The dispersive power of the material of a lens is `0.04` and the focal length of the lens is `10cm`. Find the difference in the focal length (in mm) of the lens for violet and red colour.

Explanation

To solve the problem, we need to find the difference in the focal lengths of the lens for violet and red colors based on the given dispersive power and the focal length of the lens.

### Step-by-Step Solution:

1. **Understanding Dispersive Power**:
The dispersive power (ω) is given by the formula:
\[ \omega = \frac{\mu_v - \mu_r}{\mu_{mean} - 1} \]
where:
- \(\mu_v\) is the refractive index for violet light,
- \(\mu_r\) is the refractive index for red light,
- \(\mu_{mean}\) is the mean refractive index.

Given that the dispersive power is \(0.04\), we can use this value in our calculations.

2. **Using the Lens Maker's Formula**:
The lens maker's formula relates the focal length \(f\) of a lens to its refractive index and radii of curvature:
\[ \frac{1}{f} = (\mu - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \]
For small changes in refractive index, we can differentiate this equation.

3. **Differentiating the Lens Maker's Formula**:
We differentiate the lens maker's formula to find the change in focal length \(df\) with respect to the change in refractive index \(d\mu\):
\[ df = \frac{d\mu}{(\mu - 1) \cdot \frac{1}{f^2}} \cdot \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \]
However, since we are interested in the difference in focal lengths for violet and red light, we will focus on the changes in refractive index.

4. **Calculating the Change in Focal Length**:
We know that:
\[ df = -10 \cdot \frac{(\mu_v - \mu_r)}{(\mu - 1)} \]
Substituting the dispersive power:
\[ df = -10 \cdot \frac{0.04}{1} = -0.4 \text{ cm} \]

5. **Converting to mm**:
Since we need the answer in mm, we convert:
\[ df = -0.4 \text{ cm} \times 10 = -4 \text{ mm} \]
The negative sign indicates a difference, but we are interested in the magnitude:
\[ \text{Difference in focal length} = 4 \text{ mm} \]

### Final Answer:
The difference in the focal length of the lens for violet and red color is **4 mm**.