A convex lens forms an image of a distant object at distance of 20 cm from it. On keeping another lens in contact with the first, if the image is formed at a distance of `40/3` cm from the combination, then the focal length of the second lens is

Options

A. `-20cm`
B. `-40cm`
C. `40cm`
D. `13.33cm`

Explanation

To find the focal length of the second lens in the given problem, we can follow these steps:

### Step 1: Understand the problem
We have a convex lens (Lens 1) that forms an image of a distant object at a distance of 20 cm from the lens. This means that the focal length (F1) of the first lens is 20 cm.

### Step 2: Identify the new image distance
When a second lens (Lens 2) is placed in contact with the first lens, the image is now formed at a distance of \( \frac{40}{3} \) cm from the combination of the two lenses. This distance is the effective focal length (Fc) of the combination of the two lenses.

### Step 3: Use the lens formula
The formula for the combination of two lenses in contact is given by:
\[ \frac{1}{F_c} = \frac{1}{F_1} + \frac{1}{F_2} \]
Where:
- \( F_c \) is the focal length of the combination,
- \( F_1 \) is the focal length of the first lens,
- \( F_2 \) is the focal length of the second lens.

### Step 4: Substitute known values
From the problem, we have:
- \( F_1 = 20 \) cm,
- \( F_c = \frac{40}{3} \) cm.

Now substituting these values into the lens formula:
\[ \frac{1}{\frac{40}{3}} = \frac{1}{20} + \frac{1}{F_2} \]

### Step 5: Simplify the equation
To simplify \( \frac{1}{\frac{40}{3}} \):
\[ \frac{1}{\frac{40}{3}} = \frac{3}{40} \]
So the equation becomes:
\[ \frac{3}{40} = \frac{1}{20} + \frac{1}{F_2} \]

### Step 6: Solve for \( \frac{1}{F_2} \)
Now, we need to express \( \frac{1}{20} \) in terms of a common denominator:
\[ \frac{1}{20} = \frac{2}{40} \]
Now substituting this back into the equation:
\[ \frac{3}{40} = \frac{2}{40} + \frac{1}{F_2} \]

### Step 7: Isolate \( \frac{1}{F_2} \)
Subtract \( \frac{2}{40} \) from both sides:
\[ \frac{1}{F_2} = \frac{3}{40} - \frac{2}{40} = \frac{1}{40} \]

### Step 8: Find \( F_2 \)
Taking the reciprocal gives us:
\[ F_2 = 40 \text{ cm} \]

### Conclusion
The focal length of the second lens is 40 cm.

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