An object P is focussed by a microscope M. A glass slab of thickness 2'1 cm is introduced between P and M. If the refractive index of the slab is P5, by what distance should the microscope be shifted to focus the object again ?

Explanation

To solve the problem of how much the microscope should be shifted to focus the object again after introducing a glass slab, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Thickness of the glass slab (t) = 2.1 cm
- Refractive index of the glass slab (µ) = 1.5
- Refractive index of air (µ_air) = 1 (approximately)

2. **Understand the Formula for Shift:**
The shift (S) due to the introduction of a slab can be calculated using the formula:
\[ S = t \left(1 - \frac{\mu_{air}}{\mu_{slab}}\right) \]
Where:
- \( t \) = thickness of the slab
- \( \mu_{air} \) = refractive index of air
- \( \mu_{slab} \) = refractive index of the slab

3. **Substitute the Values into the Formula:**
Substitute the known values into the formula:
\[ S = 2.1 \left(1 - \frac{1}{1.5}\right) \]

4. **Calculate the Fraction:**
First, calculate \( \frac{1}{1.5} \):
\[ \frac{1}{1.5} = \frac{2}{3} \approx 0.6667 \]

5. **Calculate \( 1 - \frac{1}{1.5} \):**
Now, calculate:
\[ 1 - 0.6667 = 0.3333 \]

6. **Calculate the Shift (S):**
Now, substitute back into the equation for S:
\[ S = 2.1 \times 0.3333 \approx 0.7 \text{ cm} \]

7. **Conclusion:**
The distance by which the microscope should be shifted to focus the object again is approximately **0.7 cm**.