The focal lengths of a convex lens for red, yellow and violet rays are 100 cm, 98 cm and 96 cm respectively. Find the dispersive power of the material of the lens.

Explanation

To find the dispersive power of the material of the lens based on the given focal lengths for red, yellow, and violet rays, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Focal length for red light, \( F_r = 100 \, \text{cm} \)
- Focal length for yellow light, \( F_y = 98 \, \text{cm} \)
- Focal length for violet light, \( F_v = 96 \, \text{cm} \)

2. **Understand the Formula for Dispersive Power:**
The dispersive power \( D \) of a material is given by the formula:
\[ D = \frac{\mu_v - \mu_r}{\mu_y - 1} \]
where \( \mu \) is the refractive index for the respective colors.

3. **Relate Focal Lengths to Refractive Indices:**
For a convex lens, the refractive index \( \mu \) can be expressed in terms of the focal length \( F \) as:
\[ \mu - 1 = \frac{k}{F} \]
where \( k \) is a constant that depends on the lens shape. Thus, we can write:
- For red: \( \mu_r - 1 = \frac{k}{F_r} \)
- For yellow: \( \mu_y - 1 = \frac{k}{F_y} \)
- For violet: \( \mu_v - 1 = \frac{k}{F_v} \)

4. **Substituting into the Dispersive Power Formula:**
Substitute the expressions for \( \mu_r, \mu_y, \) and \( \mu_v \) into the dispersive power formula:
\[ D = \frac{\left(\frac{k}{F_v} + 1\right) - \left(\frac{k}{F_r} + 1\right)}{\left(\frac{k}{F_y} + 1\right) - 1} \]
This simplifies to:
\[ D = \frac{\frac{k}{F_v} - \frac{k}{F_r}}{\frac{k}{F_y}} = \frac{k\left(\frac{1}{F_v} - \frac{1}{F_r}\right)}{\frac{k}{F_y}} \]
The \( k \) cancels out:
\[ D = \frac{F_y \left(\frac{1}{F_v} - \frac{1}{F_r}\right)}{1} \]

5. **Calculating the Values:**
Substitute the values of \( F_r, F_y, \) and \( F_v \):
\[ D = F_y \left(\frac{1}{F_v} - \frac{1}{F_r}\right) = 98 \left(\frac{1}{96} - \frac{1}{100}\right) \]

6. **Finding a Common Denominator:**
Calculate the difference:
\[ \frac{1}{96} - \frac{1}{100} = \frac{100 - 96}{9600} = \frac{4}{9600} = \frac{1}{2400} \]

7. **Final Calculation:**
Now substitute this back into the equation for \( D \):
\[ D = 98 \times \frac{1}{2400} = \frac{98}{2400} \approx 0.0408 \]

### Final Answer:
The dispersive power of the material of the lens is approximately \( D \approx 0.0408 \).