A concave mirror of focal length 15 cm is placed in water of refrective index `mu=4//3`. Water is filled upto 30 cm. An object of height 1 cm is paced at a distance of 20 cm in front of the mirror. If the size of image is n times size of object then n=

Options

A. 3
B. 2
C. 2.5
D. 5

Explanation

To solve the problem, we need to find the value of \( n \), which is the ratio of the size of the image to the size of the object. We will use the mirror formula and the magnification formula for a concave mirror.

### Step-by-Step Solution:

1. **Identify Given Values:**
- Focal length of the concave mirror, \( f = -15 \, \text{cm} \) (negative because it's a concave mirror).
- Object distance, \( u = -20 \, \text{cm} \) (negative as per the sign convention).
- Height of the object, \( h_o = 1 \, \text{cm} \).
- Refractive index of water, \( \mu = \frac{4}{3} \).
- Depth of water, \( h = 30 \, \text{cm} \).

2. **Calculate the Effective Focal Length in Water:**
The focal length of the mirror does not change with the medium, so we can use the focal length as it is:
\[ f = -15 \, \text{cm} \]

3. **Use the Mirror Formula:**
The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Rearranging gives:
\[ \frac{1}{v} = \frac{1}{f} - \frac{1}{u} \]
Substituting the values:
\[ \frac{1}{v} = \frac{1}{-15} - \frac{1}{-20} \]

4. **Calculate \( \frac{1}{v} \):**
Finding a common denominator (60):
\[ \frac{1}{v} = -\frac{4}{60} + \frac{3}{60} = -\frac{1}{60} \]
Therefore,
\[ v = -60 \, \text{cm} \]

5. **Calculate Magnification \( m \):**
The magnification \( m \) is given by:
\[ m = -\frac{v}{u} \]
Substituting the values:
\[ m = -\frac{-60}{-20} = 3 \]

6. **Relate Magnification to Size of Image:**
The magnification is also defined as the ratio of the height of the image to the height of the object:
\[ m = \frac{h_i}{h_o} \]
Therefore,
\[ h_i = m \cdot h_o = 3 \cdot 1 \, \text{cm} = 3 \, \text{cm} \]

7. **Find \( n \):**
Since \( n \) is defined as the ratio of the size of the image to the size of the object:
\[ n = m = 3 \]

### Final Answer:
\[ n = 3 \]