Focal lengths of two lenses are f and f' and dispersive powers of their materials are `omega` and `omega'`. To form achromatic combination from these, which relation is correct ?

Options

A. `omega=omega_(0)`, `omega'=2omega_(0)`, `f'=2f`
B. `omega=omega_(0)`, `omega'=2omega_(0)`, `f'=f//2`
C. `omega=omega_(0)`, `omega'=2omega_(0)`, `f'=-2f`
D. `omega=omega_(0)`, `omega'=2omega_(0)`, `f'=-f//2`

Answer

To solve the problem of finding the correct relation for forming an achromatic combination of two lenses with focal lengths \( f \) and \( f' \), and dispersive powers \( \omega \) and \( \omega' \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Concept of Achromatic Combination**:
An achromatic combination of lenses is one that eliminates chromatic aberration. This is achieved when the combination of the two lenses focuses different colors of light at the same point.

2. **Use the Formula for Achromatic Combination**:
The condition for two lenses to form an achromatic combination can be expressed as:
\[ \frac{\omega}{f} + \frac{\omega'}{f'} = 0 \]
where \( \omega \) is the dispersive power of the first lens and \( \omega' \) is the dispersive power of the second lens.

3. **Rearranging the Equation**:
Rearranging the above equation gives:
\[ \omega f' + \omega' f = 0 \]

4. **Expressing One Dispersive Power in Terms of the Other**:
If we express \( \omega' \) in terms of \( \omega \), we can write:
\[ \omega' = k \omega \quad \text{(where \( k \) is a constant)} \]
Substituting this into our equation gives:
\[ \omega f' + k \omega f = 0 \]
Dividing through by \( \omega \) (assuming \( \omega \neq 0 \)):
\[ f' + k f = 0 \]
Thus:
\[ f' = -k f \]

5. **Identifying the Correct Relation**:
If we take \( k = 2 \) (which is a common value used in optics), we find:
\[ f' = -2f \]

### Conclusion:
The correct relation for forming an achromatic combination from the two lenses is:
\[ f' = -2f \]