A convex lens and a concave lens are kept separated by a distance d. If they are placed in close contact, the focal length of the combination

Options

A. will become zero
B. will increase
C. will decrease
D. will remain constant

Explanation

To solve the problem of finding the focal length of the combination of a convex lens and a concave lens when they are placed in close contact, we can follow these steps:

### Step 1: Understand the Focal Lengths of Individual Lenses
Let:
- \( f_1 \) = focal length of the convex lens (positive value)
- \( f_2 \) = focal length of the concave lens (negative value)

### Step 2: Use the Lens Formula for Two Lenses in Contact
When two lenses are placed in contact, the formula for the combined focal length \( f_c \) of the system is given by:
\[ \frac{1}{f_c} = \frac{1}{f_1} + \frac{1}{f_2} \]

### Step 3: Substitute the Values
Substituting \( f_2 \) as a negative value (since it is a concave lens):
\[ \frac{1}{f_c} = \frac{1}{f_1} + \frac{1}{-f_2} \]
This simplifies to:
\[ \frac{1}{f_c} = \frac{1}{f_1} - \frac{1}{|f_2|} \]
where \( |f_2| \) is the absolute value of the focal length of the concave lens.

### Step 4: Rearranging the Equation
Rearranging gives:
\[ \frac{1}{f_c} = \frac{1}{f_1} - \frac{1}{|f_2|} \]

### Step 5: Finding the Combined Focal Length
To find \( f_c \), take the reciprocal:
\[ f_c = \frac{1}{\left(\frac{1}{f_1} - \frac{1}{|f_2|}\right)} \]

### Step 6: Analyze the Effect of Distance
If the distance \( d \) between the lenses is zero (i.e., they are in close contact), the focal length \( f_c \) will be determined solely by the individual focal lengths \( f_1 \) and \( f_2 \). If \( |f_2| \) is sufficiently large compared to \( f_1 \), the combined focal length \( f_c \) will be less than \( f_1 \).

### Conclusion
Thus, when the convex lens and concave lens are placed in close contact, the focal length of the combination will decrease compared to when they are separated by a distance \( d \).