A convex lens of focal length 20 cm is placed in front of a convex mirror of focal length 15 cm at a distance of 5 cm. An object of length 2 cm is placed perpendicular to the common axis at a distance of 20 cm from the lens

Options

A. The final image is formed at 163 cm from the lens
B. The final image is formed at infinity
C. The size of final image is 10.7 cm
D. The size of final image is very large

Answer

To solve the problem step by step, we will analyze the situation involving a convex lens and a convex mirror, and determine the position and size of the final image formed.

### Step 1: Identify the given data
- Focal length of the convex lens, \( f_L = 20 \, \text{cm} \)
- Focal length of the convex mirror, \( f_M = 15 \, \text{cm} \)
- Distance between the lens and the mirror, \( d = 5 \, \text{cm} \)
- Object distance from the lens, \( u_L = -20 \, \text{cm} \) (negative as per the sign convention)
- Height of the object, \( h_o = 2 \, \text{cm} \)

### Step 2: Calculate the image position formed by the convex lens
Using the lens formula:
\[ \frac{1}{f_L} = \frac{1}{v_L} - \frac{1}{u_L} \]
Rearranging gives:
\[ \frac{1}{v_L} = \frac{1}{f_L} + \frac{1}{u_L} \]
Substituting the values:
\[ \frac{1}{v_L} = \frac{1}{20} + \frac{1}{-20} = \frac{1}{20} - \frac{1}{20} = 0 \]
This implies that \( v_L \) approaches infinity, meaning the image formed by the lens is at infinity.

### Step 3: Analyze the rays after passing through the lens
Since the image formed by the lens is at infinity, the rays emerging from the lens are parallel. These parallel rays will now act as the object for the convex mirror.

### Step 4: Determine the object distance for the convex mirror
The distance from the lens to the mirror is \( 5 \, \text{cm} \). Since the image from the lens is at infinity, we consider the object for the mirror to be at infinity as well.

### Step 5: Calculate the image position formed by the convex mirror
For a convex mirror, the image formed by parallel rays (object at infinity) is at the focus of the mirror. The focal length of the convex mirror is \( f_M = 15 \, \text{cm} \), so the image will be formed at \( v_M = f_M = 15 \, \text{cm} \) behind the mirror.

### Step 6: Determine the size of the final image
Since the image formed by the convex mirror is virtual and located at the focus, the size of the image can be analyzed. The image formed by a convex mirror is always smaller than the object. However, since the object was at infinity, the final image will be highly enlarged.

### Final Result
- The final image position is at infinity.
- The size of the final image is very large.