Two transparent slabs having equal thickness but different refractive indices `mu_1 and mu_2`, are pasted side by side to form a composite slab. This slab is placed just after the double slit in a Young's experiment so that the light from one slit goes through one material and the light from the other slit goes through the other material. What should be the minimum thickness of the slab so that there is a minimum at the point `P_0` which is equidistant from the slits ?

Explanation

To solve the problem, we need to determine the minimum thickness of the composite slab formed by two transparent slabs with different refractive indices, such that there is a minimum at the point \( P_0 \) in a Young's double slit experiment.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have two transparent slabs with equal thickness \( t \) and different refractive indices \( \mu_1 \) and \( \mu_2 \).
- The slabs are placed side by side after the double slit, allowing light from one slit to pass through one slab and light from the other slit to pass through the other slab.

2. **Optical Path Difference**:
- The optical path length for light passing through the first slab (with refractive index \( \mu_1 \)) is \( \mu_1 \cdot t \).
- The optical path length for light passing through the second slab (with refractive index \( \mu_2 \)) is \( \mu_2 \cdot t \).
- The optical path difference \( \Delta x \) between the two paths is given by:
\[ \Delta x = \mu_1 \cdot t - \mu_2 \cdot t = (\mu_1 - \mu_2) \cdot t \]

3. **Condition for Minimum at Point \( P_0 \)**:
- For a minimum to occur at point \( P_0 \), the path difference must be equal to half the wavelength of light used in the experiment:
\[ \Delta x = \frac{\lambda}{2} \]

4. **Setting Up the Equation**:
- From the above, we can set up the equation:
\[ |(\mu_1 - \mu_2) \cdot t| = \frac{\lambda}{2} \]

5. **Solving for Minimum Thickness \( t \)**:
- Rearranging the equation gives us:
\[ t = \frac{\lambda}{2 |(\mu_1 - \mu_2)|} \]
- This equation provides the minimum thickness of the slab required to achieve a minimum at point \( P_0 \).

### Final Answer:
The minimum thickness \( t \) of the composite slab should be:
\[ t = \frac{\lambda}{2 |\mu_1 - \mu_2|} \]