Question

A 1.5 cm tall candle flame is placed perpendicular to the principal axis of a concave mirror of focal length 12 cm. If the distance of the flame from the pole of the mirror is 18 cm, use the mirror formula to determine the position and size of the image formed.

Answer 1

Problem Analysis:
Object: Candle flame, height = 1.5 cm.
Mirror: Concave, focal length \(f\) = 12 cm.
Object distance \(u\) = 18 cm from the pole.
Objective: Determine image position and size using the mirror formula.

Mirror Formula:
The relationship between focal length (\(f\)), image distance (\(v\)), and object distance (\(u\)) is:
\[\frac{1}{f} = \frac{1}{v} + \frac{1}{u}\]

Applying the Mirror Formula:
Given values:
- \(f = 12 \, \text{cm}\) (positive for concave mirror).
- \(u = -18 \, \text{cm}\) (negative for real object).

Substituting into the formula:
\[\frac{1}{12} = \frac{1}{v} + \frac{1}{-18}\]Solving for \(v\):
\[\frac{1}{v} = \frac{1}{12} + \frac{1}{18}\]Using the least common multiple (36):
\[\frac{1}{v} = \frac{3}{36} + \frac{2}{36} = \frac{5}{36}\]Therefore, \(v = \frac{36}{5} = 7.2 \, \text{cm}\).

The image is located 7.2 cm from the mirror's pole. A positive \(v\) indicates a real image on the same side as the object.

Image Size Calculation:
Magnification (\(M\)) is calculated as:
\[M = \frac{\text{Image height}}{\text{Object height}} = \frac{v}{u}\]Substituting known values:
\[M = \frac{7.2}{-18} = -0.4\]The negative magnification signifies an inverted image. The image height is:
\[\text{Image height} = \text{Object height} \times M = 1.5 \times (-0.4) = -0.6 \, \text{cm}\]The negative sign confirms the image is inverted.

Summary of Results:
- Image distance from the mirror's pole: 7.2 cm.
- Image size: 0.6 cm, inverted.