A convex lens forms a real image on a screen placed at a distance 60 cm from the object. When the lens is shifted towards the screen by 20 cm , another image of the object is formed on the screen. The focal length of the lens is :

Options

A. `45 cm `
B. ` 40 // 3 cm `
C. `30 cm `
D. `12 cm `

Answer

To solve the problem, we need to find the focal length of a convex lens given the distances involved when the lens is shifted. Let's break it down step by step.

### Step 1: Understand the Problem
We have a convex lens forming a real image on a screen. The distance from the object to the screen is given as 60 cm. When the lens is shifted towards the screen by 20 cm, another image is formed on the screen.

### Step 2: Define Variables
Let:
- \( A \) = distance from the object to the lens in the first position
- \( B \) = distance from the lens to the image in the first position
- According to the problem, we have:
\[ A + B = 60 \, \text{cm} \quad \text{(1)} \]

### Step 3: Analyze the Shift
When the lens is shifted towards the screen by 20 cm, the new distances become:
- New object distance = \( A - 20 \) cm
- New image distance = \( B - 20 \) cm
- The new image is still formed on the screen, so:
\[ (A - 20) + (B - 20) = 60 \quad \text{(2)} \]

### Step 4: Simplify the Equations
From equation (1):
\[ A + B = 60 \]
From equation (2):
\[ A + B - 40 = 60 \implies A + B = 100 \]
This indicates that we need to account for the shift correctly. The correct interpretation is:
\[ B - A = 20 \quad \text{(3)} \]

### Step 5: Solve the System of Equations
Now we have two equations:
1. \( A + B = 60 \)
2. \( B - A = 20 \)

We can solve these equations simultaneously.

From equation (3):
\[ B = A + 20 \]

Substituting \( B \) in equation (1):
\[ A + (A + 20) = 60 \]
\[ 2A + 20 = 60 \]
\[ 2A = 40 \implies A = 20 \, \text{cm} \]

Now substituting \( A \) back into equation (1):
\[ 20 + B = 60 \implies B = 40 \, \text{cm} \]

### Step 6: Calculate the Focal Length
Now we have:
- \( A = 20 \, \text{cm} \)
- \( B = 40 \, \text{cm} \)

Using the lens formula:
\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]
Where \( u = -A \) and \( v = B \):
\[ \frac{1}{f} = \frac{1}{40} - \frac{1}{-20} \]
\[ \frac{1}{f} = \frac{1}{40} + \frac{1}{20} \]
Finding a common denominator (which is 40):
\[ \frac{1}{f} = \frac{1}{40} + \frac{2}{40} = \frac{3}{40} \]
Thus,
\[ f = \frac{40}{3} \, \text{cm} \]

### Final Answer
The focal length of the lens is:
\[ f = \frac{40}{3} \, \text{cm} \approx 13.33 \, \text{cm} \]