A concavo-convex lens has refractive index `1.5` and the radii of curvature of its surfaces are `10 cm and 20 cm.` The concave surface is upwords and is filled with oil of refractive index `1.6.` The focal length of the combination will be

Options

A. `18.18 cm`
B. `15 cm`
C. `22 cm`
D. `28.57 cm`

Explanation

To solve the problem of finding the focal length of the concavo-convex lens filled with oil, we can follow these steps:

### Step 1: Identify the Given Values
- Refractive index of the lens (μ1) = 1.5
- Refractive index of the oil (μ2) = 1.6
- Radius of curvature of the first surface (R1) = +10 cm (convex surface)
- Radius of curvature of the second surface (R2) = -20 cm (concave surface)

### Step 2: Apply the Lensmaker's Formula
The Lensmaker's formula is given by:
\[ \frac{1}{f} = (\mu - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \]
We will apply this formula twice: once for the oil-filled surface and once for the lens material.

### Step 3: Calculate the Focal Length for the Oil Surface
For the oil surface:
\[ \frac{1}{f_1} = (1.6 - 1) \left( \frac{1}{\infty} - \frac{1}{-10} \right) \]
Since the radius of curvature for the flat surface is infinite, we can simplify:
\[ \frac{1}{f_1} = 0.6 \left( 0 + \frac{1}{10} \right) = 0.6 \cdot \frac{1}{10} = \frac{0.6}{10} = 0.06 \]

### Step 4: Calculate the Focal Length for the Lens Material
For the lens material:
\[ \frac{1}{f_2} = (1.5 - 1) \left( \frac{1}{10} - \frac{1}{-20} \right) \]
Calculating the terms:
\[ \frac{1}{f_2} = 0.5 \left( \frac{1}{10} + \frac{1}{20} \right) \]
Finding a common denominator (20):
\[ \frac{1}{f_2} = 0.5 \left( \frac{2}{20} + \frac{1}{20} \right) = 0.5 \cdot \frac{3}{20} = \frac{1.5}{20} = 0.075 \]

### Step 5: Combine the Focal Lengths
Now, we can combine the focal lengths using the formula:
\[ \frac{1}{f} = \frac{1}{f_1} + \frac{1}{f_2} \]
Substituting the values:
\[ \frac{1}{f} = 0.06 + 0.075 = 0.135 \]

### Step 6: Calculate the Focal Length
Now, we find the focal length \( f \):
\[ f = \frac{1}{0.135} \approx 7.41 \text{ cm} \]

### Final Answer
The focal length of the combination is approximately **7.41 cm**.

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