Question

An object is placed at a distance of 12 cm from a convex lens. A convex mirror of focal length 15 cm is placed on other side of lens at 8 cm as shown in the figure. Image of object coincides with the object. When the convex mirror is removed, a real and inverted image is formed at a position. The distance of the image from the object will be __________ (cm). 

 

Hint:

The condition "final image coincides with the object" in a lens-mirror system almost always means that the rays are retracing their path. This implies that the rays must strike the mirror normally (i.e., directed towards its center of curvature). This is the key to solving the first part of such problems.

Answer 1

Correct Answer: 50

Step 1: Understanding the Question:
The problem involves two parts. First, a lens-mirror combination where the final image forms at the object's location. This setup allows us to find the focal length of the lens. Second, the mirror is removed, and we need to find the position of the image formed by the lens alone and its distance from the original object.
Step 2: Key Formula or Approach:
1. For the final image to form at the object's position, the light rays must retrace their path after reflecting from the mirror.
2. For rays to retrace their path, they must strike the convex mirror normally (along the radius of curvature). This means the rays are directed towards the mirror's center of curvature.
3. The image formed by the lens (I\(_1\)) must be located at the center of curvature of the convex mirror.
4. Use the lens formula \( \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \). The radius of curvature is R = 2f.
Step 3: Detailed Explanation:
Part 1: Finding the focal length of the convex lens (f\(_l\))
Object distance for the lens, u = -12 cm.
Focal length of the convex mirror, f\(_m\) = +15 cm.
Distance between lens and mirror = 8 cm.
The center of curvature (C) of the convex mirror is at a distance R = 2f\(_m\) = 2(15) = 30 cm from its pole.
Since the mirror is 8 cm from the lens, its center of curvature is at a distance of 8 cm + 30 cm = 38 cm from the lens.
For the rays to strike the mirror normally, the image formed by the lens (I\(_1\)) must be at this point C. So, the image distance for the lens is v = +38 cm.
Now, use the lens formula to find f\(_l\):
\[ \frac{1}{f_l} = \frac{1}{v} - \frac{1}{u} = \frac{1}{38} - \frac{1}{-12} = \frac{1}{38} + \frac{1}{12} \] \[ \frac{1}{f_l} = \frac{12 + 38}{38 \times 12} = \frac{50}{456} \] So, the focal length of the lens is \( f_l = \frac{456}{50} \) cm.
Part 2: Finding the final image position with the lens alone
Now, the mirror is removed. The object is still at u = -12 cm. We find the new image position (v') using the lens formula with the calculated f\(_l\).
\[ \frac{1}{v'} = \frac{1}{f_l} + \frac{1}{u} = \frac{50}{456} + \frac{1}{-12} = \frac{50}{456} - \frac{1}{12} \] \[ \frac{1}{v'} = \frac{50 - (456/12)}{456} = \frac{50 - 38}{456} = \frac{12}{456} \] \[ v' = \frac{456}{12} = 38 \text{ cm} \] The image is formed at 38 cm on the right side of the lens. It is real and inverted.
Part 3: Distance between object and image
The object is at 12 cm to the left of the lens. The image is at 38 cm to the right of the lens.
The total distance between the object and the image is 12 cm + 38 cm = 50 cm.
Step 4: Final Answer:
The distance of the image from the object is 50 cm.