Length of Image of Rod Placed Along Axis of Concave Mirror

To find the length of the image of the rod, we determine the positions of the images of both ends of the rod using the mirror formula.

1. Identification of Given Data:

• Focal length of the concave mirror, $f = -10\text{ cm}$
• Length of the rod, $l = 10\text{ cm}$
• From the given diagram, the nearer end of the rod is at a distance of $20\text{ cm}$ from the pole of the mirror. Thus, $u_1 = -20\text{ cm}$.
• The farther end of the rod is at a distance of $u_2 = -(20 + 10)\text{ cm} = -30\text{ cm}$ from the pole of the mirror.

2. Mirror Formula: The mirror formula is given by: $\frac{1}{v} + \frac{1}{u} = \frac{1}{f}$

3. Image of Nearer End ($u_1 = -20\text{ cm}$): $\frac{1}{v_1} + \frac{1}{-20} = \frac{1}{-10}$ $\frac{1}{v_1} = -\frac{1}{10} + \frac{1}{20} = -\frac{1}{20}$ $v_1 = -20\text{ cm}$

4. Image of Farther End ($u_2 = -30\text{ cm}$): $\frac{1}{v_2} + \frac{1}{-30} = \frac{1}{-10}$ $\frac{1}{v_2} = -\frac{1}{10} + \frac{1}{30} = \frac{-3 + 1}{30} = -\frac{2}{30} = -\frac{1}{15}$ $v_2 = -15\text{ cm}$

5. Calculation of Length of the Image: The length of the image is the distance between the images of the two ends: $\text{Length of image} = |v_1 - v_2| = |-20 - (-15)| = 5\text{ cm}$

Thus, the length of the image is $5\text{ cm}$, which corresponds to Option B.