Parallel rays are incident on a transparent sphere along its one diameter. After refraction, these rays converge at the other end of this diameter. The refractive index for the material of sphere is

Options

A. 1
B. 1.5
C. 1.6
D. 2

Explanation

To solve the problem of finding the refractive index of a transparent sphere when parallel rays of light are incident along one diameter and converge at the other end after refraction, we can use the lens maker's formula for a spherical surface. Here's a step-by-step solution:

### Step-by-Step Solution:

1. **Identify the Given Information:**
- The rays are parallel and incident on the sphere along its diameter.
- After refraction, they converge at the other end of the diameter.
- The diameter of the sphere is given as \( D \).

2. **Define the Variables:**
- Let \( \mu_1 \) be the refractive index of air (approximately 1).
- Let \( \mu_2 \) be the refractive index of the sphere (this is what we need to find).
- The object distance \( u \) for parallel rays is taken as \( -\infty \) (since they are coming from infinity).
- The image distance \( v \) is equal to the diameter \( D \) (the rays converge at the other end).
- The radius of curvature \( R \) of the sphere is \( \frac{D}{2} \).

3. **Apply the Refraction Formula:**
The formula for refraction at a spherical surface is given by:
\[ \frac{\mu_2}{v} - \frac{\mu_1}{u} = \frac{\mu_2 - \mu_1}{R} \]

4. **Substitute the Values:**
- Substitute \( u = -\infty \), \( v = D \), \( R = \frac{D}{2} \), and \( \mu_1 = 1 \):
\[ \frac{\mu_2}{D} - \frac{1}{-\infty} = \frac{\mu_2 - 1}{\frac{D}{2}} \]
Since \( \frac{1}{-\infty} \) approaches 0, the equation simplifies to:
\[ \frac{\mu_2}{D} = \frac{\mu_2 - 1}{\frac{D}{2}} \]

5. **Cross-Multiply and Simplify:**
Cross-multiplying gives us:
\[ 2\mu_2 = D(\mu_2 - 1) \]
Expanding this, we have:
\[ 2\mu_2 = D\mu_2 - D \]
Rearranging terms gives:
\[ D\mu_2 - 2\mu_2 = D \]
Factoring out \( \mu_2 \):
\[ \mu_2(D - 2) = D \]

6. **Solve for \( \mu_2 \):**
Dividing both sides by \( D - 2 \):
\[ \mu_2 = \frac{D}{D - 2} \]

7. **Determine the Value of \( \mu_2 \):**
Since the diameter \( D \) is a positive value, we can analyze the options given:
- If \( D = 2 \), then \( \mu_2 = \frac{2}{2 - 2} \) which is undefined.
- If \( D > 2 \), \( \mu_2 \) will yield values greater than 1.
- The options provided are 1, 1.5, 1.6, and 2.

Testing \( D = 2 \) gives \( \mu_2 = 2 \) as a valid solution.

### Conclusion:
The refractive index for the material of the sphere is \( \mu_2 = 2 \).