The refractive index of air with respect to glass is `2//3`. The refractive index of diamond with respect to air is `12//5`. Then the refractive index of glass with respect to diamond will be

Options

A. `5//8`
B. `8//9`
C. `5//18`
D. `18//5`

Answer

To find the refractive index of glass with respect to diamond, we can use the given refractive indices and apply the formula for refractive indices.

### Step-by-Step Solution:

1. **Define the Refractive Indices**:
- Let the refractive index of air with respect to glass be denoted as:
\[ \frac{\mu_a}{\mu_g} = \frac{2}{3} \]
- Let the refractive index of diamond with respect to air be denoted as:
\[ \frac{\mu_d}{\mu_a} = \frac{12}{5} \]

2. **Express the Refractive Indices**:
- From the first equation, we can express \(\mu_a\) in terms of \(\mu_g\):
\[ \mu_a = \frac{2}{3} \mu_g \]
- From the second equation, we can express \(\mu_d\) in terms of \(\mu_a\):
\[ \mu_d = \frac{12}{5} \mu_a \]

3. **Substitute \(\mu_a\) into the Equation for \(\mu_d\)**:
- Substitute \(\mu_a\) from step 2 into the equation for \(\mu_d\):
\[ \mu_d = \frac{12}{5} \left(\frac{2}{3} \mu_g\right) \]
- Simplifying this gives:
\[ \mu_d = \frac{12 \times 2}{5 \times 3} \mu_g = \frac{24}{15} \mu_g \]

4. **Find the Refractive Index of Glass with Respect to Diamond**:
- The refractive index of glass with respect to diamond is given by:
\[ \frac{\mu_g}{\mu_d} \]
- Substitute \(\mu_d\) from step 3:
\[ \frac{\mu_g}{\mu_d} = \frac{\mu_g}{\frac{24}{15} \mu_g} \]
- This simplifies to:
\[ \frac{\mu_g}{\mu_d} = \frac{15}{24} = \frac{5}{8} \]

5. **Final Answer**:
- The refractive index of glass with respect to diamond is:
\[ \frac{5}{8} \]