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 find the focal length of a plano-concave lens when its plane surface is polished, we can follow these steps:

### Step 1: Understand the Given Information
We are given that the focal length of the plano-concave lens is \( f = -10 \, \text{cm} \). The negative sign indicates that it is a diverging lens.

### Step 2: Use the Lensmaker's Formula
The Lensmaker's formula for a lens is given by:
\[ \frac{1}{f} = (\mu - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \]
Where:
- \( f \) is the focal length of the lens,
- \( \mu \) 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 3: Assign Values for the Plano-Concave Lens
For a plano-concave lens:
- The radius of curvature of the concave surface \( R_1 = -20 \, \text{cm} \) (since it is concave, it is negative),
- The radius of curvature of the plane surface \( R_2 = \infty \) (since it is flat).

### Step 4: Substitute Values into the Formula
Substituting the values into the Lensmaker's formula:
\[ \frac{1}{-10} = (\mu - 1) \left( \frac{1}{-20} - 0 \right) \]
This simplifies to:
\[ \frac{1}{-10} = (\mu - 1) \left( -\frac{1}{20} \right) \]

### Step 5: Solve for the Refractive Index \( \mu \)
Rearranging gives:
\[ \mu - 1 = \frac{1/10}{1/20} = 2 \]
Thus,
\[ \mu = 3 \]

### Step 6: Calculate the Focal Length After Polishing
When the plane surface is polished, the lens becomes a biconvex lens with both surfaces contributing to the focal length. The new configuration will have:
- \( R_1 = -20 \, \text{cm} \) (concave surface),
- \( R_2 = 20 \, \text{cm} \) (convex surface).

Now, we will use the Lensmaker's formula again:
\[ \frac{1}{f'} = (\mu - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \]
Substituting the values:
\[ \frac{1}{f'} = (3 - 1) \left( \frac{1}{-20} - \frac{1}{20} \right) \]
This simplifies to:
\[ \frac{1}{f'} = 2 \left( -\frac{1}{20} - \frac{1}{20} \right) = 2 \left( -\frac{2}{20} \right) = -\frac{4}{20} = -\frac{1}{5} \]

### Step 7: Solve for \( f' \)
Thus:
\[ f' = -5 \, \text{cm} \]
The focal length of the polished plano-concave lens is \( 5 \, \text{cm} \) (positive, indicating it is now a converging lens).

### Final Answer
The focal length when the plane surface is polished is \( 5 \, \text{cm} \).

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