There is a sphere of radius R and refractive index `mu`. At what distance from the surface of sphere should an object be placed so that real image is formed at the same distance from the sphere ?

Options

A. `(R)/(mu-1)`
B. `(R)/(mu)`
C. `(R)/(2-mu)`
D. `(2R)/(mu-1)`

Explanation

To solve the problem of finding the distance from the surface of a sphere at which an object should be placed to form a real image at the same distance from the sphere, we can follow these steps:

### Step 1: Understand the Problem
We have a sphere with a radius \( R \) and a refractive index \( \mu \). We need to find the object distance \( u \) such that the real image is formed at the same distance \( v \) from the sphere.

### Step 2: Set Up the Sign Convention
In optics, we typically use the following sign convention:
- Distances measured in the direction of the incident light (towards the surface) are considered negative.
- Distances measured in the direction of the outgoing light (away from the surface) are considered positive.

### Step 3: Apply the Lens Maker's Formula
For a spherical surface, the lens maker's formula can be expressed as:
\[ \frac{\mu_2}{v} - \frac{\mu_1}{u} = \frac{1}{R} \]
Where:
- \( \mu_2 \) is the refractive index of the medium where the image is formed (for air, \( \mu_2 = 1 \)).
- \( \mu_1 \) is the refractive index of the sphere (given as \( \mu \)).
- \( R \) is the radius of the sphere.

### Step 4: Substitute Values
Since we want the image to be formed at the same distance as the object, we can denote the distance from the surface of the sphere to the object as \( x \). Therefore, we can write:
- \( u = -x \) (since it's measured towards the sphere)
- \( v = x \) (since it's measured away from the sphere)

Substituting these into the lens maker's formula gives:
\[ \frac{1}{x} - \frac{\mu}{-x} = \frac{1}{R} \]

### Step 5: Simplify the Equation
This simplifies to:
\[ \frac{1}{x} + \frac{\mu}{x} = \frac{1}{R} \]
Combining the fractions on the left side:
\[ \frac{1 + \mu}{x} = \frac{1}{R} \]

### Step 6: Solve for \( x \)
Rearranging the equation to solve for \( x \):
\[ x = R(1 + \mu) \]

### Step 7: Find the Distance from the Surface of the Sphere
Since \( x \) is the distance from the center of the sphere, the distance from the surface of the sphere is:
\[ \text{Distance from surface} = x - R = R(1 + \mu) - R = R\mu \]

### Final Answer
Thus, the object should be placed at a distance of \( R\mu \) from the surface of the sphere.

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