also tan
d
l n d tan
Example: Two objects A and B when placed one after another infront of a concave mirror of focal length 10 cm from images of same size. Size of object A is four times that of B. If object
A is placed at a distance of 50 cm from the mirror, what should be the distance of B from the mirror (a)10 cm
(b) 20 cm
Solution : (b) By using
(c) 30 cm
(d) 40 cm
10 u B I O f uB 1 1 I f u B 20 cm . A B O f u IB O A f uA 1 4 10 50
Example: A square of side 3 cm is placed at a distance of 25 cm from a concave mirror of focal length 10 cm. The center of the square is at the axis of the mirror and the plane is normal to the axis. The area enclosed by the image of the wire is (a)4 cm2
(b) 6 cm2
Solution: (a)
By using m 2
(c) 16 cm2 Ai ; where Ao
Hence from given values m
m
(d) 36 cm2
f f u
10 2 and 10 25 3
2
A o 9 cm 2
2 2 Ai 9 4 cm 3
Example: A concave mirror of focal length 100 cm is used to obtain the image of the sun which subtends an angle of 30'. The diameter of the image of the sun will be (a)1.74 cm
(b) 0.87 cm
(c) 0.435 cm
(d) 100 cm
solution: Diameter of image of sun d f
Image of
30 d 100 60 180
d
sun
F
d 0.87 cm .
Example:A thin rod of length f / 3 lies along the axis of a concave mirror of focal length f. One end of its magnified image touches an end of the rod. The length of the image is (a)f
(b)
1 f 2
(c) 2 f
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So by using
f 5f 3 3
1 1 1 f v u
Length of image
2f
1 1 1 5 v f f v 5f 2 3
f/3
u = 2f – (f/3)
A A'
5 f f 2f 2 2
F
C v
Example: A concave mirror is placed on a horizontal table with its axis directed vertically upwards. Let O be the pole of the mirror and C its centre of curvature. A point object is placed at C. It has a real image, also located at C. If the mirror is now filled with water, the image will be (a)Real, and will remain at C (b)Real, and located at a point between C and (c)Virtual and located at a point between C and O (d)Real, and located at a point between C and O Solution : (d)
Object
C
C
Object
image Image
O
O
Initially
Finally
Example: A small plane mirror placed at the center of a spherical screen of radius R. A beam of light is falling on the mirror. If the mirror makes n revolution. per second, the speed of light on the screen after reflection from the mirror will be (a)4nR
(b) 2nR
(c)
nR 2
(d)
nR 4
Solution : (a) When plane mirror rotates through an angle , the reflected ray rotates through an angle 2. So spot on the screen will make 2n revolution per second Speed of light on screen v R 2 (2n)R 4nR
Position, size and nature of image formed by the spherical mirror
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Location of the object
(a) Concave
C
F
P
F
Location of the image
Magnification, Size of the image
Nature Real virtual
Erect inverted
At infinity i.e. u = ∞
At focus i.e. v = f
m 2f)
Between f and 2f i.e. f < v < 2f
m < 1, diminished
Real
inverted
At centre of curvature i.e. v = 2f
m = 1, same size as that of the object
Real
inverted
Between centre of curvature and focus : F < u < 2f
Away from the centre of curvature v > 2f
m > 1, magnified
Real
inverted
At focus i.e. u = f
At infinity i.e. v = ∞
m = ∞, magnified
Real
inverted
Between pole and focus u < f
v>u
m > 1 magnified
Virtual
erect
At infinity i.e. u = ∞
At focus i.e., v = f
m < 1, diminished
Virtual
erect
Between pole and focus
m < 1, diminished
Virtual
erect
At centre of P curvature u = 2f
(b) Convex
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C Anywhere
between infinity and pole
EXERCISES
Level 1 1. An object is initially at a distance of 100 cm from a plane mirror. If the mirror approaches the object at a speed of 5 cm/s, then after 6 s the distance between the object and its image will be (a)
60 cm (b)
140 cm
(c)
170 cm
(d)
150 cm
2. An object placed in front of a plane mirror is displaced by 0.4 m along a straight line at an angle of 30o to mirror plane. The change in the distance between the object and its image is (a)
0.20 m (b)
0.40 m (c)
0.25 m (d)
0.80 m
3. A convex mirror of focal length 10 cm is placed in water. The refractive index of water is 4/3. What will be the focal length of the mirror in water (a)10 cm
(b) 40/3 cm
(c) 30/4 cm
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(d) None of these
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Carbohydrates Page 21 4. A ray of light travels from A to B with uniform speed. On its way it is reflected by the surface XX'. The path followed by the ray to take least time is (a)
1
(b)
2
(c)
3
(d)
4
A
X
1
B
2
3
4
X
5. Two mirrors, each 1.6m long are facing each other. The distance between the mirrors is 20 cm. a light ray incident on one end of one of the mirrors at an angle of incidence of How many times is the ray reflected before it reaches the other end? 6. An object is 375 mm from a concave mirror of 250 mm focal length. Find the image distance. If the object is moved 5 mm farther from the mirror, how far does the image move? 7. A object 28 mm high is 0.48 m from a convex mirror with a radius of curvature of 0.82m. locate the image. Is it real or virtual? It is erect or inverted? What is its size? 8. A man runs towards a mirror at a speed 15 m/s. The speed of the image relative to the man is (a) 15 ms 1 (b) 30 ms 1 (c) 35 ms 1 (d) 20 ms 1 9.A man is 180 cm tall and his eyes are 10 cm below the top of his head. In order to see his entire height right from toe to head, he uses a plane mirror kept at a distance of 1 m from him. The minimum length of the plane mirror required is (a) 180 cm (b) 90 cm (c) 85 cm(d) 170 cm 10.A small object is placed 10 cm infront of a plane mirror. If you stand behind the object 30 cm from the object and look at its image, the distance focused for your eye will be (a) 60 cm (b) 20 cm (c) 40 cm (d) 80 cm 11. A plane mirror produces a magnification of (a) – 1 (b) +1 (c) Zero(d)Between 0 and + 12. Two plane mirrors are parallel to each other an spaced 20 cm apart. An object is kept in between them at 15 cm from A. Out of the following at which point an image is not formed in mirror A (distance measured from mirror A) (a) 15 cm
(b) 25 cm
(c) 45 cm
(d) 55 cm
13. A diminished virtual image can be formed only in (a) Plane mirror (b) A concave mirror (c) A convex mirror (d) Concave-parabolic mirror 14. A diminished virtual image can be formed only in (a) Plane mirror (b) A concave mirror (c) A convex mirror (d) Concave-parabolic mirror
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Carbohydrates Page 22 15. Match List I with List II and select the correct answer using the codes given below the lists [SCRA 1998] List I List II (Position of the object) (Magnification) (I) An object is placed at focus before a convex mirror (A) Magnification is – (II) An object is placed at centre of curvature before a concave mirror (B) Magnification is 0.5 (III) An object is placed at focus before a concave mirror (C) Magnification is + 1 (IV) An object is placed at centre of curvature before a convex mirror (D) Magnification is –1 (E) Magnification is 0.33 Codes : (a) I-B, II-D, III-A, IV-E (b) I-A, II-D, III-C, IV-B (c) I-C, II-B, III-A, IV-E (d) I-B, II-E, III-D, IV-C 16. All of the following statements are correct except (a) The magnification produced by a convex mirror is always less than one (b) A virtual, erect, same-sized image can be obtained using a plane mirror (c) A virtual, erect, magnified image can be formed using a concave mirror (d) A real, inverted, same-sized image can be formed using a convex mirror 17. A convex mirror is used to form the image of an object. Then which of the following statements is wrong (a) The images lies between the pole and the focus (b) The image is diminished in size (c) The images is erect (d) The image is real 18. A dice is placed with its one edge parallel to the principal axis between the principal focus and the centre of the curvature of a concave mirror. Then the image has the shape of (a) Cube (b)Cuboid (c)Barrel shaped (d)Spherical 19. An object of length 6cm is placed on the principal axis of a concave mirror of focal length f at a distance of 4 f. The length of the image will be (a)
2 cm (b)
12 cm (c)
4 cm (d)
1.2 cm
20. A point object is placed at a distance of 30 cm from a convex mirror of focal length 30cm. The image will form at (a)
Infinity
(b)
Focus (c)
Pole
(d)
15 cm behind the mirror
21. A concave mirror of focal length 15 cm forms an image having twice the linear dimensions of the object. The position of the object when the image is virtual will be (a)
22.5 cm
(b)
7.5 cm
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(c)
30 cm (d)
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Carbohydrates
Page 23
Level -2 1. The figure shows two rays A and B being reflected by a mirror and going as A' and B'. The A
mirror is
A
B
(a)
Plane
(b)
Concave
(c)
Convex
(d)
May be any spherical mirror
B
2. A point source of light B is placed at a distance L in front of the centre of a mirror of width d hung vertically on a wall. A man walks in front of the mirror along a line parallel to the mirror at a distance 2L from it as shown. The greatest distance over which he can see the image of the light source in the mirror is (a)
d/2
(b)
d
(c)
2d
(d)
3d
3. A man having a height of 6 m, want to see his full height in mirror. He observes an image of 2m height of his whole body erect, then used mirror is (a)
Concave
(b)
Convex
(c)
Plane (d)
None of these
4. The focal length of a convex mirror is 20 cm its radius of curvature will be (a)
10 cm (b)
20 cm (c)
30 cm (d)
40 cm
5. Under which of the following conditions will a convex mirror of focal length f produce an image that is erect, diminished and virtual (a)
Only when 2f > u > f (b)
Only when u = f
(c)
Only when u < f
Always
(d)
6. As the position of an object reflected in a concave shell mirror of 0.25m focal length is varied, the position of the image varies. Plot the image distance as a function of the object distance, letting the latter change from
Where is the image real? Where virtual?
7. If it is possible, under what condition will the image in a concave mirror be (a) real (b) virtual (c) erect (d) inverted (e) magnified (f) reduced? 8. Repeat question 7 for a convex mirror? 9. A U-shaped wire is placed before a concave mirror having radius of curvature 20 cm as shown in figure. Find the length of the image EDUDIGM 1B Panditya Road, Kolkata 29
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Carbohydrates 10. A point source S is placed midway between two converging mirrors having equal focal length
Page 24
as shown in figure. Find the values
of d for which only one image is formed 11. 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
of the particle from the mirror.
12. 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 mas
The block is pushed at
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
(b) at a time
13. A gun of mass M fires a bullet of mas m with a horizontal speed V. the gum is fitted with a concave mirror of focal length
facing towards the receding bullet. Find the speed of
separation of the bullet and the image just after the gum was fired 14. A mass m=50 g is dropped on a vertical spring of spring constant 500 from a height
cm as shown in figure. The mass sticks to the spring and
executes simple harmonic oscillates after that. A concave mirror of focal length 12cm 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. 15. Two concave mirrors of equal radii of curvature R are fixed on a stand facing opposite directions. The whole system has a mass 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
, the
seperation between A and the mirrors is 2R and also the seperation between B and the mirrors is 2R. the block B moves towards the mirror at a speed
. All collisions which take
place are elastic. Taking the original position of the mirrors-stand system to be
and
X-axis along AB, find the position of the image of A and B at
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Carbohydrates (a)
Page 25 (b)
(c)
16. consider the situation shown in figure. The elevator is going up with an acceleration of
and the focal length of the mirror is 12.0 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
when the distance
of B from the mirror is 42.0 cm. find the distance between the image of the block B and the mirror at
Take
17. Two plane mirrors are at right angles to each other. A man stands between them and combs his hair with his right hand. In how many of the images will he be seen using his right hand (a) None (b) 1 (c) 2 (d) 3 18. Two plane mirrors A and B are aligned parallel to each other, as shown in the figure. A light ray is incident at an angle of 30o at a point just inside one end of A. The plane of incidence coincides with the plane of the figure. The maximum number of times the ray undergoes reflections (including the first one) before it emerges out is (a) 28
B
0.2m
30o
(b) 30 (c) 32
A
(d) 34
19. A point object O is placed between two plan mirrors as shown is fig. The distance of the first three images formed by mirror M 2 from it are M M2
1
(a) 2 mm, 8 mm, 18 mm
O
(b) 2 mm, 18 mm, 28 mm (c) 2 mm, 18 mm, 22 mm
10mm
2mm
(d) 2 mm, 18 mm, 58 mm 20.A plane mirror is placed at the bottom of the tank containing a liquid of refractive index . P is a small object at a height h above the mirror. An observer O-vertically above P outside the liquid see P and its image in the mirror. The apparent distance between these two will be (a) 2 h (b)
2h
(c)
2h 1
(d)
1 h 1
O
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P
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h
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Carbohydrates Page 26 21. In an experiment of find the focal length of a concave mirror a graph is drawn between the magnitudes of u and v. The graph looks like v
(a)
v
v
v
(b) u
u
u
u
22. A short linear object of length l lies along the axis of a concave mirror of focal length f at a distance u form the pole of the mirror. The size of the image is approximately equal to (a)
u l f
1/2
f
(b) l u f
f
2
(c)
f l u
1/2
f
(d)
f l u
f
2
23.A point object is moving on the principal axis of a concave mirror of focal length 24 cm towards the mirror. When it is at a distance of 60 cm from the mirror, its velocity is 9 cm/sec. What is the velocity of the image at that instant (a) 5 cm/sec towards the mirror (b)4cm/sec towards the mirror (c) 4 cm/sec away from the mirror (d) 9 cm/sec away from the mirror 24. A thin rod of 5 cm length is kept along the axis of a concave mirror of 10 cm focal length such that its image is real and magnified and one end touches the rod. Its magnification will be (a) 1 (b)
2 (c)
3 (d)
4
25.A luminous object is placed 20 cm from surface of a convex mirror and a plane mirror is set so that virtual images formed in two mirrors coincide. If plane mirror is at a distance of 12 cm from object, then focal length of convex mirror, is (a) 5 cm
(b) 10 cm
(c) 20 cm
(d) 40 cm
26. A vehicle has a driving mirror of focal length 30 cm. Another vehicle of dimension 2 4 1.75 m 3 is 9 m away from the mirror of first vehicle. Position of the second vehicle as seen in the mirror of first vehicle is (a) 30 cm 9m
(b) 60 cm (c) 90 cm (d) 9 cm
27. A concave mirror of radius of curvature 60 cm is placed at the bottom of tank containing water upto a height of 20 cm. The mirror faces upwards with its axis vertical. Solar light falls normally on the surface of water and the image of the sun is formed. If a
w
4 3
then with the observer in air, the distance of the image from the surface of water is
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Carbohydrates Page 27 (a) 30 cm (b) 10 cm (c) 7.5 cm above (d) 7.5 cm below 28.A concave mirror forms an image of the sun at a distance of 12 cm from it (a) The radius of curvature of this mirror is 6 cm (b) To use it as a shaving mirror, it must be held at a distance of 8-10 cm from the face (c) If an object is kept at a distance of 12 cm from it, the image formed will be of the same size as the object (d) All the above a alternatives are correct 29. A small piece of wire bent into an L shape with upright and horizontal portions of equal lengths, is placed with the horizontal portion along the axis of the concave mirror whose radius of curvature is 10 cm. If the bend is 20 cm from the pole of the mirror, then the ratio of the lengths of the images of the upright and horizontal portions of the wire is (a) 1 : 2 (b) 3 : 1 (c) 1 : 3 (d) 2 : 1 30.As the position of an object (u) reflected from a concave mirror is varied, the position of the image (v) also varies. By letting the u changes from 0 to the graph between v versus u will be v
v
v
(a)
(b) u
v
(c) u
(d) u
u
31. A concave mirror of focal length 10 cm and a convex mirror of focal length 15 cm are placed facing each other 40 cm apart. A point object is placed between the mirrors, on their common axis and 15 cm from the concave mirror. Find the position and nature of the image produced by the successive reflections, first at concave mirror and then at convex mirror (a) 2 cm (b) 4 cm (c) 6 cm (d) 8 cm
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