1. Aim

To determine the focal length of a convex mirror using a convex lens of known focal length.

2. Apparatus Used

• Convex mirror
• Convex lens of known focal length
• Optical bench with holders
• Screen
• Object (illuminated pin or cross-wire)
• Meter scale or ruler
• Clamps and stands
• White sheet of paper
• Source of light (preferably a bright LED)

3. Diagram

4. Theory

This experiment utilizes the principles of ray optics and image formation to determine the focal length of a convex mirror. Since direct measurement of the focal length of a convex mirror is challenging (as the focus lies behind the mirror), we use an indirect method involving a convex lens of known focal length.

The method is based on the fact that when an object is placed beyond the focus of a convex lens, a real image is formed. If a convex mirror is placed in the path of these rays before they converge to form the real image, the rays will be reflected and appear to come from a virtual image formed behind the mirror.

By adjusting the positions of the lens and mirror, we can create a situation where the reflected rays form an image at the same position as the object. In this case, the rays are incident normally on the mirror, and the position of the lens becomes significant for our calculations.

When the final image coincides with the object, we have the following relationships:

where:

• $d_1$ is the distance from the lens to the object/final image
• $d_2$ is the distance from the lens to the mirror
• $f_m$ is the focal length of the convex mirror

We can also relate this to the lens equation:

where:

• $u$ is the object distance from the lens
• $v$ is the image distance from the lens
• $f_l$ is the focal length of the convex lens

Through appropriate positioning and measurements, we can determine the focal length of the convex mirror.

5. Formula

The focal length of the convex mirror ($f_m$) can be calculated using:

where:

• $d_1$ is the distance from the lens to the object/final image
• $d_2$ is the distance from the lens to the mirror

Alternatively, using the lens formula and the positions measured:

Once $d_1$ and $d_2$ are determined experimentally, $f_m$ can be calculated.

6. Procedure

1. Set up the optical bench and ensure it's stable and horizontal.
2. Place the object (illuminated pin or cross-wire) at one end of the optical bench.
3. Mount the convex lens on a holder and place it on the optical bench at a distance greater than its focal length from the object.
4. Adjust the position of the screen to obtain a sharp real image of the object.
5. Remove the screen and place the convex mirror beyond the lens such that the concave side faces the lens.
6. Adjust the position of the mirror so that the reflected rays retrace their path and form an image at the position of the object itself.
7. To verify this, place a small screen (like a white card with a pinhole) near the object. If the position is correct, a clear image will be visible on the screen.
8. Measure and record the distance $d_1$ between the object and the lens.
9. Measure and record the distance $d_2$ between the lens and the mirror.
10. Repeat the experiment 3-5 times by changing the position of the lens and mirror to get more accurate results.
11. Calculate the focal length of the convex mirror using the formula $f_m = \frac{d_1 + d_2}{2}$ for each set of readings.
12. Calculate the average focal length and report it as the final result.

7. Observation Table

S.No.	Distance between object and lens ($d_1$) cm	Distance between lens and mirror ($d_2$) cm	Focal length of convex mirror ($f_m = \frac{d_1 + d_2}{2}$) cm
1
2
3
4
5

8. Calculations

For each observation, calculate the focal length of the convex mirror using:

Example calculation for the first observation (assuming measured values):

Let $d_1 = 15$ cm and $d_2 = 25$ cm

Calculate the mean focal length:

9. Result

The focal length of the given convex mirror is __________ cm.

The percentage error in the measurement is __________ %.

10. Precautions

1. The optical bench should be placed on a stable surface and properly leveled.
2. All optical components (lens, mirror, object) should be aligned along the same horizontal axis.
3. The object should be well-illuminated to get clear images.
4. The lens and mirror surfaces should be clean and free from dust or fingerprints.
5. Parallax errors should be avoided while taking measurements.
6. The observer's eye should be perpendicular to the scale while noting down readings to avoid parallax.
7. The room should be moderately dark to better observe the formed images.
8. When verifying that the image coincides with the object, use a small screen with a pinhole to avoid interference with the light path.
9. Handle the optical components carefully to prevent scratches or damage.

11. Sources of Error

1. Spherical aberration in the lens can lead to blurred images and imprecise measurements.
2. Imperfect alignment of the optical components can introduce systematic errors.
3. Parallax errors while reading measurements from the scale.
4. The convex mirror might not be perfectly convex, leading to deviations in the calculated focal length.
5. Temperature variations can slightly alter the focal lengths of both the lens and mirror.
6. Difficulty in precisely determining when the reflected image coincides with the object.
7. Limitations in the precision of the measuring instruments used.
8. Human errors in reading and recording measurements.
9. Imperfect adjustment of the positions of the optical components.

12. Viva Voice Questions

1. Q: Why do we use a convex lens to determine the focal length of a convex mirror?

   A: Since the focal point of a convex mirror lies behind the mirror and is not directly accessible, we use a convex lens to create a situation where we can indirectly measure the focal length by analyzing the path of light rays.

2. Q: What happens if the distance between the lens and the mirror is less than the focal length of the lens?

   A: If the distance is less than the focal length of the lens, the lens will not form a real image, and the method described in this experiment would not work because it relies on the formation of a real image by the lens.

3. Q: Why must the image coincide with the object in this experiment?

   A: When the image coincides with the object, it means the rays after reflection from the mirror are retracing their path. This condition gives us a specific geometric relationship that allows us to calculate the focal length of the mirror.

4. Q: How does the curvature of a convex mirror relate to its focal length?

   A: The focal length of a convex mirror is directly proportional to its radius of curvature. Specifically, the focal length is half of the radius of curvature ($f = \frac{R}{2}$).

5. Q: Can this method be used to find the focal length of a concave mirror as well?

   A: Yes, a similar method can be used for a concave mirror, but the setup and calculations would be slightly different since a concave mirror can form real images.

6. Q: What are the practical applications of convex mirrors?

   A: Convex mirrors are used as rear-view mirrors in vehicles, security mirrors in stores and at blind corners, and for decorative purposes. They provide a wider field of view than flat mirrors, though the images are smaller and distorted.

7. Q: Why does a convex mirror always form a virtual, erect, and diminished image?

   A: In a convex mirror, incident rays diverge after reflection, making it impossible for them to converge to form a real image. The reflected rays appear to come from behind the mirror (forming a virtual image). The image is erect and diminished due to the geometry of reflection from the curved surface.

8. Q: How would the results change if we used a lens with a different focal length?

   A: The distances $d_1$ and $d_2$ would change, but the calculated focal length of the mirror should remain the same if the experiment is conducted correctly.