Optics

# flat mirrors

Flat mirrors are flat surfaces, polished and without curvature, capable of promoting regular reflection of light When light rays are reflected by plane mirrors , the angle of the reflected rays is equal to the angle of the incident rays, moreover, the incident and reflected rays lie in the same plane.

## Image formation by plane mirrors

When two or more reflected light rays cross each other, image formation occurs . In the case of plane mirrors, the reflected light rays do not cross each other, instead, their extensions cross to form images. The extensions of the reflected light rays, as illustrated in the following image, cross each other “behind” the mirror, at a distance equal to that in which the object in the image is and with the same size.

In plane mirrors, images are formed by extensions of reflected rays. In order for us to see our reflection on a reflective surface, it must promote regular light reflection , that is, reflect light rays with an angle equal to the incident angle. In addition, it is also necessary that the incident and reflected rays are contained in the same plane.

Almost all surfaces reflect light, white walls, for example, are able to do so, however, in a diffuse way. Because of this, it is possible to see them, however, we cannot see our reflection, since the reflected light rays do not have the same angles as the incident rays. The following figure brings a diagram showing the difference between a regular reflection and a diffuse reflection , note:

Images such as those formed on plane mirrors are called virtual images . Among their main characteristics are these: they are formed by the crossing of extensions of light rays , therefore, they are formed behind the mirror; they are always direct , that is, they have the same vertical orientation as their objects; in addition, as they are virtual, they cannot be projected onto a screen, unlike real images, which can be projected.

## Characteristics of images formed by plane mirrors

When we look into a flat mirror, like the ones we have in bathrooms and home furniture, our image is formed behind the mirror at the same distance we are from the mirror’s reflective surface . Therefore, if you are 2 m from a plane mirror, your image is formed 2 m behind the plane mirror, at a distance of 4 m from you.

Because they are the same distance from the mirror as the object, the images conjugated by plane mirrors are the same size as their objects. Also, when we approach a plane mirror, our image translates towards us, with the same speed. Thus, the speed with which we approach an image produced by a plane mirror is equal to the sum of the speed of the image and the speed of the object .

For example: if you are at a distance of 4 m from a plane mirror and you approach it at 1 m/s, you will reach it in 4 seconds. However, you are 8 m from your image, which will also meet you in 4 seconds, so your speed relative to this image must be 8 m/s.

In addition to translation, it is possible that we rotate an object in front of a mirror, in which case the rotation angle of the image is always twice as large as the rotation angle applied to the object. This is because when the object is rotated by some angle, clockwise, its reflection will be rotated by the same angle, but counterclockwise .

## enantiomorphism

Despite having the same vertical orientation, the images formed in plane mirrors are horizontally inverted , as if they had been rotated 180º, because of this characteristic, we say that the images produced by plane mirrors are enantiomorphic.

Enantiomorphism is a characteristic of an object that is not superimposable with its reflection , as in the case of the left hand with the right hand : placing one on top of the other, each thumb will point to one side: despite having identical shapes, the hands are arranged in a different order. The same is true of fresh ink, used to write something on the pages of a notebook. If we close the notebook, when we open it, it will be possible to see the “reflection” of the ink, printed on the other page.

Despite being present in all images conjugated by plane mirrors, enantiomorphism can only be observed in chiral objects. Chiral objects are those that have a similar structure, however, cannot be superimposed. Here are some examples of chiral objects:

• Shoes : cannot be superimposed, it is not possible, for example, to wear the right shoe on the left foot.
• Letters: have the image reflected in a chiral way, generally. There are some exceptions for cases where the letters have bilateral symmetry, such as the letters H, I, M, O, T, U, X and Y. Note that when reflected these letters will have a mirror image identical to them, which can be perfectly fitted over them.

## Association of plane mirrors

When two plane mirrors are joined together, we are aligning the normal directions with their surfaces at some angle, as when we open the doors of a mirrored cabinet towards us, or even when we enter an elevator that has mirrors on its inner walls.

According to the angle formed between the directions perpendicular to the surface of each mirror, it is possible to determine the number of images that will be formed . The formula that allows us to calculate the number of images formed by two mirrors aligned with an angle α is shown below, check it out:

N – number of conjugated images

α – angle formed between the normal lines and the surface of each mirror

Analyzing the formula, it is possible to see that the number of images is inversely proportional to the angle formed between the mirrors. When this angle is equal to 0º, as when we place a mirror in front of the other, the number of images, formed by successive reflections, is infinitely large .

## Exercises on plane mirrors

Question 1) (UNIFOR – adapted) The angle between two plane mirrors is 20°. An object of negligible dimensions is placed in such a position that it will obtain several images formed by the set of mirrors. The number of images formed by the conjugation of these mirrors will be equal to:

a) 8

b) 9

c) 10

d) 17

e) 18

Template: Letter D

Resolution:

Just use the formula that calculates the number of images formed according to the angle formed between the mirrors, note:

According to the statement, the angle between the mirrors is 20º, so we will do the following calculation:

With the previous calculation, we found that, in all, 17 images are formed, so the correct alternative is the letter D.

Question 2) (Mackenzie) An extended object of height h is fixed and placed frontally in front of a reflecting surface of a plane mirror, at a distance of 120.0 cm. Approaching the mirror of the object at a distance of 20.0 cm, the conjugate image, in this condition, is distant from the object in:

a) 100.0 cm

b) 120.0 cm

c) 200.0 cm

d) 240.0 cm

e) 300.0 cm

Template: Letter C

Resolution:

Initially, the object was 120.0 cm from the mirror, after approaching 20.0 cm, its distance from the mirror becomes 100.0 cm. As we know, the object and its image are the same distance from the mirror surface, so the distance between them is 200.0 cm.

Question 3) A boy is 2.0 m away from a plane mirror when he starts to move towards that mirror with a speed of 1.5 m/s. The instant the boy starts to move, the mirror image is:

a) 2.0 m away from the mirror, with a speed of 3.0 m/s relative to the boy.

b) 2.0 m away from the mirror, with a speed of 1.5 m/s relative to the boy.

c) a distance greater than 2.0 m from the mirror, with a speed of 3.0 m/s relative to the mirror.

d) a distance of less than 2.0 m from the mirror, with a speed of 1.5 m/s relative to the mirror.

Template: Letter A

Resolution:

Both the image and the boy are 2.0 m from the mirror, and both the object and its image are moving at a speed of 1.5 m/s, so the relative speed of approach is : 3.0 m/sec.