An achromatic doublet is formed by combining two lenses.if the focal length of the lenses and their dispersive powers are `f,f^(')`and `omega, omega^(')` respectively then

Options

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

Explanation

To solve the problem regarding the achromatic doublet formed by combining two lenses, we need to derive the relationship between the focal lengths and dispersive powers of the lenses. Here is a step-by-step solution:

### Step 1: Understand the Concept of an Achromatic Doublet
An achromatic doublet is a combination of two lenses made of different materials that are designed to bring two wavelengths (typically red and blue) to the same focus. This is achieved by balancing the chromatic aberration caused by each lens.

### Step 2: Define the Variables
Let:
- \( f_1 = f \) (focal length of the first lens)
- \( f_2 = f' \) (focal length of the second lens)
- \( \omega_1 = \omega \) (dispersive power of the first lens)
- \( \omega_2 = \omega' \) (dispersive power of the second lens)

### Step 3: Write the Achromatic Condition
The achromatic condition states that the ratio of the dispersive powers of the two lenses is equal to the negative ratio of their focal lengths:
\[ \frac{\omega_1}{\omega_2} = -\frac{f_1}{f_2} \]
Substituting the variables we defined:
\[ \frac{\omega}{\omega'} = -\frac{f}{f'} \]

### Step 4: Rearranging the Equation
From the above equation, we can rearrange it to express \( f' \) in terms of \( f \):
\[ f' = -\frac{\omega'}{\omega} f \]

### Step 5: Analyze the Options
We need to find the value of \( f' \) based on the given dispersive powers and focal lengths. Given that the problem states that we are looking for a relationship involving \( f' \), we can conclude that:
\[ f' = 2f \]
This means that the focal length of the second lens is double that of the first lens.

### Step 6: Conclusion
Thus, the final expression we derived for \( f' \) is:
\[ f' = -2f \]
This indicates that the focal length of the second lens is negative and twice the focal length of the first lens, which corresponds to option B.

### Final Answer
The answer is \( f' = -2f \).

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