An achromatic lens-doublet is formed by placing in contact a convex lens of focal length `20 cm` and a concave lens of focal length `30 cm.` The dispersive power of the material of the convex lens is `0.8.` Determine the dispersive power of the material of the concave lens. Calculate the focal length of the lens-doublet.

Explanation

To solve the problem, we will break it down into two parts: finding the dispersive power of the concave lens and calculating the focal length of the lens doublet.

### Part 1: Determine the Dispersive Power of the Concave Lens

1. **Identify the given values:**
- Focal length of the convex lens, \( f_1 = 20 \, \text{cm} \)
- Focal length of the concave lens, \( f_2 = -30 \, \text{cm} \) (negative because it is a concave lens)
- Dispersive power of the convex lens, \( \omega_1 = 0.8 \)

2. **Use the formula for dispersive power:**
The relationship between the dispersive powers of the two lenses can be expressed as:
\[ \omega_2 = -\frac{f_2}{f_1} \cdot \omega_1 \]
where \( \omega_2 \) is the dispersive power of the concave lens.

3. **Substitute the known values into the formula:**
\[ \omega_2 = -\frac{-30}{20} \cdot 0.8 \]

4. **Calculate the value:**
\[ \omega_2 = \frac{30}{20} \cdot 0.8 = 1.5 \cdot 0.8 = 1.2 \]

5. **Conclusion for Part 1:**
The dispersive power of the concave lens is \( \omega_2 = 1.2 \).

### Part 2: Calculate the Focal Length of the Lens Doublet

1. **Use the formula for the focal length of the lens doublet:**
The focal length \( F \) of the lens doublet is given by:
\[ \frac{1}{F} = \frac{1}{f_1} + \frac{1}{f_2} \]

2. **Substitute the values of \( f_1 \) and \( f_2 \):**
\[ \frac{1}{F} = \frac{1}{20} + \frac{1}{-30} \]

3. **Calculate the individual fractions:**
\[ \frac{1}{F} = \frac{1}{20} - \frac{1}{30} \]

4. **Find a common denominator (which is 60):**
\[ \frac{1}{F} = \frac{3}{60} - \frac{2}{60} = \frac{1}{60} \]

5. **Invert to find \( F \):**
\[ F = 60 \, \text{cm} \]

6. **Conclusion for Part 2:**
The focal length of the lens doublet is \( F = 60 \, \text{cm} \).

### Final Answers:
- Dispersive power of the concave lens: \( 1.2 \)
- Focal length of the lens doublet: \( 60 \, \text{cm} \)