A concave mirror is placed at the bottom of an empty tank with face upwards and axis vertical. When sunlight falls normally on the mirror, it is focussed at distance of `32cm` from the mirror. If the tank filled with water `(mu=4//3)` up to a height of `20cm`, then the sunlight will now get focussed at

Explanation

To solve the problem step by step, let's break it down:

### Step 1: Understand the Initial Setup
We have a concave mirror placed at the bottom of an empty tank. When sunlight falls normally on the mirror, it focuses at a distance of 32 cm from the mirror.

### Step 2: Determine the Focal Length of the Mirror
Since the sunlight focuses at a distance of 32 cm when the tank is empty, this distance is equal to the focal length (f) of the concave mirror. Therefore:
- Focal length, \( f = 32 \, \text{cm} \)

### Step 3: Fill the Tank with Water
The tank is filled with water up to a height of 20 cm. We need to find the new position where the sunlight will focus after the water is added.

### Step 4: Calculate the Actual Depth Below Water Surface
The distance from the water surface to the mirror is:
- Actual depth, \( h = 20 \, \text{cm} \)

### Step 5: Calculate the Distance from the Water Surface to the Focus Point
The distance from the mirror to the point where the sunlight was originally focused (O) is 32 cm. Therefore, the distance from the water surface to the focus point (O) is:
- Distance from water surface to mirror, \( BO = 32 \, \text{cm} - 20 \, \text{cm} = 12 \, \text{cm} \)

### Step 6: Apply the Concept of Apparent Depth
When light passes from a denser medium (water) to a rarer medium (air), it bends away from the normal. The relationship between the actual depth and the apparent depth can be described using the refractive index (\( \mu \)):
\[ \frac{\mu_{\text{water}}}{\mu_{\text{air}}} = \frac{\text{Actual Depth}}{\text{Apparent Depth}} \]
Given that:
- \( \mu_{\text{water}} = \frac{4}{3} \)
- \( \mu_{\text{air}} = 1 \)

### Step 7: Set Up the Equation
Let \( BI \) be the apparent depth (the new focus point). The actual depth \( BO \) is 12 cm. Thus, we can write:
\[ \frac{\frac{4}{3}}{1} = \frac{12 \, \text{cm}}{BI} \]

### Step 8: Solve for Apparent Depth
Rearranging the equation gives:
\[ BI = \frac{12 \, \text{cm} \times 1}{\frac{4}{3}} = 12 \, \text{cm} \times \frac{3}{4} = 9 \, \text{cm} \]

### Conclusion
When the tank is filled with water, the sunlight will now focus at a distance of **9 cm** from the mirror.

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