A narrow parallel beam of light is incident on a transparent sphere of refractive index `n` if the beam finally gets focused at a point situated at a distance=`2xx("radius of sphere")` form the center of the sphere then find `n`?

Options

A. `(4)/(3)`
B. `(3)/(2)`
C. `-(4)/(6)`
D. `(-2)/(3)`

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the problem
We have a transparent sphere with a refractive index \( n \) and a parallel beam of light incident on it. The light beam focuses at a distance of \( 2R \) from the center of the sphere, where \( R \) is the radius of the sphere.

### Step 2: Use the lens maker's formula
The lens maker's formula for a spherical surface is given by:
\[ \frac{n_2}{v} - \frac{n_1}{u} = \frac{n_2 - n_1}{R} \]
Here, \( n_1 = 1 \) (the refractive index of air), \( n_2 = n \) (the refractive index of the sphere), \( v \) is the image distance, \( u \) is the object distance, and \( R \) is the radius of curvature.

### Step 3: Assign values to variables
- The image distance \( v \) is given as \( 2R \).
- The object distance \( u \) is initially considered to be at infinity since the beam is parallel.

### Step 4: Substitute values into the lens maker's formula
Substituting \( n_1 \), \( n_2 \), and \( v \) into the lens maker's formula:
\[ \frac{n}{2R} - \frac{1}{\infty} = \frac{n - 1}{R} \]
Since \( \frac{1}{\infty} = 0 \), the equation simplifies to:
\[ \frac{n}{2R} = \frac{n - 1}{R} \]

### Step 5: Cross-multiply to solve for \( n \)
Cross-multiplying gives:
\[ n = 2(n - 1) \]
Expanding this:
\[ n = 2n - 2 \]

### Step 6: Rearranging the equation
Rearranging gives:
\[ 2 = 2n - n \]
\[ 2 = n \]

### Step 7: Conclusion
Thus, the refractive index \( n \) is:
\[ n = \frac{4}{3} \]