The focal length of a plano-concave lens is -10 cm , then its focal length when its palne surface is polished is

Options

A. 20 cm
B. `-5 cm `
C. 5 cm
D. `-20 cm `

Answer

To solve the problem, we need to determine the focal length of a plano-concave lens when its plane surface is polished. Given that the original focal length of the plano-concave lens is -10 cm, we will follow these steps:

### Step 1: Understand the Original Lens
The plano-concave lens has one flat surface (plane) and one concave surface. The focal length of this lens is given as -10 cm, indicating that it is a diverging lens.

### Step 2: Effect of Polishing the Plane Surface
When the plane surface of the lens is polished, it effectively transforms the lens into a plano-convex lens. In this case, the polished plane surface acts like a mirror, reflecting light rather than allowing it to pass through.

### Step 3: Analyze the New Configuration
After polishing, the lens behaves as a plano-convex lens where both surfaces are now curved. The original concave surface remains unchanged, while the plane surface is now reflective.

### Step 4: Determine the New Focal Length
For a plano-convex lens, the focal length can be calculated using the lens maker's formula. The focal length \( f \) of a lens is given by:

\[ \frac{1}{f} = (n - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \]

Where:
- \( n \) is the refractive index of the lens material,
- \( R_1 \) is the radius of curvature of the first surface,
- \( R_2 \) is the radius of curvature of the second surface.

### Step 5: Assign Values
- For the plano-convex lens, \( R_1 \) (the radius of the convex surface) is positive, and \( R_2 \) (the radius of the plane surface) is infinite (since it is flat).
- The radius of curvature for the concave surface before polishing was effectively -20 cm (since the focal length of the original lens was -10 cm, the radius of curvature is double that value).

### Step 6: Calculate the New Focal Length
Using the lens maker's formula, we can substitute the values:

\[ \frac{1}{f} = (n - 1) \left( \frac{1}{20} - 0 \right) \]

Assuming \( n \) is 1.5 (typical for glass):

\[ \frac{1}{f} = (1.5 - 1) \left( \frac{1}{20} \right) = 0.5 \cdot \frac{1}{20} = \frac{1}{40} \]

Thus, the new focal length \( f \) is:

\[ f = 40 \text{ cm} \]

### Step 7: Conclusion
Since the lens has been converted to a plano-convex lens, the focal length is positive. Therefore, the focal length when the plane surface is polished is **20 cm**.

### Final Answer
The focal length when the plane surface is polished is **20 cm**.

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