Question

A 5 cm long pencil is placed along the principal axis of a concave mirror of focal length 20 cm such that its nearest end is at a distance of 25 cm from the mirror. Calculate the length of the image of the pencil.

Hint:

For extended objects along principal axis: Compute image position of both ends separately using mirror formula.

Answer 1

Concept: For a concave mirror, the mirror formula is: \[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \] Magnification is: \[ m = \frac{dv}{du} \approx \frac{f^2}{(u - f)^2} \] For a small object along principal axis, image length is obtained using longitudinal magnification.

Step 1: Given data
\[ f = -20 \text{ cm}, \quad u_1 = -25 \text{ cm} \] Pencil length = 5 cm, so far end: \[ u_2 = -30 \text{ cm} \]

Step 2: Find image positions using mirror formula For near end: \[ \frac{1}{-20} = \frac{1}{v_1} + \frac{1}{-25} \] \[ \frac{1}{v_1} = -\frac{1}{20} + \frac{1}{25} = \frac{-5 + 4}{100} = -\frac{1}{100} \] \[ v_1 = -100 \text{ cm} \] For far end: \[ u_2 = -30 \] \[ \frac{1}{-20} = \frac{1}{v_2} + \frac{1}{-30} \] \[ \frac{1}{v_2} = -\frac{1}{20} + \frac{1}{30} = \frac{-3 + 2}{60} = -\frac{1}{60} \] \[ v_2 = -60 \text{ cm} \]

Step 3: Image length
\[ \text{Image length} = |v_2 - v_1| \] \[ = |-60 - (-100)| = 40 \text{ cm} \] Final Answer: \[ \boxed{40 \ \text{cm}} \]