A spherical mirror is placed 10 cm below the level of water. A point oject is placed in air 30 cm above the water surface on the axis of the mirror such that two images seen by an observer above the water surface coincide. The images are formed by partial reflection at the water surface and due to emerging light after feflection from the mirror. Find the focal length of the mirror. `(mu_("water")=(4)/(3))`

Options

A. `(100)/(3)cm`
B. 75 cm
C. `(100)/(8)cm`
D. 50 cm

Answer

To solve the problem, we need to find the focal length of a spherical mirror based on the given conditions. Let's break this down step by step.

### Step 1: Understanding the Setup
- We have a spherical mirror placed 10 cm below the water surface.
- A point object is placed 30 cm above the water surface.
- The refractive index of water (μ) is given as \( \frac{4}{3} \).

### Step 2: Calculate the Actual Depth of the Object
The apparent depth of the object when viewed from above the water surface can be calculated using the formula:
\[ d_{\text{apparent}} = \frac{d_{\text{actual}}}{\mu} \]
Given that the object is 30 cm above the water surface, the actual depth (d_actual) can be calculated as follows:
\[ d_{\text{apparent}} = 30 \text{ cm} \]
\[ d_{\text{actual}} = d_{\text{apparent}} \times \mu = 30 \times \frac{4}{3} = 40 \text{ cm} \]

### Step 3: Determine the Position of the Object Relative to the Mirror
Since the mirror is 10 cm below the water surface, the distance from the water surface to the mirror is 10 cm. Therefore, the distance from the mirror to the actual position of the object is:
\[ \text{Distance from mirror to object} = d_{\text{actual}} - \text{Distance from water surface to mirror} = 40 \text{ cm} - 10 \text{ cm} = 30 \text{ cm} \]
Thus, the object distance (u) is -30 cm (the negative sign indicates that the object is in front of the mirror).

### Step 4: Understanding the Image Formation
The observer sees two images that coincide. One image is formed by the reflection at the water surface, and the other is formed by the reflection from the mirror.

### Step 5: Using the Mirror Formula
The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Where:
- \( f \) is the focal length of the mirror.
- \( v \) is the image distance from the mirror.
- \( u \) is the object distance from the mirror.

### Step 6: Finding the Image Distance (v)
Since the two images coincide, we can assume that the image formed by the mirror is at the same position as the image formed by the water surface reflection.

The image formed by the water surface can be considered as being at the same distance as the object but on the opposite side of the water surface. Therefore, the image distance (v) can be calculated as:
\[ v = \text{Distance from water surface to mirror} + \text{Distance from water surface to object} = 10 \text{ cm} + 30 \text{ cm} = 40 \text{ cm} \]
Thus, \( v = 40 \text{ cm} \).

### Step 7: Substitute Values into the Mirror Formula
Now we can substitute the values into the mirror formula:
\[ \frac{1}{f} = \frac{1}{40} + \frac{1}{-30} \]
Calculating the right-hand side:
\[ \frac{1}{f} = \frac{1}{40} - \frac{1}{30} \]
Finding a common denominator (120):
\[ \frac{1}{f} = \frac{3}{120} - \frac{4}{120} = \frac{-1}{120} \]
Thus,
\[ f = -120 \text{ cm} \]

### Final Answer
The focal length of the mirror is \( -120 \text{ cm} \).

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