To find focal length of a convex mirror using convex lens

To determine the focal length of a convex mirror, we can use a convex lens in an experimental setup. A convex mirror always forms a virtual, diminished, and upright image, while a convex lens can form a real image of an object. By placing an object in front of a convex lens, we first obtain a real image on a screen. Then, by positioning the convex mirror behind the convex lens, the mirror reflects the image back through the lens, forming a secondary image. This secondary image appears to come from behind the mirror, and by adjusting the distances until the image on the screen coincides with the original object position, we can apply the lens formula to calculate the focal length of the convex mirror. This experiment combines the properties of lenses and mirrors to find the focal length of the convex mirror, which is typically more challenging to measure directly.

Aim

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

Apparatus

An optical bench with four uprights (two fixed uprights in the middle, two outer uprights with lateral movement), a convex lens (20 cm focal length), a convex mirror, a lens holder, a mirror holder, two optical needles, (one thin, one thick) a knitting needle, and a half metre scale.

Theory

Focal length of a convex mirror $f=\frac{R}{2}$
where R= radius of curvature of the mirror.

Experiment Arrangement and Ray Diagram

An auxiliary convex lens L is introduced between the convex mirror M and object needle 0 as shown in ray diagram (a). Keeping the object needle at a distance of about 1.5 times the rough focal length of a convex lens, the position of the convex mirror behind the convex lens is so
adjusted that a real and inverted image of object needle O is formed at O itself. Under such conditions, the light rays are incident normally over the convex mirror to retrace their path. In the absence of a convex mirror, these rays would have met at the centre of curvature C of the convex mirror. The distance PC gives the radius of curvature R of the mirror.

To locate the position of C, the convex mirror is removed (without disturbing the object needle O and convex lens L ). An image needle I is put behind the convex lens and moved to a position at which there is no parallax between the tip of the inverted image of the O needle and the tip of the I needle. Position of image needle I gives the position of the centre of curvature C of the mirror M ray diagram (b)

Then PC= PI=R

and $f=\frac{R}{2}=\frac{P I}{2}$

f= focal length

R= radius of curvature of the mirror

Procedure

1. The rough focal length of the convex lens is determined by focusing on a distant object.

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

To set the object needle
5 . Take the thin optical needle as an object needle (O). Mount it in outer laterally moveable upright near zero end.
6. Move the object needle upright and clamp it at a distance (in full cm) nearly 1.5 times the obtained
the rough focal length of the lens.

7. Adjust the height of the object needle to make its tip lie on a horizontal line through the optical centre of the lens.
8. Note the position of the index mark on the base of the object needle upright.

To set the convex mirror
9. Clamp the holder with a convex mirror in the second fixed upright near the lens, keeping the reflecting surface of the mirror towards the lens.
10. Adjust the height of the mirror to make its pole lie on a horizontal line through the optical centre of the lens.

11. Make the mirror surface vertical and perpendicular to the length of the optical bench (the principal
axes of mirror and lens must coincide.)
12. Move towards the zero end of the optical bench (where the object needle is mounted).
13. Closing left eye, keep open right eye about 30 cm away from the tip of the object needle.

14. See the inverted image of the object needle (formed by reflection from the convex mirror).
15. Keep the eye in a position at which the tips of the inverted image and the object needle are seen
simultaneously.
16. Adjust the height of the needle so that the two tips are seen in one line with the right open eye.

17. Move the eye towards the right. The tips will get separated. The tips have parallax.
18. Move the convex mirror back and forth till tip to tip parallax is removed.
19. Note the position of the index mark on the base of the convex mirror upright.

To set the image needle
20. Remove the convex mirror, keeping it upright in its position.
21. With the left eye closed, see with the right open eye from the other end of the optical bench. An inverted and enlarged image of the object needle will be seen. The tip of the image must lie in the middle of the lens.

22. Mount the thick optical needle (image needle) in the fourth upright near the other end of the optical
bench.
23. Adjust the height of the image needle so that its tip is seen in line with the tip of the image when seen with the right open eye.
24. Move the eye towards the right. The tips will get separated. The image tip and the image needle tip have parallax.

25. Remove the parallax tip to tip.
26. Note the position of the index mark on the base of the image needle upright.
27. Record the position of the index marks.

To determine index correction
28. Find index correction for the distance between the pole of the convex mirror and the tip of the image needle.

To get more observations
29. Move the object needle upright towards the lens by 2 cm to get more observation and Repeat the experiment.
30. Move the object needle upright away from the lens (from a position of observation 2 ) by 2 cm to get
more observation and Repeat the experiment.
31 Get more and more observations from the experiment and on the basis of observations find out the focal length of the given convex mirror.

Observations

Rough focal length of the convex lens = .....
The actual length of the knitting needle = .....

Observed distance between image needle I and back of the convex mirror y =...

So Index correction =(x-y)=...

Calculations

1. Write down both observed PI and corrected PI
2. Divide corrected PI by 2 to get f
3. Find the mean of values of f recorded

Result

The focal length of the given convex mirror=....cm

Solved Examples Based on Finding Focal Length of a Convex Mirror Using Convex Lens

Example 1: A convex spherical mirror has a focal length of -20 cm. An object is placed 10 cm in front of the mirror on the mirror axis, where is the mirror located.

1) 6.7 cm in front of the mirror

2) 20 cm behind the mirror

3) 6.7 cm behind the mirror

4) 20 cm in front of the mirror

Solution

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

$\begin{aligned} & \mathrm{PC}=\mathrm{PI}-\mathrm{R} \\ & f=\frac{R}{2}=\frac{P I}{2} \\ & \mathrm{f}=-20 \mathrm{~cm}, \mathrm{u}=10 \mathrm{~cm} \\ & \frac{1}{v}=\frac{1}{f}-\frac{1}{u}=-\frac{1}{20}-\frac{1}{10}=\frac{3}{-20} \\ & v=\frac{-20}{3} \mathrm{~cm}=-6.7 \mathrm{~cm}\end{aligned}$

$\therefore$ Image id 6.7 cm behind the mirror

Hence, the answer is the option (3).

Example 2: You are provided with a convex lens and a distant object. Using the parallax method, determine the focal length of the convex lens. The lens is placed in front of the object, and you observe the apparent shift when viewing the object through the lens from different angles.
Given:
• Distance between the object and the lens, Do = 50 cm
• Apparent shift of the object, d = 2 cm
• Distance between your eyes and the lens, De = 25 cm

Assumptions:
• The lens is thin and the lens thickness is negligible.
• The lens is properly aligned and centred.

1) 14.12 cm

2) 17.87 cm

3) 16.67 cm

4) 15.26 cm

Solution

Apply the parallax method formula to calculate the focal length (f) of the convex lens:

$
f=\frac{D_o \times D_e}{D_o+D_e} f=\frac{50 \mathrm{~cm} \times 25 \mathrm{~cm}}{50 \mathrm{~cm}+25 \mathrm{~cm}}
$
Substitute the given values into the formula:

$
\begin{aligned}
& f=\frac{50 \mathrm{~cm} \times 25 \mathrm{~cm}}{50 \mathrm{~cm}+25 \mathrm{~cm}} \\
& f=16.67 \mathrm{~cm}
\end{aligned}
$
The focal length of the convex lens using the parallax method is approximately 16.67 cm.

Hence, the answer is the option (3).

Example 3: You are provided with a convex mirror, a concave mirror, and a convex lens. Using the parallax method, determine the focal lengths of these optical devices. For each case, the object is placed at a certain distance, and you observe the apparent shift when viewing the object through the optical device from different angles.

Given:

• Distance between the object and the optical device, Do = 60 cm
• Apparent shift of the object, d = 3 cm
• Distance between your eyes and the optical device, De = 30 cm

Assumptions:
• The optical devices are properly aligned and centred.
• For convex mirror and lens, the object is virtual.

1) 20 cm, -20 cm , 20 cm

2)-20 cm, -20 cm , 20 cm

3) 20 cm, 20 cm , 20 cm

4)20 cm, -20 cm , -20 cm

Solution

For Convex Mirror:
1. Apply the parallax method formula to calculate the focal length (f) of the convex mirror:

$
f=\frac{D_o \cdot D e}{D_o-D_e}
$

2. Substitute the given values into the formula:

$
f=\frac{60 \mathrm{~cm} \cdot 30 \mathrm{~cm}}{60 \mathrm{~cm}-30 \mathrm{~cm}}
$

3. Calculate the focal length:

$
f=-20 \mathrm{~cm}
$
For Concave Mirror:
1. Apply the same parallax method formula to calculate the focal length ( $f$ ) of the concave mirror:

$
f=\frac{D_o \cdot D e}{D_o-D_e}
$

2. Substitute the given values into the formula:

$
f=\frac{60 \mathrm{~cm} .30 \mathrm{~cm}}{60 \mathrm{~cm}-30 \mathrm{~cm}}
$

3. Calculate the focal length:

f = −20 cm

For Convex Lens:

1. Apply the parallax method formula to calculate the focal length (f) of the convex lens:
$
f=\frac{D_o \cdot D e}{D_o+D_e}
$

2. Substitute the given values into the formula:

$
f=\frac{60 \mathrm{~cm} \cdot 30 \mathrm{~cm}}{60 \mathrm{~cm}+30 \mathrm{~cm}}
$

3. Calculate the focal length:

$
\mathrm{f}=20 \mathrm{~cm}
$

The focal lengths of the optical devices, determined using the parallax method, are as follows:
• For the convex mirror: f = 20 cm
• For the concave mirror: f = -20 cm
• For the convex lens: f = 20 cm
The parallax method involves observing the apparent shift in the position of the object when viewed through the optical device from different angles. By measuring the distances and applying the parallax formula, you can calculate the focal lengths of the devices.
Hence, the answer is the option(1).

Example 4: You are provided with a convex mirror. To determine its focal length using the parallax method, follow these steps:
1. Set up the convex mirror on a stand and place an object at a distance of 60 cm in front of the mirror.
2. Observe the object through the convex mirror while moving your head from side to side, noting the apparent shift in the position of the object.

3. Measure the distance between your eyes and the mirror to be 25 cm.
Given:
• Distance between the object and the convex mirror, Do = 60 cm
• Apparent shift of the object, d = 6 cm
• Distance between your eyes and the convex mirror, De = 25 cm

Assumptions:
• The convex mirror is properly aligned and positioned.

1) -30.2 cm

2) -33.7 cm

3) -37.5 cm

4) -29.1 cm

Solution

Apply the parallax method formula to calculate the focal length (f) of the convex mirror:
$
f=\frac{D_o \cdot D_e}{D_o-D_e}
$
Substitute the given values into the formula:

$
f=\frac{60 \mathrm{~cm} \cdot 25 \mathrm{~cm}}{60 \mathrm{~cm}-25 \mathrm{~cm}}
$

Calculate the focal length:

f = −37.5 cm
The focal length of the convex mirror, determined using the parallax method, is approximately −37.5 cm.

Hence, the answer is the option (3).

Summary

To find the focal length of a convex mirror, we use a convex lens to project a real image onto a screen. By placing the convex mirror behind the lens, the mirror reflects the image back through the lens, forming a secondary image. By adjusting the setup so that the secondary image overlaps with the original object's position, we can calculate the focal length of the convex mirror using the lens formula. This method provides an effective way to measure the focal length of a convex mirror, demonstrating the interplay between lenses and mirrors in optical systems.