A sphere of glass `(mu = 1.5)` is of `20 cm` diameter. A parallel beam enters it from one side. Where will it get focussed on the other side ?

Answer

To solve the problem of where a parallel beam of light will focus after passing through a glass sphere, we can follow these steps:

### Step 1: Identify the parameters
- The refractive index of the glass sphere, \( \mu = 1.5 \).
- The diameter of the sphere is \( 20 \, \text{cm} \), so the radius \( r = 10 \, \text{cm} \).

### Step 2: Understand the situation
A parallel beam of light enters the sphere, which means that the object distance \( u \) can be considered as approaching infinity (\( u = \infty \)) for the first surface of the sphere.

### Step 3: Apply the lens maker's formula
Using the lens maker's formula for refraction at a spherical surface, we have:

\[ \frac{\mu_2}{v} - \frac{\mu_1}{u} = \frac{\mu_2 - \mu_1}{r} \]

Where:
- \( \mu_1 = 1 \) (the refractive index of air),
- \( \mu_2 = 1.5 \) (the refractive index of glass),
- \( r = 10 \, \text{cm} \) (the radius of the sphere),
- \( u = \infty \).

### Step 4: Substitute the values into the formula
Since \( u = \infty \), \( \frac{\mu_1}{u} = 0 \). Thus, the equation simplifies to:

\[ \frac{1.5}{v} = \frac{1.5 - 1}{10} \]

This simplifies to:

\[ \frac{1.5}{v} = \frac{0.5}{10} \]

### Step 5: Solve for \( v \)
Cross-multiplying gives:

\[ 1.5 \cdot 10 = 0.5 \cdot v \]

\[ 15 = 0.5v \]

Dividing both sides by \( 0.5 \):

\[ v = \frac{15}{0.5} = 30 \, \text{cm} \]

### Step 6: Determine the object distance for the second surface
Now, we need to find where the light converges after passing through the second surface of the sphere. The light will now be treated as coming from a distance of \( 30 \, \text{cm} \) from the second surface, which means \( u = -10 \, \text{cm} \) (since we take the radius as positive).

### Step 7: Apply the lens maker's formula again for the second surface
Using the same formula:

\[ \frac{\mu_2}{v} - \frac{\mu_1}{u} = \frac{\mu_2 - \mu_1}{r} \]

Substituting the values:

\[ \frac{1}{v} - \frac{1.5}{-10} = \frac{1 - 1.5}{-10} \]

This simplifies to:

\[ \frac{1}{v} + \frac{1.5}{10} = \frac{-0.5}{-10} \]

\[ \frac{1}{v} + 0.15 = 0.05 \]

### Step 8: Solve for \( v \)
Rearranging gives:

\[ \frac{1}{v} = 0.05 - 0.15 = -0.10 \]

Thus:

\[ v = \frac{1}{-0.10} = -10 \, \text{cm} \]

### Final Result
The negative sign indicates that the focus is on the same side as the incoming light, at a distance of \( 5 \, \text{cm} \) from the second surface of the sphere.

### Summary
The parallel beam of light will focus at a distance of \( 5 \, \text{cm} \) from the second surface of the glass sphere.

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