An object is placed at a distance of 10 cm from a convex mirror of focal length 15 cm. Find the position and nature of the image.
The correct answer and explanation is:
Given:
Object distance, u=−10u = -10 cm (since object distances are taken negative for mirrors when measured against the incoming light direction)
Focal length, f=+15f = +15 cm (convex mirror focal length is positive)
Using the mirror formula: 1f=1v+1u\frac{1}{f} = \frac{1}{v} + \frac{1}{u}
Rearranged to find image distance vv: 1v=1f−1u\frac{1}{v} = \frac{1}{f} – \frac{1}{u}
Substitute the values: 1v=115−1−10=115+110\frac{1}{v} = \frac{1}{15} – \frac{1}{-10} = \frac{1}{15} + \frac{1}{10}
Finding a common denominator (30): 1v=230+330=530=16\frac{1}{v} = \frac{2}{30} + \frac{3}{30} = \frac{5}{30} = \frac{1}{6}
Therefore, v=6 cmv = 6 \, \text{cm}
The positive value of vv indicates that the image is formed on the same side as the reflected rays (virtual image side).
Nature of the image:
- Position: The image is formed 6 cm behind the convex mirror.
- Type: Since the image distance is positive, the image is virtual.
- Size: To find magnification mm, use:
m=vu=6−10=−0.6m = \frac{v}{u} = \frac{6}{-10} = -0.6
The negative sign for magnification indicates the image is upright (erect). The magnitude less than 1 shows the image is diminished (smaller than the object).
Explanation
When an object is placed in front of a convex mirror, the reflected rays diverge after reflection, making the image appear behind the mirror. The focal length of a convex mirror is always positive, indicating it is a diverging mirror.
Using the mirror formula, the image position can be found by calculating the reciprocal of the focal length and object distance. The image distance comes out positive, showing that the image forms on the virtual side of the mirror, which is behind it.
Convex mirrors always produce virtual, erect, and diminished images regardless of the object’s position. This makes convex mirrors useful in applications like vehicle side mirrors where a wider field of view is needed and the images are smaller but upright and virtual.
In this specific case, placing the object 10 cm from the mirror with a focal length of 15 cm results in the image forming 6 cm behind the mirror. The image is virtual and upright, but smaller than the object by 60%. This is consistent with the general behavior of convex mirrors.