Plane mirrors: equation, behavior, graphs

Ideally, a perfect mirror does not absorb any light and reflects all of the incident light, regardless of its intensity, at least in the visible light range. the image formed in a plane mirror is

Remember that light is an electromagnetic wave with a wide spectrum of wavelengths, of which a small fraction is visible to the human eye. This range is between 400 and 700 nanometers, where one nanometer equals 1 × 10 ^-9 m.

In practice, common bathroom mirrors are far from perfect, although they serve everyday grooming purposes. These mirrors are made of glass, to which a layer of polished metal adheres to the bottom, which acts as a mirror surface.

Apart from that, mirrors are used as part of optical devices: telescopes, microscopes, polarimeters, car rear view mirrors, periscopes and even decorative elements.

The image formed in a plane mirror is

The image of an object that is formed in the flat mirror is characterized by:

-Being virtual, that is, it is an image from which no light emanates, however the eye does not perceive the difference.

-It is the same size as the object.

-It is formed at the same distance, behind the plane of the mirror, of which the object is in front of it.

-Being a right image, that is to say that its orientation is the same as that of the object, but with specular symmetry , which means that its relationship is the same as that between the right hand and the left hand. the image formed in a plane mirror is

In addition, the image produced by the mirror can serve as an object for a second mirror, as we will see later. This is the principle of the periscope, an instrument used to look at objects that are not at the same height as the observer’s eyes.

Equation

Reflection in plane mirrors is governed by a very simple equation, called the law of reflection , which is stated as follows:

The angle of incidence of a ray of light θ _i is equal to the angle of reflection θ _r .

Both angles are always measured taking as reference the normal to the surface, that is, the line perpendicular to the plane of the mirror. And furthermore, the incident ray, the reflected ray and the normal line are in the same plane.

Mathematical expressions for the plane mirror

In mathematical form the equation is written:

θ _i = θ _r

When parallel rays strike the mirror surface, so are the reflected rays. Similarly, any normal to the mirror is parallel to another normal.

As a consequence, as we shall see below, the distance at which the object is with respect to the surface of the mirror d _i , is the same at which the image is formed on the opposite side d _o .

Therefore:

| d _i | = | d _or |

The absolute value bars are placed, since by convention the distance from the virtual image to the mirror is taken negative, while the distance between the object and the mirror is positive.

Behaviour

Let’s see how the flat mirror behaves before a point source, like the flame of a lit candle. In the figure below, two rays are drawn, ray 1 which is directed directly towards the mirror and is reflected in the same direction, and ray 2, which is obliquely incident, with angle θ _i and reflected at angle θ _r .

In addition, the optical axis is represented , which is defined as a normal to the mirror plane. Since the mirror is flat, many normals can be drawn to it, unlike the spherical mirror, in which a single optical axis is drawn. the image formed in a plane mirror is

Extending the rays by means of discontinuous lines, we see that they intersect at point P ‘, behind the mirror. From that point, at a distance d _o , from the mirror, the observer’s eye interprets that the image of the flame comes from.

The mirror also reflects the rest of the candle, a large object of finite size. To each point of it corresponds a point in the image, thus being determined two congruent right triangles, whose common height is h = h ‘ , the height of the candle.

In this way, the image has the same height as the real object and the same orientation. And it can also be seen that the object and its image are in the same relationship as the open palms of the hands when viewed from the front.

Mirror combinations

As we said at the beginning, an image from one mirror can serve as the object to produce a second image in another mirror.

Let the object P, whose image P ₁ ´is formed in mirror 1. Object P is also reflected in mirror 2 and forms another image, called P ₂ ´.

Additionally, P _{1 ´s} serves as an object for mirror 2 to form an image P ₃ ´ at the point indicated in the following figure.

Also P ₂ ´ can act as an object so that mirror 1 forms its image in the same place as P ₃ ´. Well, this interesting principle is the foundation of imaging in refractive telescopes, for example.

The periscope

The periscope is often used to observe objects on the surface of the water from a submerged position, and generally to view objects that are at a height that is above the observer.

In this way events can be viewed above the heads of a crowd. There are also binoculars that have the possibility of adding periscopes to them.

A simple periscope consists of two flat mirrors inclined 45º with respect to the vertical and assembled inside a tube.

In the figure it can be seen that the light beam falls at an angle of 45º with respect to the upper mirror, being deviated 90º and directed to the lower mirror, being deviated again 90º to reach the observer’s eye. the image formed in a plane mirror is

Graphics

The graphs consist of ray tracing to show the formation of the images. Whether they are point or large objects, to locate the image in the mirror, it is enough to trace two rays that originate from the point in question.

In the figure above, two rays are traced to determine the place where the image of the flame is formed, one that falls perpendicular to the mirror and the other at an angle. Both fulfill the law of reflection. Then the respective reflections were prolonged and the point where they meet corresponds to the point where the image is formed.