Two thin lenses of powers `2D` and `3D` are placed in contact. An object is placed at a distance of 30 cm from the combination The distance in cm of the image from the combination is

Options

A. 30
B. 40
C. 50
D. 60

Explanation

To solve the problem step by step, we will follow the concepts of lens power and the lens formula.

### Step 1: Understanding the Powers of the Lenses
We have two thin lenses with powers:
- Power of lens 1, \( P_1 = 2D \)
- Power of lens 2, \( P_2 = 3D \)

### Step 2: Finding the Total Power of the Combination
When two lenses are placed in contact, the total power \( P \) of the combination is the sum of the individual powers:
\[ P = P_1 + P_2 = 2D + 3D = 5D \]

### Step 3: Finding the Focal Length of the Combination
The focal length \( f \) of a lens is related to its power \( P \) by the formula:
\[ f = \frac{100}{P} \quad \text{(in cm)} \]
Substituting the total power:
\[ f = \frac{100}{5} = 20 \text{ cm} \]

### Step 4: Applying the Lens Formula
The lens formula is given by:
\[ \frac{1}{v} - \frac{1}{u} = \frac{1}{f} \]
Where:
- \( v \) is the image distance from the lens,
- \( u \) is the object distance from the lens (taken as negative in lens convention).

Given that the object distance \( u = -30 \) cm (since the object is placed in front of the lens), we can substitute the values into the lens formula:
\[ \frac{1}{v} - \frac{1}{-30} = \frac{1}{20} \]

### Step 5: Rearranging the Lens Formula
Rearranging the equation gives:
\[ \frac{1}{v} + \frac{1}{30} = \frac{1}{20} \]

### Step 6: Finding a Common Denominator
To combine the fractions, we find a common denominator:
\[ \frac{1}{v} = \frac{1}{20} - \frac{1}{30} \]
The common denominator of 20 and 30 is 60, so we rewrite the fractions:
\[ \frac{1}{20} = \frac{3}{60}, \quad \frac{1}{30} = \frac{2}{60} \]
Thus:
\[ \frac{1}{v} = \frac{3}{60} - \frac{2}{60} = \frac{1}{60} \]

### Step 7: Solving for the Image Distance \( v \)
Taking the reciprocal gives:
\[ v = 60 \text{ cm} \]

### Conclusion
The distance of the image from the combination of the lenses is **60 cm**.

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