A concavo-convex lens is made of glass of refractive index `1.5.` The radii of curvature of its two surfaces are `30 cm` and `50 cm.` Its focal length when placed in a liquid of refractive index `1.4` is

Options

A. `200 cm`
B. `500 cm`
C. `800 cm`
D. `1050 cm`

Explanation

To find the focal length of a concavo-convex lens made of glass with a refractive index of 1.5 when placed in a liquid with a refractive index of 1.4, we can use the lens maker's formula. The formula for the focal length \( f \) of a lens in a medium is given by:

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

Where:
- \( \mu_{\text{relative}} = \frac{\mu_{\text{lens}}}{\mu_{\text{liquid}}} \)
- \( R_1 \) is the radius of curvature of the first surface (convex surface, positive)
- \( R_2 \) is the radius of curvature of the second surface (concave surface, negative)

### Step-by-step solution:

1. **Identify the given values:**
- Refractive index of the lens \( \mu_{\text{lens}} = 1.5 \)
- Refractive index of the liquid \( \mu_{\text{liquid}} = 1.4 \)
- Radius of curvature of the first surface \( R_1 = 30 \, \text{cm} \)
- Radius of curvature of the second surface \( R_2 = -50 \, \text{cm} \) (negative because it is concave)

2. **Calculate the relative refractive index:**
\[ \mu_{\text{relative}} = \frac{\mu_{\text{lens}}}{\mu_{\text{liquid}}} = \frac{1.5}{1.4} = \frac{15}{14} \]

3. **Apply the lens maker's formula:**
\[ \frac{1}{f} = \left( \frac{15}{14} - 1 \right) \left( \frac{1}{30} - \frac{1}{-50} \right) \]

4. **Simplify the equation:**
\[ \frac{1}{f} = \left( \frac{15}{14} - \frac{14}{14} \right) \left( \frac{1}{30} + \frac{1}{50} \right) \]
\[ = \left( \frac{1}{14} \right) \left( \frac{5 + 3}{150} \right) = \left( \frac{1}{14} \right) \left( \frac{8}{150} \right) \]

5. **Calculate \( \frac{1}{f} \):**
\[ \frac{1}{f} = \frac{8}{2100} = \frac{4}{1050} \]

6. **Find the focal length \( f \):**
\[ f = \frac{1050}{4} = 262.5 \, \text{cm} \]

### Final Answer:
The focal length of the concavo-convex lens when placed in a liquid of refractive index 1.4 is \( 262.5 \, \text{cm} \).

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