A thin converging lens of refractive index 1.5 has a focal power of 5 D. When this lens is immersed in a liquid, it acts as a diverging lens of focal length 100 cm. The refractive index of the liquid is

Options

A. `11/6`
B. `9/5`
C. ``5/3
D. 2

Explanation

To solve the problem, we need to find the refractive index of the liquid in which a thin converging lens is immersed. Here are the steps to derive the solution:

### Step 1: Understand the Given Data
- The refractive index of the lens (glass) \( \mu_g = 1.5 \).
- The focal power of the lens in air is \( P = 5 \, D \) (diopters).
- The focal length of the lens in air can be calculated using the formula:
\[ P = \frac{1}{f} \]
Therefore, the focal length \( f \) in air is:
\[ f = \frac{1}{P} = \frac{1}{5} \, m = 0.2 \, m = 20 \, cm \]

### Step 2: Apply the Lensmaker's Formula in Air
The lensmaker's formula for a thin lens is given by:
\[ P = \left( \frac{\mu_g}{\mu_a} - 1 \right) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \]
Where:
- \( \mu_a = 1 \) (refractive index of air)
- \( R_1 \) and \( R_2 \) are the radii of curvature of the lens surfaces.

Substituting the values we have:
\[ 5 = \left( \frac{1.5}{1} - 1 \right) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \]
This simplifies to:
\[ 5 = (0.5) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \]
Thus,
\[ \frac{1}{R_1} - \frac{1}{R_2} = 10 \]
Let’s denote this equation as (1).

### Step 3: Apply the Lensmaker's Formula in the Liquid
When the lens is immersed in a liquid, it behaves as a diverging lens with a focal length of \( -100 \, cm \) (or \( -1 \, m \)):
\[ P' = \frac{1}{f'} = -1 \, D \]
Using the lensmaker's formula again:
\[ -1 = \left( \frac{\mu_g}{\mu} - 1 \right) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \]
Substituting the known values:
\[ -1 = \left( \frac{1.5}{\mu} - 1 \right)(10) \]
This simplifies to:
\[ -1 = 10 \left( \frac{1.5}{\mu} - 1 \right) \]
Thus,
\[ -1 = \frac{15}{\mu} - 10 \]
Rearranging gives:
\[ \frac{15}{\mu} = 9 \implies \mu = \frac{15}{9} = \frac{5}{3} \]

### Step 4: Conclusion
The refractive index of the liquid is:
\[ \mu = \frac{5}{3} \]

### Final Answer
The refractive index of the liquid is \( \frac{5}{3} \).

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