The Optical Path of a ray of light of a given wavelength travelling a distance of 3 cm in flint glass having refractive index 1.6 is same as that on travelling a distance x cm through a medium having refractive index 1.25. Determine the value of x.

Answer

To solve the problem, we need to equate the optical path lengths in the two different media. The optical path length (OPL) is given by the product of the refractive index (μ) and the distance (d) traveled in that medium.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Refractive index of flint glass (μ_f_g) = 1.6
- Distance traveled in flint glass (d_f_g) = 3 cm
- Refractive index of the second medium (μ_m) = 1.25
- Distance traveled in the second medium (d_m) = x cm (this is what we need to find)

2. **Write the Formula for Optical Path Length:**
The optical path length in the first medium (flint glass) can be expressed as:
\[ OPL_{f_g} = μ_{f_g} \cdot d_{f_g} \]
The optical path length in the second medium can be expressed as:
\[ OPL_m = μ_m \cdot d_m \]

3. **Set the Optical Path Lengths Equal:**
Since the optical path lengths are equal, we can set up the equation:
\[ μ_{f_g} \cdot d_{f_g} = μ_m \cdot d_m \]
Substituting the known values:
\[ 1.6 \cdot 3 = 1.25 \cdot x \]

4. **Calculate the Left Side:**
Calculate \(1.6 \cdot 3\):
\[ 1.6 \cdot 3 = 4.8 \]

5. **Set Up the Equation:**
Now we have:
\[ 4.8 = 1.25 \cdot x \]

6. **Solve for x:**
To find x, divide both sides by 1.25:
\[ x = \frac{4.8}{1.25} \]

7. **Calculate the Value of x:**
Performing the division:
\[ x = 3.84 \text{ cm} \]

### Final Answer:
The value of x is 3.84 cm.

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