# To Find Focal Length of A Convex Mirror Using A Convex lens

## Aim/Aim of Experiment

To find the focal length of a convex mirror using a convex lens.

## Apparatus/Material Required

• Optical bench with four Uprights
• Convex Lens (f=21 cm)
• Convex Mirror
• A Lens holder
• Mirror Holder
• Two optical needle
• A knitting needle
• A half meter scale
• A spirit level.

## Theory

A convex mirror always forms a virtual & erect image, therefore its focal length cannot be found directly as for a concave mirror. To find the focal length of a convex mirror we use indirect method, which can be determined by introducing a convex lens in between the object and the convex mirror.

As we know that a convex lens formed a real image of an object ‘O’ located at C. place the convex mirror and the position of convex mirror behind convex lens is so adjusted that a real and inverted image of object needle O, is formed at O itself. Then the light rays are incident normally over the convex mirror to retrace their path. In the absence of convex mirror, these rays would have met at centre of curvature C of the convex mirror, The distance PC gives the radius of curvature R of the mirror,

i.e. PC=PI=R, and f=R/2 = PI/2 = PC/2,

So, the focal length of a convex mirror, f=R/2, where R is the radius of curvature of the convex mirror.

## Procedure

To determine rough focal length of Convex Lens:

• Determine rough focal length in a convex lens by getting a sharp image on a screen.

To Set the Convex Lens:

1. Clamp the holder with lens in a fix upright at the 50 cm mark.
2. Adjust the lens such that its surface is vertical and perpendicular to the length of the optical bench.
3. Keep the upright fixed in this position throughout.

To set the Object Needle:

1. Take the thin optical needle as the object needle (O). Mount it on the outer laterally movable upright near the zero end.
2. Move the object needle upright and clamp it at a distance (in full cm) nearly about 1.5 times the obtained rough focal length of the lens.
3. To make its tip lie on horizontal line through the optical centre of the lens, Adjust height of the object needle.
4. On the base of the object needle upright, Note down its position of the index mark.

To set the Convex Mirror:

1. Clamp the holder with convex mirror in second fixed upright near the lens upright, keeping reflecting surface of the mirror towards lens.
2. Adjust the height of the mirror to make its pole lie on horizontal line through the optical centre of the lens.
3. See the inverted image of the object needle (formed by reflection from the convex mirror).
4. Keep the eye in a position at which the tips of the inverted image and the object needle are seen simultaneously.
5. Adjust the height of the needle to make the two tips are seen in one line with right open eye.
6. Move the eye towards right, the tips will separate and there will be parallax in the tip.
7. Move the convex mirror back and forth until the end-to-end parallax is removed.
8. Note the position of the index mark on the base of the convex mirror upright, Record it in observation table.

To set the Image Needle:

1. Remove the convex mirror, keeping upright in its position.
2. An inverted and enlarged image of the object needle will be seen. Tip of the image must lie in the middle of the lens.
3. Mount the thick optical needle (image needle) in the fourth upright near the other end of the optical bench.
4. Adjust the height of the image needle so that its tip is seen in line with the tip of the image when seen with right open eye.
5. Repeat the step 6 and 7 (To set the convex mirror).
6. Note the position of the index mark on base of the image needle upright, Record it in observation table.

To determine Index Correction:

• To find index correction substrate the observed length of index needle (y) from the actual length of the knitting needle (x).

To get more observations:

1. Move the object needle 2 cm towards lens and the two cm away from the lens.
2. Repeat the experiment to get more observations.

## Observations

• Rough focal length of Convex lens = 21 cm.
• Actual length of the knitting needle, x = 10 cm.
• Observed distance between image needle (I) and back of the convex mirror (Observed length of the index needle), y = 20 cm.
• Index Correction, e = (x-y) = 0 cm.
• Observation Table for focal length of Convex Mirror:
• Position of – (PO)
• Radius of Curvature – (R)
• Mean value of focal length, (Mean f) = 19.4 cm.

## Calculations

1. Write observed PI in column 3a and corrected PI in column 3b.
2. Divide corrected PI by 2 and write in column 4.
3. Find mean of values of recorded in column 4.

## Result

The focal length of given Convex Mirror using Convex Lens is, f = 19.94 cm.

## Precautions

1. The principal axis of the lens should be horizontal and parallel to the center line of the lens on the optical bench.
2. All the uprights should be vertical.
3. The tip of the needle, the center of the mirror and the center of the lens should be at same height.
4. While removing the parallax, the eye should be kept at least 30 cm from the needle.
5. Make sure tip to tip parallax should be removed.
6. Index correction should be applied between the image needle I and the back surface of the convex mirror.
7. The given convex mirror should be placed near the convex lens.
8. For a set of observations, the position of the lens alone and for a convex lens the needle (O) upright should not change when the parallax is removed.

## Sources of Error

• The upright may not be vertical.
• Parallax removal may not be perfect.
• The focal length of convex lens may not be small.

## Viva Voice Questions with Answers

1. What is a mirror, Define it.

Answer: It is a fine polished surface which reflects most of the light that is incident on it.

2. Describe different types of mirrors.

Answer: There are two types of mirrors:

• Plane mirrors,
• Spherical mirrors.

3. Define a spherical mirror.

Answer: A mirror whose surface is cut out of a spherical shell, is called a spherical mirror.

4. Give types of a spherical mirror.

Answer: There are two types of spherical mirrors:

1. Concave mirror: Its outer convex surface is polished and inner concave surface reflects.
2. Convex mirror: Its inner concave surface is polished and outer convex surface reflects.

5. Mention three special rays.

1. Incident on the mirror parallel to principal axis. After reflection from the mirror, it actually passes through mirror focus F (in case of a concave mirror) or appears to come from it (in case of a convex mirror).
2. Incident on the mirror through focus F (in case of a concave mirror) or in direction of focus F (in case of convex mirror). After reflection from the mirror, it goes parallel to the principal axis.
3. Incident in the mirror through centre of curvature C (in case of a concave mirror) or in direction of centre of curvature C (in case of a convex mirror). After reflection from the mirror, it returns back along the path of incidence.

6. Define sign convention.

Answer: It is a convention, which fixes the signs of different distances measured. The sign convention followed is the new cartesian sign convention.

7. Give rules of sign convention.

1. All distances are measured from the pole of the mirror (along the principal axis).
2. The distance measured in the same direction as the direction of incident light,are taken as positive.
3. The distances measured opposite to the direction of incident light, are taken as negative.
4. The distances measured above the principal axis, are taken positive and the distance taken below the principal axis, are taken negative.

8. Give facts obtained from sign convention.

Answer: According to above mentioned rules of sign convention:

1. Radius of curvature and focal length of a concave mirror are taken as negative and the same for convex mirror are taken positive.
2. The distance of an object is always negative.
3. The distance of real image is negative, while that of a virtual image is positive.
4. The size of object is always positive but size of real image is always negative while size of virtual image is positive for mirrors.

9. Define and give mirror formula.

Answer: The equation relating the object distance (u), the image distance (v) and the mirror focal length (f), is called the mirror formula. It is also called Gaussian formula.

The mirror formula is, 1/v+1/u=1/f.

10. Describe various assumptions made in derivation of mirror formula.

1. The mirror has a small aperture.
2. The point object lies on to the principal axis and placed perpendicular.
3. The incident rays make small angles with the mirror surface or the principal axis.

11. Define image, real image and virtual image.

Answer: (a) Image: When rays of light starting from a point object, after reflection (or refraction), meet at a point or appear to come from a point, then this point is called the image of the point object.

(b) Real Image: If the reflected or refracted rays actually meet at the point, then the image is real.

(c) Virtual Image: If the reflected or refracted rays appear to come from the point, then the image is virtual.

12. Give distinction between real and virtual image.

Answer: Real image of a big object (combination of real images of its different points) is always inverted. It can be obtained on a screen. Virtual image of a big object (combination of virtual images of its different points) is always erect. It cannot be obtained on a screen.

13. Define parallax. How is it removed?

Answer: The relative side way shift between two objects, at unequal distance from the eye, when the eye is moved sideways, is called parallax. When two objects which are placed at different distances from eye, the nearer object moves in a direction opposite to that eye and the farther one in the direction of the eye. Once their relative positions are known, they can be brought to the same position by; shifting them suitably. When two objects occupy the same position in space with respect to the eye, then the apparent shift disappears and parallax is said to be removed.

14. Define index correction.

Answer: The difference of the actual distance and the observed distance, is called index correction. It is also called bench-correction.

15. Define and explain spherical aberration.

Defination: The defect or drawback of a mirror due to which it makes a spread or bulging image of a point object, is called spherical aberration. It is due to large aperture of the mirror.

Explanation: For rays incident on the mirror near the axis (axial rays), f= R/2. For rays incident on the mirror near its periphery (marginal rays) F < R/2. The focus does not remain a point. Hence, the spherical aberration arises mainly due to marginal rays.

16. Why is a mirror silvered at the back surface?

Answer: To avoid spoiling of mirror in rough handling.

17. Is silvering done by depositing silver coating or some other material?

Answer: In cheap mirrors silvering is done by deposition of mercuric oxide and in good quality mirrors silvering is done by deposition of silver nitrate.

18. What is the relation between focal length and radius of curvature of a spherical mirror?

19. What is the radius of curvature of a plane mirror?

20. Can we find the rough focal length of a convex mirror? If not, why?

Answer: No, we cannot find the rough focal length of convex mirror. It is so because the image formed by a convex mirror is always virtual and cannot be obtained on a screen.

21. What is index error?

Answer: The difference between the actual distance between the pole of a mirror (optical centre, of a lens) and the point object (or image) and observed distance measured on optical bench is called index error. It is also called bench error.

• Index error = observed distances – actual distance.
• Index correction = actual distance – observed distance.

22. What is the value of the radius of curvature and magnification of a plane mirror?

23. At what distance from the needle should the eye be placed while removing parallel?

24. What are paraxial and marginal rays?

Answer: The rays which are very close to the principal axis or subtend a very small angle are called paraxial rays. The rays which are away or subtended large angle with principal axis are called marginal rays.

25. ow the spherical aberration is removed?

Answer: It is removed by taking small portion (aprature) of a lens or mirror or by using parabolic mirrors.

26. Is chromatic aberration takes place in mirror? Why?

Answer: No. Since it arises due to refraction in which lens provides dispersion like prism.

27. Is the focal length of a mirror change due to change in medium or colour of light.

Answer. No. Since focal length of a mirror does not depend the external medium and wavelength of light.

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