A travelling microscope is focussed on to a scratch on the bottom of a beaker. Water of refractive index `(4)/(3)` is poured in it. Then the microscope is to be lifted through 2 cm focus it again. Find the depth of water in the beaker.

Options

A. 4 cm
B. 8 cm
C. `(8)/(7)cm`
D. `(8)/(3) cm`

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the problem
We have a scratch at the bottom of a beaker, and a microscope is focused on it. When water is poured into the beaker, the microscope needs to be lifted by 2 cm to refocus on the scratch. We need to find the actual depth of the water in the beaker.

### Step 2: Define the variables
- Let \( D \) be the actual depth of the water in the beaker.
- The refractive index of water \( \mu = \frac{4}{3} \).
- The apparent depth \( d' \) when viewed through the water can be calculated using the formula:
\[ d' = \frac{D}{\mu} \]

### Step 3: Set up the equation
When the microscope is lifted by 2 cm, the relationship between the actual depth \( D \), the apparent depth \( d' \), and the distance the microscope is lifted can be expressed as:
\[ D - d' = 2 \text{ cm} \]

Substituting \( d' \) from the previous step:
\[ D - \frac{D}{\mu} = 2 \]

### Step 4: Simplify the equation
Factor out \( D \):
\[ D \left(1 - \frac{1}{\mu}\right) = 2 \]

### Step 5: Substitute the value of \( \mu \)
Substituting \( \mu = \frac{4}{3} \):
\[ D \left(1 - \frac{1}{\frac{4}{3}}\right) = 2 \]

This simplifies to:
\[ D \left(1 - \frac{3}{4}\right) = 2 \]
\[ D \left(\frac{1}{4}\right) = 2 \]

### Step 6: Solve for \( D \)
Multiplying both sides by 4:
\[ D = 2 \times 4 = 8 \text{ cm} \]

### Conclusion
The actual depth of the water in the beaker is \( 8 \text{ cm} \).

---