A parallel beam of light enters a clear plastic bead `2.50 cm` in diameter and index `1.44.` At what point beyond the bead are these rays brought to a focus?

Explanation

To solve the problem of finding the point beyond the bead where the rays of light are brought to a focus, we can use the lens maker's formula for refraction at a spherical surface. Here’s a step-by-step solution:

### Step 1: Identify Given Values
- Diameter of the bead (d) = 2.50 cm
- Radius of the bead (R) = d/2 = 2.50 cm / 2 = 1.25 cm
- Refractive index of the bead (μ2) = 1.44
- Refractive index of air (μ1) = 1.00
- Object distance (u) = ∞ (since the light rays are parallel)

### Step 2: Use the Refraction Formula
The refraction formula for a spherical surface is given by:

\[ \frac{\mu_2}{V} - \frac{\mu_1}{u} = \frac{\mu_2 - \mu_1}{R} \]

### Step 3: Substitute the Known Values
Substituting the known values into the formula:

\[ \frac{1.44}{V} - \frac{1.00}{\infty} = \frac{1.44 - 1.00}{1.25} \]

Since \( \frac{1.00}{\infty} = 0 \), we can simplify this to:

\[ \frac{1.44}{V} = \frac{0.44}{1.25} \]

### Step 4: Solve for V
Now, we can rearrange the equation to solve for \( V \):

\[ V = \frac{1.44 \times 1.25}{0.44} \]

Calculating the right-hand side:

\[ V = \frac{1.80}{0.44} \approx 4.09 \text{ cm} \]

### Final Answer
The rays are brought to a focus approximately **4.09 cm** beyond the bead.

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