# Convex Mirror: Concept, Characteristics, Examples

The convex or divergent mirror is a curved mirror, almost always spherical in shape and with the reflecting surface on the outer side of the sphere, such as Christmas tree ornaments. Thanks to convex mirrors it is possible to achieve a great variety of images depending on where the object is located and that is the reason why they have so many uses.

For example, the mirrors that are placed in the streets to facilitate the transit of vehicles in narrow intersections are convex, since they produce an image with a wide visual field.

The images thus formed are diverse, depending on the place where the object is placed. In the above image parallel rays from a distant source of as shown Sol .

The rays are reflected according to the law of reflection, which indicates that the angle of incidence of the ray is the same with which it is reflected. As we can see, the reflected rays separate – they do not cross – when they leave the specular surface, which is why this kind of mirror is also known as divergent .

When the reflections extend behind the mirror – dashed lines in the figure – they intersect at a point F called the focus.

## Features of convex mirrors

The convex mirror has the following characteristics (see image above):

-The notable points of the mirror are:

• C the center, which coincides with the center of the sphere to which the mirror belongs.
• F the focus, where the rays reflected from behind the mirror converge.
• Its vertex P, which corresponds to the center of the spherical surface and is collinear with C and F.

-It has an optical axis or main axis , which is the line perpendicular to the specular surface. Rays hitting just the optic axis are reflected in the same direction.

-The center of the sphere to which the mirror belongs is at point C and r is its radius. AC is known as the center of curvature , while r is the radius of curvature and indicates how curved the mirror is: the smaller r , the more pronounced the convex shape.

-The point of intersection of the reflected rays is known as the focal point of the mirror. The distance between F and P is approximately r / 2:

f = r / 2

This expression is valid for mirrors whose size is much smaller than their radius of curvature.

-The image that is formed is smaller and also virtual, since it is located behind the mirror, as we will see below.

## Image formation in the convex mirror

In order to know what the image that is formed in the convex mirror is like, the ray treatment is used, which consists of representing the light rays that leave the object by means of straight lines.

These rays are reflected on the mirror surface and the reflected rays are also drawn. The ray method is applicable to any kind of mirror, not just convex ones.

By prolonging the reflected rays, they intersect at a certain point, and that is precisely where the image is formed. The extensions of the reflected rays that come from an extended object such as a tree are shown in the figure below by dashed lines.

In the lower figure, three rays are drawn from the object, very particular and easy to draw, as well as their reflections:

-Ray 1, striking parallel to the optical axis.

-Ray 2, which impacts in such a way that the prolongation of the reflected ray passes precisely through the focus of the mirror, that is, point F. This ray is reflected in a direction parallel to the optical axis.

-Finally ray 3, which arrives perpendicular to the spherical surface, and for this reason is reflected in the same direction.

In principle, this procedure is applied to each point of the tree, but the information obtained from the 3 rays drawn is enough to find the image of the object: it is formed behind the mirror, it is straight and smaller than the original.

## Examples and applications

Many highly polished spherical surfaces act as convex mirrors, for example shiny and silver Christmas ornaments, as well as shiny new steel spoons.

Also convex mirrors have many practical applications, for example:

### Mirrors to prevent traffic accidents

Convex mirrors on streets and avenues help prevent accidents, as they allow you to see traffic coming from corners.

### Mirrors for surveillance

Convex mirrors are often used in stores and banks to detect thieves, as well as to avoid collisions between people and forklifts moving through aisles and between shelves.

### Rear view mirrors

Cars and motorcycles have convex rear-view mirrors, which produce slightly smaller images, but cover more field of view than flat mirrors.

### Cassegrain telescope

One of the mirrors of the Cassegrain reflecting telescope, the secondary mirror, is convex, although it is not spherical and serves to reflect the image towards the main mirror of the telescope.

## Convex mirror equations

Let us consider the right triangles in the following figure, determined by ray 1, which comes from the top of the arrow, its reflection and its extension.

The original image has height y, while the height of the virtual image is y ‘  . It is true that:

tan θ = y / d o = y ‘/ d i

### Mirror magnification

The ratio between the height of the image and the height of the object is the magnification of the mirror , which is called that, even if the image obtained is smaller than the real object. We denote it by m :

m = y ‘/ y = d i / d o

### Relationship between the object and its image in the convex mirror

Now let’s consider this other figure, where the AVF region can be considered roughly like a right triangle, since the curvature of the mirror is not very accentuated. Thus:

AV ≈ h o

So:

tan α = h

$tan\:&space;\alpha&space;=\frac{h_{i}}{f-d_{i}}=\frac{h_{o}}{f}$$\frac{f}{f-d_{_{i}}}&space;=\frac{h_{o}}{h_{i}}$$\frac{h_{i}}{h_{o}}=\frac{f-d_{i}}{f}=\frac{d_{i}}{d_{o}}$

1- (d i / f) = d i / d o

When dividing everything by d i :

$\frac{1}{d_{i}}-\frac{1}{f}=\frac{1}{d_{o}}$$\frac{1}{d_{o}}-\frac{1}{d_{i}}=-\frac{1}{f}$

Therefore, as f and d i are behind the mirror, a minus sign is placed before them, while for the distance d or that is not necessary, since it is in front of the mirror. Thus the previous equation is:

$\frac{1}{d_{o}}+\frac{1}{d_{i}}=\frac{1}{f}$

Concave mirror .

## References

1. Bauer, W. 2011. Physics for Engineering and Sciences. Volume 2. Mc Graw Hill.
2. Giambattista, A. 2010. Physics. 2nd. Ed. McGraw Hill.
3. Katz, D. 2017. Physics for Scientists and Engineers. Cengage Learning.
4. Thomas, W. 2008. Conceptual Physics. McGraw Hill.
5. Tippens, P. 2011. Physics: Concepts and Applications. 7th Edition. McGraw Hill.