A narrow pencil of parallel light is incident normally on a solid transparent sphere of radius r. What should be the refractive index if the pencil is to be focussed (a) at the surface of the sphere, (b) at the centre of the sphere.

Explanation

To solve the problem, we need to determine the refractive index of a solid transparent sphere when a narrow pencil of parallel light is incident normally on its surface. We will consider two cases: (a) when the light is focused at the surface of the sphere and (b) when the light is focused at the center of the sphere.

### Given:
- Radius of the sphere, \( r \)
- Refractive index of air, \( \mu_1 = 1 \)
- Object distance, \( u = -\infty \) (since the light rays are coming from infinity)

### Case (a): Focusing at the Surface of the Sphere
1. **Identify the Image Distance**:
- Since the light is focused at the surface of the sphere, the image distance \( v = 2r \) (the distance from the center of the sphere to the surface is \( r \), and the light converges to the surface).

2. **Apply the Lens Maker's Formula**:
The formula for refraction at a curved surface is given by:
\[ \frac{\mu_2}{v} - \frac{\mu_1}{u} = \frac{\mu_2 - \mu_1}{R} \]
Here, \( R = r \) (the radius of the sphere).

3. **Substituting Values**:
Substitute \( v = 2r \), \( u = -\infty \), and \( R = r \):
\[ \frac{\mu_2}{2r} - \frac{1}{-\infty} = \frac{\mu_2 - 1}{r} \]
The term \( \frac{1}{-\infty} \) becomes \( 0 \), so we have:
\[ \frac{\mu_2}{2r} = \frac{\mu_2 - 1}{r} \]

4. **Simplifying the Equation**:
Multiply through by \( 2r \):
\[ \mu_2 = 2(\mu_2 - 1) \]
Expanding gives:
\[ \mu_2 = 2\mu_2 - 2 \]
Rearranging leads to:
\[ \mu_2 - 2\mu_2 = -2 \implies -\mu_2 = -2 \implies \mu_2 = 2 \]

### Conclusion for Case (a):
The refractive index \( \mu_2 \) for the light to be focused at the surface of the sphere is \( 2 \).

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### Case (b): Focusing at the Center of the Sphere
1. **Identify the Image Distance**:
- For this case, the image is formed at the center of the sphere, so the image distance \( v = r \).

2. **Apply the Lens Maker's Formula Again**:
Using the same formula:
\[ \frac{\mu_2}{v} - \frac{\mu_1}{u} = \frac{\mu_2 - \mu_1}{R} \]
Substitute \( v = r \), \( u = -\infty \), and \( R = r \):
\[ \frac{\mu_2}{r} - \frac{1}{-\infty} = \frac{\mu_2 - 1}{r} \]
Again, the term \( \frac{1}{-\infty} \) becomes \( 0 \):
\[ \frac{\mu_2}{r} = \frac{\mu_2 - 1}{r} \]

3. **Simplifying the Equation**:
Multiply through by \( r \):
\[ \mu_2 = \mu_2 - 1 \]
Rearranging gives:
\[ 0 = -1 \]
This is not possible.

### Conclusion for Case (b):
The equation cannot be satisfied, indicating that it is not possible for the light to be focused at the center of the sphere.

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### Summary of Results:
- (a) The refractive index \( \mu_2 \) for focusing at the surface of the sphere is \( 2 \).
- (b) It is not possible to focus the light at the center of the sphere.