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

Step 1: Understanding the problem:
- The object is a candle flame that is 1.5 cm tall.
- The concave mirror has a focal length (f) of 12 cm.
- The distance of the object (u) from the pole of the mirror is 18 cm.
We need to determine the position and size of the image formed by the concave mirror using the mirror formula.

Step 2: 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 from the pole of the mirror.
- \( u \) is the object distance from the pole of the mirror.

Step 3: Substituting the given values into the mirror formula:
We are given:
- Focal length \( f = 12 \, \text{cm} \) (since it's a concave mirror, \( f \) is positive).
- Object distance \( u = -18 \, \text{cm} \) (object distance is always negative for real objects).

Substitute these values into the mirror formula:
\[ \frac{1}{12} = \frac{1}{v} + \frac{1}{-18} \] Now solve for \( v \):
\[ \frac{1}{v} = \frac{1}{12} + \frac{1}{18} \] Taking the LCM of 12 and 18, which is 36, we get:
\[ \frac{1}{v} = \frac{3}{36} + \frac{2}{36} = \frac{5}{36} \] Thus, \( v = \frac{36}{5} = 7.2 \, \text{cm} \).

So, the image is formed at a distance of 7.2 cm from the pole of the concave mirror.
Since \( v \) is positive, the image is real and formed on the same side as the object.

Step 4: Calculating the size of the image:
To calculate the size of the image, we use the magnification formula:
\[ \text{Magnification} (M) = \frac{\text{Image height}}{\text{Object height}} = \frac{v}{u} \] Substitute the known values:
\[ M = \frac{7.2}{-18} = -0.4 \] The magnification is negative, indicating that the image is inverted. Now, calculate the image height:
\[ \text{Image height} = \text{Object height} \times M = 1.5 \times (-0.4) = -0.6 \, \text{cm} \] The negative sign indicates that the image is inverted.

Step 5: Conclusion:
- The image is formed at a distance of 7.2 cm from the pole of the mirror.
- The size of the image is 0.6 cm, and it is inverted.