Construction of images in spherical mirrors
In order to construct an image of any point light source in a spherical mirror, it is enough to construct a path any two rays emanating from this source and reflected from the mirror. The intersection point of the reflected rays themselves will give a real image of the source, and the intersection point of the extensions of the reflected rays will give an imaginary one.
Characteristic rays. To construct images in spherical mirrors, it is convenient to use certain characteristic rays, the course of which is easy to construct.
1. Beam 1 , incident on the mirror parallel to the main optical axis, reflected, passes through the main focus of the mirror in a concave mirror (Fig. 3.6, A); in a convex mirror, a continuation of the reflected ray passes through the main focus 1 ¢ (Fig. 3.6, b).
2. Beam 2 , passing through the main focus of a concave mirror, having been reflected, goes parallel to the main optical axis - a ray 2 ¢ (Fig. 3.7, A). Ray 2 , incident on a convex mirror so that its continuation passes through the main focus of the mirror, having been reflected, it also goes parallel to the main optical axis - a ray 2 ¢ (Fig. 3.7, b).
Rice. 3.7 
3. Consider a ray 3 , passing through center concave mirror - point ABOUT(Fig. 3.8, A) and beam 3 , incident on a convex mirror so that its continuation passes through the center of the mirror - the point ABOUT(Fig. 3.8, b). As we know from geometry, the radius of a circle is perpendicular to the tangent to the circle at the point of contact, so the rays 3 in Fig. 3.8 fall on the mirrors under right angle, that is, the angles of incidence of these rays are zero. This means that the reflected rays 3 ¢ in both cases coincide with the falling ones.
Rice. 3.8 
4. Beam 4 , passing through pole mirrors - point R, is reflected symmetrically relative to the main optical axis (rays 4¢ in Fig. 3.9), since the angle of incidence is equal to the angle of reflection.
Rice. 3.9 
STOP! Decide for yourself: A2, A5.
Reader: Once I took an ordinary tablespoon and tried to see my image in it. I saw the image, but it turned out that if you look at convex part of a spoon, then the image direct, and if on concave, That inverted. I wonder why this is so? After all, a spoon, I think, can be considered as some kind of spherical mirror.
Task 3.1. Construct images of small vertical segments of the same length in a concave mirror (Fig. 3.10). The focal length is set. It is considered known that the images of small straight segments perpendicular to the main optical axis in a spherical mirror also represent small straight segments perpendicular to the main optical axis.
Solution.
1. Case a. Note that in this case all objects are in front of the main focus of the concave mirror.
 Rice. 3.11 |
We will construct images only of the top points of our segments. To do this, draw through all the upper points: A, IN And WITH one common beam 1 , parallel to the main optical axis (Fig. 3.11). Reflected beam 1 F 1 .
Now from the points A, IN And WITH let's send out rays 2 , 3 And 4 through the main focus of the mirror. Reflected rays 2 ¢, 3 ¢ and 4 ¢ will go parallel to the main optical axis.
Points of intersection of rays 2 ¢, 3 ¢ and 4 ¢ with beam 1 ¢ are images of points A, IN And WITH. These are the points A¢, IN¢ and WITH¢ in fig. 3.11.
To get images segments it is enough to omit from the points A¢, IN¢ and WITH¢ perpendiculars to the main optical axis.
As can be seen from Fig. 3.11, all images turned out valid And upside down.
Reader: What do you mean – valid?
Author: The image of objects happens valid And imaginary. We already became acquainted with the virtual image when we studied a plane mirror: the virtual image of a point source is the point at which they intersect continuation rays reflected from the mirror. The actual image of a point source is the point at which the themselves rays reflected from the mirror.
Note that what further there was an object from the mirror, so smaller it turned out his image and that closer this is the image to mirror focus. Note also that the image of a segment whose lowest point coincided with center mirrors - dot ABOUT, happened symmetrical object relative to the main optical axis.
I hope you now understand why, looking at your reflection in the concave surface of a tablespoon, you saw yourself reduced and inverted: after all, the object (your face) was clearly before the main focus of a concave mirror.
2. Case b. In this case, the objects are between the main focus and the surface of the mirror.
The first ray is the ray 1
, as in the case A, let us pass through the upper points of the segments - points A And IN 1
¢ will pass through the main focus of the mirror - the point F 1 (Fig. 3.12).
Now let's use the rays 2 And 3 emanating from points A And IN and passing through pole mirrors - point R. Reflected rays 2 ¢ and 3 ¢ make the same angles with the main optical axis as the incident rays.
As can be seen from Fig. 3.12, reflected rays 2 ¢ and 3 ¢ do not intersect with reflected beam 1 ¢. Means, valid images in this case No. But continuation reflected rays 2 ¢ and 3 ¢ intersect with continuation reflected beam 1 ¢ at points A¢ and IN¢ behind the mirror, forming imaginary dot images A And IN.
Dropping perpendiculars from points A¢ and IN¢ to the main optical axis, we obtain images of our segments.
As can be seen from Fig. 3.12, the images of the segments turned out straight And enlarged, and what closer subject to the main focus, the more his image and theme further This is the image from the mirror.
STOP! Decide for yourself: A3, A4.
Problem 3.2. Construct images of two small identical vertical segments in a convex mirror (Fig. 3.13).
 |  |
Rice. 3.13 Fig. 3.14
Solution. Let's send out a beam 1 through the upper points of the segments A And IN parallel to the main optical axis. Reflected beam 1 ¢ will go so that its continuation intersects the main focus of the mirror - the point F 2 (Fig. 3.14).
Now let's send rays onto the mirror 2 And 3 from points A And IN so that the continuations of these rays pass through center mirrors - point ABOUT. These rays will be reflected so that the reflected rays 2 ¢ and 3 ¢ coincide with the incident rays.
As we see from Fig. 3.14, reflected beam 1 ¢ does not intersect with reflected rays 2 ¢ and 3 ¢. Means, valid dot images A And B no. But continuation reflected beam 1 ¢ intersects with continuations reflected rays 2 ¢ and 3 ¢ at points A¢ and IN¢. Therefore, the points A¢ and IN¢ – imaginary dot images A And IN.
To build images segments drop the perpendiculars from the points A¢ and IN¢ to the main optical axis. As can be seen from Fig. 3.14, the images of the segments turned out straight And reduced. And what? closer the object to the mirror, the more his image and theme closer it's towards the mirror. However, even a very distant object cannot produce an image distant from the mirror beyond the main focus of the mirror.
I hope it is now clear why, looking at your reflection in the convex surface of the spoon, you saw yourself reduced, but not inverted.
STOP! Decide for yourself: A6.
If the reflecting surface of the mirror is flat, then it is a type of flat mirror. Light is always reflected from a flat mirror without scattering according to the laws of geometric optics:
- The angle of incidence is equal to the angle of reflection.
- The incident ray, the reflected ray, and the normal to the mirror surface at the point of incidence lie in the same plane.
One thing to remember is that a glass mirror has a reflective surface (usually a thin layer of aluminum or silver) placed on its back. It is covered with a protective layer. This means that although the main reflected image is formed on this surface, light will also be reflected from the front surface of the glass. A secondary image is formed, which is much weaker than the main one. It is generally invisible in everyday life, but poses serious problems in the field of astronomy. For this reason, all astronomical mirrors have a reflective surface applied to the front side of the glass.
Image Types
There are two types of images: real and imaginary.
The real is formed on the film of a video camera, camera or on the retina of the eye. Light rays pass through a lens or lens, converge when falling on a surface, and at their intersection form an image.
Imaginary (virtual) is obtained when rays, reflected from a surface, form a divergent system. If you complete the continuation of the rays in the opposite direction, then they will definitely intersect at a certain (imaginary) point. It is from these points that a virtual image is formed, which cannot be recorded without the use of a flat mirror or other optical instruments (magnifying glass, microscope or binoculars).
Image in a plane mirror: properties and construction algorithm
For a real object, the image obtained using a plane mirror is:
- imaginary;
- straight (not inverted);
- the dimensions of the image are equal to the dimensions of the object;
- the image is at the same distance behind the mirror as the object in front of it.
Let's construct an image of some object in a plane mirror.
Let's use the properties of a virtual image in a plane mirror. Let's draw an image of a red arrow on the other side of the mirror. Distance A is equal to distance B, and the image is the same size as the object.
A virtual image is obtained at the intersection of the continuation of reflected rays. Let's depict light rays coming from an imaginary red arrow to the eye. Let us show that the rays are imaginary by drawing them with a dotted line. Continuous lines extending from the surface of the mirror show the path of the reflected rays.
Let's draw straight lines from the object to the points of reflection of the rays on the surface of the mirror. We take into account that the angle of incidence is equal to the angle of reflection.
Plane mirrors are used in many optical instruments. For example, in a periscope, flat telescope, graphic projector, sextant and kaleidoscope. A dental mirror for examining the oral cavity is also flat.
Flat mirror- This is a flat surface that specularly reflects light.
The construction of an image in mirrors is based on the laws of rectilinear propagation and reflection of light.
Let's build an image of a point source S(Fig. 16.10). From the source the light goes in all directions. A beam of light falls on the mirror SAB, and the image is created by the entire beam. But to construct an image, it is enough to take any two rays from this beam, for example SO And S.C.. Ray SO falls perpendicular to the mirror surface AB(angle of incidence is 0), so the reflected one will go in the opposite direction OS. Ray S.C. will be reflected at an angle \(~\gamma=\alpha\). Reflected rays OS And SK diverge and do not intersect, but if they fall into a person’s eye, then the person will see the image S 1 which represents the point of intersection continuation reflected rays.
The image obtained at the intersection of reflected (or refracted) rays is called actual image.
The image obtained when not the reflected (or refracted) rays themselves intersect, but their continuations, is called virtual image.
Thus, in a plane mirror the image is always virtual.
Can be proven (consider triangles SOC and S 1 OC), which is the distance SO= S 1 O, i.e. the image of point S 1 is located from the mirror at the same distance as the point S itself. It follows that to construct an image of a point in a plane mirror, it is enough to lower a perpendicular to the plane mirror from this point and extend it to the same distance behind the mirror ( Fig. 16.11).
When constructing an image of an object, the latter is represented as a collection of point light sources. Therefore, it is enough to find an image of the extreme points of the object.
The image A 1 B 1 (Fig. 16.12) of the object AB in a flat mirror is always virtual, straight, the same dimensions as the object, and symmetrical relative to the mirror.
Let us find the connection between the optical characteristic and the distances that determine the position of the object and its image.
Let the object be a certain point A located on the optical axis. Using the laws of light reflection, we will construct an image of this point (Fig. 2.13).
Let us denote the distance from the object to the pole of the mirror 
(AO), and from pole to image 
(OA).
Consider the triangle APC, we find that 
From the triangle APA, we obtain that 
. Let us exclude the angle from these expressions 
, since it is the only one that does not rely on OR.
,
or
(2.3)
Angles ,,are based on OR. Let the beams under consideration be paraxial, then these angles are small and, therefore, their values in radian measure are equal to the tangent of these angles:
;
;
, where R=OC, is the radius of curvature of the mirror.
Let us substitute the resulting expressions into equation (2.3)
Since we previously found out that the focal length is related to the radius of curvature of the mirror, then
(2.4)
Expression (2.4) is called the mirror formula, which is used only with the sign rule:
Distances 
,
,
are considered positive if they are counted along the ray, and negative otherwise.
Convex mirror.
Let's look at several examples of constructing images in convex mirrors.
The focus of a convex mirror is imaginary. Convex mirror formula
.
The sign rule for d and f remains the same as for a concave mirror.
The linear magnification of an object is determined by the ratio of the height of the image to the height of the object itself
. (2.5)
Thus, regardless of the location of the object relative to the convex mirror, the image always turns out to be virtual, straight, reduced and located behind the mirror. While the images in a concave mirror are more varied, they depend on the location of the object relative to the mirror. Therefore, concave mirrors are used more often.
Having considered the principles of constructing images in various mirrors, we have come to understand the operation of such various instruments as astronomical telescopes and magnifying mirrors in cosmetic devices and medical practice, we are able to design some devices ourselves.
Specular reflection, diffuse reflection
Flat mirror.
The simplest optical system is a flat mirror. If a parallel beam of rays incident on a flat surface between two media remains parallel after reflection, then the reflection is called mirror, and the surface itself is called a plane mirror (Fig. 2.16).
If the reflecting surface is rough, then the reflection wrong and the light scatters, or diffusely reflected (Fig. 2.19)
Diffuse reflection is much more pleasing to the eye than reflection from smooth surfaces, called correct reflection.
Lenses.
Lenses, like mirrors, are optical systems, i.e. capable of changing the path of a light beam. Lenses can be different in shape: spherical, cylindrical. We will focus only on spherical lenses.
A transparent body bounded by two spherical surfaces is called lens.
Converging lenses . Focus A converging lens is the point at which rays parallel to the optical axis intersect after refraction in the lens. The focus of the converging lens is real. The focus lying on the main optical axis is called the main focus. Any lens has two main focuses: the front (from the side of the incident rays) and the back (from the side of the refracted rays). The plane in which the foci lie is called the focal plane. The focal plane is always perpendicular to the main optical axis and passes through the main focus. The distance from the center of the lens to the main focus is called the main focal length F (Fig. 2.21).
To construct images of any luminous point, one should trace the course of any two rays incident on the lens and refracted in it until they intersect (or intersect their continuation). The image of extended luminous objects is a collection of images of its individual points. The most convenient rays used in constructing images in lenses are the following characteristic rays:
Figure 2.25 demonstrates the construction of an image of point A of object AB.
In addition to the listed rays, when constructing images in thin lenses, rays parallel to any secondary optical axis are used. It should be borne in mind that rays incident on a collecting lens in a beam parallel to the secondary optical axis intersect the rear focal surface at the same point as the secondary axis.
Thin lens formula:
, (2.6)
where F is the focal length of the lens; D is the optical power of the lens; d is the distance from the object to the center of the lens; f is the distance from the center of the lens to the image. The sign rule will be the same as for a mirror: all distances to real points are considered positive, all distances to imaginary points are considered negative.
The linear magnification given by the lens is
, (2.7)
where H is the image height; h is the height of the object.
Diffusing Lenses . Rays incident on a diverging lens in a parallel beam diverge so that their extensions intersect at a point called imaginary focus.
Rules for the path of rays in a diverging lens:
2) the beam traveling along the optical axis does not change its direction.
Diverging lens formula:
(the rule of signs remains the same).
Figure 2.27 shows an example of imaging in diverging lenses.
Reflection of light- this is a phenomenon in which the incidence of light on the interface between two media MN part of the incident light flux, having changed the direction of its propagation, remains in the same medium. Incident beamA.O.– a ray showing the direction of light propagation. Reflected beamO.B.- a ray showing the direction of propagation of the reflected part of the light flux.
Angle of incidence– the angle between the incident beam and the perpendicular to the reflecting surface.
Reflection angle 
- the angle between the reflected beam and the perpendicular to the interface at the point of incidence of the beam.
The law of light reflection: 1) the incident and reflected rays lie in the same plane with the perpendicular established at the point of incidence of the ray to the interface between the two media; 2) the angle of reflection is equal to the angle of incidence.
A mirror whose surface is a plane is called a plane mirror. Mirror reflection- This is a directional reflection of light.
If the interface between media is a surface whose uneven dimensions are greater than the wavelength of light incident on it, then mutually parallel light rays incident on such a surface do not retain their parallelism after reflection, but are scattered in all possible directions. This reflection of light is called absent-minded or diffuse.
Real Image- this is the image that is obtained when the rays intersect.
Virtual image- this is the image that is obtained by continuing the rays.
Construction of images in spherical mirrors.
Spherical mirror MK called the surface of a spherical segment that specularly reflects light. If light is reflected from the inner surface of a segment, then the mirror is called concave, and if from the outer surface of the segment – convex. A concave mirror is collecting and convex - scattering.
Center of the sphere C, from which a spherical segment is cut to form a mirror is called optical center of the mirror, and the vertex of the spherical segment O- his pole; R – radius of curvature of a spherical mirror.
Any straight line passing through the optical center of a mirror is called optical axis(KC; M.C.). The optical axis passing through the pole of the mirror is called main optical axis (O.C.). Rays coming near the main optical axis are called paraxial.
Full stop F, in which paraxial rays intersect after reflection and incident on a spherical mirror parallel to the main optical axis are called main focus.
The distance from the pole to the main focus of a spherical mirror is called focalOF.
Any ray incident along one of its optical axes is reflected from the mirror along the same axis.
Formula for a concave spherical mirror:
, Where d– distance from the object to the mirror (m), f– distance from the mirror to the image (m).
Formula for the focal length of a spherical mirror:
or 
The value D, the reciprocal of the focal length F of a spherical mirror, is called optical power.
/diopter/.
The optical power of a concave mirror is positive, while that of a convex mirror is negative.
The linear magnification Г of a spherical mirror is the ratio of the size of the image it creates H to the size of the imaged object h, i.e. 
.