A convex lens of focal length 100 cm and a concave lens of focal length 10 cm are placed coaxialiy at a separation of 90 cm. If a parallel beam of light is incident on convex lens, then after passing through the two lenses the beam

Options

A. Converges
B. Diverges
C. Remains parallel
D. Disappears

Answer

To solve the problem of a parallel beam of light passing through a convex lens and a concave lens placed coaxially at a separation of 90 cm, we can follow these steps:

### Step 1: Identify the Focal Lengths
- The focal length of the convex lens (f1) is +100 cm (positive because it is a converging lens).
- The focal length of the concave lens (f2) is -10 cm (negative because it is a diverging lens).

### Step 2: Use the Lens Formula
The formula for the combined focal length (F) of two lenses in contact is given by:
\[ \frac{1}{F} = \frac{1}{f_1} + \frac{1}{f_2} \]
However, since the lenses are separated by a distance (d), we need to modify the formula to account for this separation:
\[ \frac{1}{F} = \frac{1}{f_1} + \frac{1}{f_2} - \frac{d}{f_1 \cdot f_2} \]
Where:
- \(d = 90 \, \text{cm}\)

### Step 3: Substitute the Values
Substituting the values into the modified lens formula:
\[ \frac{1}{F} = \frac{1}{100} + \frac{1}{-10} - \frac{90}{100 \cdot (-10)} \]

### Step 4: Calculate Each Term
- First term: \(\frac{1}{100} = 0.01\)
- Second term: \(\frac{1}{-10} = -0.1\)
- Third term: \(-\frac{90}{100 \cdot (-10)} = \frac{90}{1000} = 0.09\)

### Step 5: Combine the Terms
Now, combine the terms:
\[ \frac{1}{F} = 0.01 - 0.1 + 0.09 \]
Calculating this gives:
\[ \frac{1}{F} = 0.01 - 0.1 + 0.09 = 0 \]

### Step 6: Calculate Focal Length
If \(\frac{1}{F} = 0\), then:
\[ F = \infty \]
This means the effective focal length of the combination of the two lenses is infinite.

### Step 7: Conclusion
Since the effective focal length is infinite, a parallel beam of light incident on the convex lens will remain parallel after passing through both lenses.

### Final Answer
The beam will remain parallel after passing through the two lenses.

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