One end of a long glass rod `(mu=1.5)` is formed into a convex surface of radius `6.0 cm.` An object is positioned in air along the axis of the rod. Find the image positions corresponding to object distances of (a) `20.0 cm,` (b) `10.0 cm,` (c) `3.0 cm` from the end of the rod.

Explanation

To solve the problem, we will use the refraction formula for a spherical surface, which is given by:

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

Where:
- \( \mu_1 \) is the refractive index of air (approximately 1.0),
- \( \mu_2 \) is the refractive index of glass (1.5),
- \( U \) is the object distance (taken as negative in this case),
- \( V \) is the image distance,
- \( R \) is the radius of curvature (positive for a convex surface).

Given:
- \( R = 6.0 \, \text{cm} \)
- \( \mu_1 = 1.0 \)
- \( \mu_2 = 1.5 \)

### Step-by-Step Solution

1. **Set Up the Refraction Equation**:
\[ \frac{1.5}{V} - \frac{1.0}{-D} = \frac{1.5 - 1.0}{6} \]
Simplifying this gives:
\[ \frac{1.5}{V} + \frac{1.0}{D} = \frac{0.5}{6} \]
Which can be rewritten as:
\[ \frac{1.5}{V} + \frac{1.0}{D} = \frac{1}{12} \]

2. **Rearranging the Equation**:
\[ \frac{1.5}{V} = \frac{1}{12} - \frac{1.0}{D} \]
\[ V = \frac{1.5}{\frac{1}{12} - \frac{1.0}{D}} \]

3. **Calculate Image Distances for Each Object Distance**:

**(a) For \( D = 20.0 \, \text{cm} \)**:
\[ V_1 = \frac{1.5}{\frac{1}{12} - \frac{1.0}{20}} = \frac{1.5}{\frac{5}{60} - \frac{3}{60}} = \frac{1.5}{\frac{2}{60}} = \frac{1.5 \times 60}{2} = 45 \, \text{cm} \]

**(b) For \( D = 10.0 \, \text{cm} \)**:
\[ V_2 = \frac{1.5}{\frac{1}{12} - \frac{1.0}{10}} = \frac{1.5}{\frac{5}{60} - \frac{6}{60}} = \frac{1.5}{-\frac{1}{60}} = -90 \, \text{cm} \]

**(c) For \( D = 3.0 \, \text{cm} \)**:
\[ V_3 = \frac{1.5}{\frac{1}{12} - \frac{1.0}{3}} = \frac{1.5}{\frac{5}{60} - \frac{20}{60}} = \frac{1.5}{-\frac{15}{60}} = -6 \, \text{cm} \]

### Final Results
- For \( D = 20.0 \, \text{cm} \), \( V_1 = 45 \, \text{cm} \)
- For \( D = 10.0 \, \text{cm} \), \( V_2 = -90 \, \text{cm} \)
- For \( D = 3.0 \, \text{cm} \), \( V_3 = -6 \, \text{cm} \)