A candle flame 1.6 cm high is imaged in a ball bearing of diameter 0.4 cm. If the ball bearing is 20 cm away from the flame, find the location and the height of the image.

Explanation

To solve the problem step by step, we will follow these steps:

### Step 1: Identify the given data
- Height of the candle flame (object height, \( h_o \)) = 1.6 cm
- Diameter of the ball bearing = 0.4 cm
- Radius of the ball bearing (\( r \)) = Diameter / 2 = 0.4 cm / 2 = 0.2 cm
- Object distance (\( u \)) = -20 cm (negative as per sign convention for mirrors)

### Step 2: Calculate the focal length of the convex mirror
The focal length (\( f \)) of a convex mirror is given by:
\[ f = \frac{r}{2} \]
Substituting the value of \( r \):
\[ f = \frac{0.2 \, \text{cm}}{2} = 0.1 \, \text{cm} \]

### Step 3: Use the mirror formula to find the image distance (\( v \))
The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Substituting the known values:
\[ \frac{1}{0.1} = \frac{1}{v} + \frac{1}{-20} \]
Rearranging gives:
\[ \frac{1}{v} = \frac{1}{0.1} + \frac{1}{20} \]
Calculating the right-hand side:
\[ \frac{1}{v} = 10 + 0.05 = 10.05 \]
Thus,
\[ v = \frac{1}{10.05} \approx 0.0995 \, \text{cm} \]

### Step 4: Determine the height of the image (\( h_i \))
The magnification (\( m \)) for mirrors is given by:
\[ m = \frac{h_i}{h_o} = -\frac{v}{u} \]
Substituting the known values:
\[ m = -\frac{0.0995}{-20} = \frac{0.0995}{20} \]
Now, substituting for \( h_i \):
\[ h_i = m \cdot h_o = \left(\frac{0.0995}{20}\right) \cdot 1.6 \]
Calculating \( h_i \):
\[ h_i = \frac{0.0995 \cdot 1.6}{20} \approx 0.00796 \, \text{cm} \]

### Step 5: Summarize the results
- The image distance \( v \approx 0.0995 \, \text{cm} \) (positive indicates that the image is formed on the same side as the object).
- The height of the image \( h_i \approx 0.00796 \, \text{cm} \).

### Final Answers
- Location of the image: \( \approx 0.0995 \, \text{cm} \) from the ball bearing.
- Height of the image: \( \approx 0.00796 \, \text{cm} \).