The refractive index of a lens material is 1.5 and focal length f. Due to some chemical changes in the material, its refractive index has increased by 2%. The percentage change in its focal length is

Options

A. `+4.5%`
B. `-4.5%`
C. `+5.6%`
D. `-5.67%`

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the relationship between refractive index and focal length
The lens maker's formula relates the focal length (F) of a lens to its refractive index (μ) and the radii of curvature (r1 and r2) of the lens surfaces. The formula is given by:

\[ \frac{1}{F} = (\mu - 1) \left( \frac{1}{r_1} + \frac{1}{r_2} \right) \]

### Step 2: Define the initial conditions
Given:
- Initial refractive index, \( \mu = 1.5 \)
- Focal length, \( F \)

### Step 3: Calculate the new refractive index
The refractive index increases by 2%. Therefore, the new refractive index \( \mu' \) can be calculated as:

\[ \mu' = \mu + 0.02 \cdot \mu = 1.5 + 0.02 \cdot 1.5 = 1.5 \times 1.02 = 1.53 \]

### Step 4: Write the new focal length using the new refractive index
Using the lens maker's formula for the new focal length \( F' \):

\[ \frac{1}{F'} = (\mu' - 1) \left( \frac{1}{r_1} + \frac{1}{r_2} \right) \]

### Step 5: Set up the ratio of the old and new focal lengths
Dividing the original and new equations gives:

\[ \frac{F}{F'} = \frac{\mu' - 1}{\mu - 1} \]

Substituting the values of \( \mu \) and \( \mu' \):

\[ \frac{F}{F'} = \frac{1.53 - 1}{1.5 - 1} = \frac{0.53}{0.5} \]

### Step 6: Solve for the new focal length
Rearranging gives:

\[ F' = F \cdot \frac{0.5}{0.53} \]

### Step 7: Calculate the percentage change in focal length
The percentage change in focal length can be calculated using the formula:

\[ \text{Percentage change} = \frac{F - F'}{F} \times 100 \]

Substituting \( F' \):

\[ \text{Percentage change} = \frac{F - \left(F \cdot \frac{0.5}{0.53}\right)}{F} \times 100 \]

This simplifies to:

\[ \text{Percentage change} = \left(1 - \frac{0.5}{0.53}\right) \times 100 \]

Calculating the fraction:

\[ 1 - \frac{0.5}{0.53} = 1 - 0.943396 = 0.056604 \]

Thus,

\[ \text{Percentage change} \approx 0.056604 \times 100 \approx 5.66\% \]

Since the focal length decreases, we can express it as:

\[ \text{Percentage change} \approx -5.66\% \]

### Final Answer
The percentage change in the focal length is approximately **-5.66%**.

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