A rod of length 10 cm lies along the principal axis of a concave mirror of focal length 10 cm in such a way that the end closer to the pole is 20 cm away from it. Find the length of the image.

Options

A. `10 cm`
B. `15 cm`
C. `2.5 cm`
D. `5 cm`

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the setup
We have a concave mirror with a focal length (F) of -10 cm (the negative sign indicates that it is a concave mirror). A rod of length 10 cm is positioned such that one end (point A) is 20 cm from the mirror's pole, and the other end (point B) is 10 cm from point A.

### Step 2: Determine the object distances
- For point A (the end of the rod closer to the mirror), the object distance (U_A) is -20 cm (the negative sign indicates that the object is in front of the mirror).
- For point B (the end of the rod further from the mirror), the object distance (U_B) is -30 cm (20 cm + 10 cm).

### Step 3: Use the mirror formula
The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Where:
- \( f \) is the focal length of the mirror,
- \( v \) is the image distance,
- \( u \) is the object distance.

### Step 4: Calculate the image distance for point A
Using the mirror formula for point A:
\[ \frac{1}{-10} = \frac{1}{v_A} + \frac{1}{-20} \]
Rearranging gives:
\[ \frac{1}{v_A} = \frac{1}{-10} + \frac{1}{20} \]
Calculating the right-hand side:
\[ \frac{1}{v_A} = -\frac{1}{10} + \frac{1}{20} = -\frac{2}{20} + \frac{1}{20} = -\frac{1}{20} \]
Thus,
\[ v_A = -20 \text{ cm} \]

### Step 5: Calculate the image distance for point B
Using the mirror formula for point B:
\[ \frac{1}{-10} = \frac{1}{v_B} + \frac{1}{-30} \]
Rearranging gives:
\[ \frac{1}{v_B} = \frac{1}{-10} + \frac{1}{-30} \]
Calculating the right-hand side:
\[ \frac{1}{v_B} = -\frac{1}{10} - \frac{1}{30} = -\frac{3}{30} - \frac{1}{30} = -\frac{4}{30} = -\frac{2}{15} \]
Thus,
\[ v_B = -\frac{15}{2} \text{ cm} = -7.5 \text{ cm} \]

### Step 6: Calculate the length of the image
The length of the image of the rod is the difference between the image distances of points A and B:
\[ \text{Length of image} = v_A - v_B = (-20) - (-7.5) = -20 + 7.5 = -12.5 \text{ cm} \]
Since the length cannot be negative, we take the absolute value:
\[ \text{Length of image} = 12.5 \text{ cm} \]

### Final Answer
The length of the image of the rod is 12.5 cm.