A luminous object is separated from a screen by distance d. A convex lends is placed between the object and the screeen such that it forms a distinct image on the screen. The maximum possible focal length of this convex lens is.

Options

A. 4d
B. 2d
C. `(d)/(2)`
D. `(d)/(4)`

Explanation

To solve the problem, we need to determine the maximum possible focal length of a convex lens placed between a luminous object and a screen, separated by a distance \( d \).

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have a luminous object and a screen separated by a distance \( d \).
- A convex lens is placed between the object and the screen, forming a distinct image on the screen.

2. **Identifying Variables**:
- Let \( a \) be the distance between the object and the lens.
- Let \( b \) be the distance between the lens and the screen.
- According to the problem, we know that \( a + b = d \).

3. **Using the Lens Formula**:
- The lens formula relates the object distance \( u \), image distance \( v \), and focal length \( f \) of the lens:
\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]
- Here, \( u = -a \) (object distance is taken as negative in lens formula) and \( v = b \).

4. **Substituting Values**:
- Substitute \( u \) and \( v \) into the lens formula:
\[ \frac{1}{f} = \frac{1}{b} + \frac{1}{a} \]

5. **Expressing Distances**:
- From \( a + b = d \), we can express \( b \) as \( b = d - a \).
- Substitute \( b \) in the lens formula:
\[ \frac{1}{f} = \frac{1}{(d - a)} + \frac{1}{a} \]

6. **Finding Maximum Focal Length**:
- To find the maximum focal length, we need to consider the conditions for distinct image formation.
- The maximum focal length occurs when the lens is positioned such that the object and image distances are equal, which means \( a = b \).
- Therefore, \( a = b = \frac{d}{2} \).

7. **Substituting Back**:
- Now substituting \( a = \frac{d}{2} \) into the lens formula:
\[ \frac{1}{f} = \frac{1}{\frac{d}{2}} + \frac{1}{\frac{d}{2}} = \frac{2}{\frac{d}{2}} = \frac{4}{d} \]
- Therefore, the focal length \( f \) becomes:
\[ f = \frac{d}{4} \]

### Final Answer:
The maximum possible focal length of the convex lens is \( \frac{d}{4} \).

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