Image Formation By Spherical Mirrors

Spherical mirrors, including concave and convex mirrors, play a crucial role in the formation of images in various optical devices. Understanding how these mirrors form images helps us grasp the fundamental principles of reflection and optics. In everyday life, concave mirrors are used in shaving mirrors to provide a magnified view of the face, while convex mirrors serve as essential safety tools in vehicle rearview mirrors by offering a wider field of view.

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This Story also Contains

1. Image formation by spherical mirrors
2. Rules For Ray Diagrams
3. Image Formation by Concave Mirror
4. Image Formation by Convex Mirror:
5. Solved Examples Based on Image Formation by Spherical Mirrors
6. Summary

In this article, we are going to study image formation by spherical mirrors in which we will cover sign conversion, rules for ray diagrams and some of the solved examples.

Image formation by spherical mirrors

Image formation by spherical mirrors depends on the position of the object relative to the mirror's surface and principal axis. Both concave and convex mirrors have unique properties that influence the characteristics of the images they produce, including their nature, size, and orientation.

Sign conventions

• In the cartesian sign convention direction of the incident, the ray is taken as +ve.
• All the measurements are measured from the pole.
• If the incident ray is travelling from left to right the distance, measurement along the right direction will be taken as positive.
• We can treat this direction as +ve x-axis direction and the rest can be decided on the basis of the graph that we use in mathematics. Like upward direction will be taken as +ve as it is +ve y-axis. And downward as -ve
• Height above the principle axis is taken as positive and below it is taken as negative.
• Angles measured from the normal in an anti-clockwise sense are positive, while those in a clockwise sense are negative.

Rules For Ray Diagrams

The position of the image formed by spherical mirrors can be found by taking two rays of light coming from a point on the object which intersect each other to form an image. The following are the rules which are used for obtaining images formed by spherical mirrors.

Rule 1: A ray of light that runs parallel to the principal axis, after reflection, passes through the principal focus F of a concave mirror or appears to pass through the principal focus of a convex mirror.

Rule 2: A ray of light passing through the centre of curvature in a concave mirror or a ray of light going towards the centre of curvature of a convex mirror is reflected back along the same path.

Rule 3: A ray of light passing through the principal focus of a concave mirror or appearing to pass through the principal focus of a convex mirror becomes parallel to the principal axis after reflection.

Rule 4: A ray incident at the pole is reflected back making the same angle as the principal axis.

Image Formation by Concave Mirror

1: For a real object very far away from the mirror, the real image is formed at the focus.

2: For a real object close to the mirror but outside of the center of curvature, the real image is formed between C and f. The image is inverted and smaller than the object.

3: For a real object at C, the real image is formed at C. The image is inverted and the same size as the object.

4: For a real object between C and f, a real image is formed outside of C. The image is inverted and larger than the object.

5: For a real object at f, no image is formed. The reflected rays are parallel and never converge.

6: For a real object between f and the mirror, a virtual image is formed behind the mirror. The position of the image is found by tracing the reflected rays back behind the mirror to where they meet. The image is upright and larger than the object.

Image Formation by Convex Mirror:

1. When the object is at infinity, a point-sized image is formed at the principal focus behind the convex mirror.

Properties of image: The image is highly diminished, virtual and erect.

2. When the object is between infinity and the pole of a convex mirror, a diminished, virtual and erect image is formed between the pole and focus behind the mirror.

Properties of image: The image is diminished, virtual and erect.

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Solved Examples Based on Image Formation by Spherical Mirrors

Example 1: An object is at a distance of 10 cm from a mirror and the image of the object is at a distance of 30 cm from the mirror on the same side as the object. The nature of the mirror and its focal length are:

1) Convex, 15cm

2) Concave, 1.5cm

3) Convex, 7.5cm

4) Concave, 7.5cm

Solution:

Sign Convention

1) All distances are measured from the pole.

2) Distance measured in the direction of incident rays is taken as positive.

3) Distance measured in the direction opposite to that of incident rays is taken as negative.

4) Distance above the principal axis as positive and below the principal axis as negative.

u = -10 cm

v = - 30cm from sign convention

from mirror formula

$\begin{aligned} & \frac{1}{f}=\frac{1}{v}+\frac{1}{u}=-\frac{1}{30}-\frac{1}{10}=\frac{-4}{30} \\ & \mathrm{f}=-7.5 \mathrm{~cm}\end{aligned}$

since focal length is negative hence its mirror concave.

Hence, The answer is the option (4).

Example 2: What type of image will form when the object is between infinity and the pole of a convex mirror?

1) A diminished, Real and erect image is formed between the pole and the focus behind the mirror.

2) A diminished, virtual and erect image is formed between the pole and the focus behind the mirror.

3) A diminished, virtual and inverted image is formed between the pole and the focus behind the mirror.

4) an enlarged, virtual and erect image is formed between the pole and the focus behind the mirror.

Solution:

When the object is between infinity and the pole of a convex mirror, a diminished, virtual and erect image is formed between the pole and focus behind the mirror.

Properties of image: The image is diminished, virtual and erect.

Example 3: What type of image will form for a real object between the Center of curvature(C) and focus(f) in the concave mirror?

1) For a real object between C and f, a real image is at C. The image is inverted and larger than the object.

2) For a real object between C and f, a real image is formed outside of C. The image is inverted and larger than the object.

3) For a real object between C and f, a real image is formed outside of C. The image is erect and larger than the object.

4) For a real object between C and f, a real image is formed outside of C. The image is inverted and diminished than the object.

Solution:

For a real object between C and f, a real image is formed outside of C. The image is inverted and larger than the object.

Example 4: A short straight object of height 100 cm lies before the central axis of a spherical mirror whose focal length has absolute value $|f|=40$ cm. The image of an object produced by the mirror is of height 25 cm and has the same orientation as the object. One may conclude from the information :

1) The image is virtual, opposite side of a convex mirror.

2) The image is real, same side as a concave mirror

3) The image is real, same side as the convex mirror.

4) The image is virtual, opposite side of the concave mirror.

Solution

Since the orientation is the same image is virtual. Since the image is smaller the mirror has to be convex.

Hence, the answer is option (1).

Example 5: The direction of a ray of light incident on a concave mirror is shown by PQ while directions in which the ray would travel after reflection is shown by four rays marked $1,2,3$ and 4 (figure). Which of the four rays correctly shows the direction of the reflected ray?

1) 1
2) 2
3) 3
4) 4

Solution:

The PQ ray of light passes through focus F and incident on the concave mirror, after reflection the ray, should become parallel to the principal axis as shown by the ray - 2 in the figure.

Hence, the answer is option (2).

Summary

Depending on its position in relation to the mirror, a spherical mirror which is like the concave and convex mirrors, creates an image. When the object is placed beyond the focal point, real and inverted images are formed by the concave mirror while virtual and upright images are formed when the object lies between the mirror and the focal point. In relation to the object’s distance, size, and character vary with distance.