A concave mirror of radius of curvature 40cm forms an image of an object placed on the principal axis at a distance 45 cm in front of it. Now if the system (including object) is completely immersed in water `(mu=1.33)` , then

Options

A. the image will shift towards the mirror
B. the magnification will reduce
C. the image will shift away from the mirror and magnification will increase.
D. the position of the image and magnification will not change.

Explanation

To solve the problem, we need to analyze the situation step by step.

### Step 1: Understand the given data
- Radius of curvature (R) of the concave mirror = 40 cm
- Object distance (u) = -45 cm (the negative sign is used because the object is in front of the mirror)
- Refractive index of water (µ) = 1.33

### Step 2: Calculate the focal length (F) of the concave mirror
The focal length (F) of a concave mirror is given by the formula:
\[ F = \frac{R}{2} \]
Substituting the value of R:
\[ F = \frac{40 \, \text{cm}}{2} = 20 \, \text{cm} \]

### Step 3: Use the mirror formula to find the image distance (V)
The mirror formula is given by:
\[ \frac{1}{F} = \frac{1}{V} + \frac{1}{U} \]
Substituting the known values:
\[ \frac{1}{20} = \frac{1}{V} + \frac{1}{-45} \]

### Step 4: Solve for V
Rearranging the equation:
\[ \frac{1}{V} = \frac{1}{20} + \frac{1}{45} \]

To add these fractions, we need a common denominator. The least common multiple of 20 and 45 is 180. Therefore:
\[ \frac{1}{20} = \frac{9}{180} \]
\[ \frac{1}{45} = \frac{4}{180} \]

Now, substituting these values back:
\[ \frac{1}{V} = \frac{9}{180} + \frac{4}{180} = \frac{13}{180} \]

Now, taking the reciprocal to find V:
\[ V = \frac{180}{13} \approx 13.85 \, \text{cm} \]
(Note: V is positive, indicating that the image is formed on the same side as the object.)

### Step 5: Analyze the effect of immersion in water
The mirror formula does not depend on the refractive index of the medium. Therefore, when the entire system is immersed in water, the distances (u and V) remain unchanged.

### Conclusion
Since the mirror formula remains valid regardless of the medium, the position of the image and the magnification will not change when the system is immersed in water.

### Final Answer
The correct option is **D**: The position of the image and the magnification will not change.

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