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Lesson type : presentation of new material, consolidation of knowledge and skills.

Technology: informational - developing, developing problem - search, personality - oriented.

Textbook. Myakishev G.Ya., Bukhovtsev B.B., Charugin V.M.- M., "Education", 2008

Equipment: a computer, a multimedia projector, a screen, electronic educational publications, a mobile device for connecting to the Internet, equipment for obtaining an image of an object using a collecting lens.

Target: — to study the actions of collecting and diffusing lenses;

- to familiarize students with obtaining images with lenses.

Tasks . -Educational: to provide students with an idea of ​​the path of rays in lenses and methods of constructing images in them.

— Developing: to develop students' creative and imaginative thinking, the ability to independently solve logical problems, find non-standard methods of solution, creative activity and cognitive interest;

-Educational: development of cognitive interest in the study of physical phenomena and education of information culture; to learn to argue their versions and choose one of all the proposed versions - the optimal one, to continue the formation of a sense of duty and responsibility for their own results in studies.

The result of the formation of cognitive universal educational actions will be the skills:

• Give examples of experiments proving the analogy of light refraction at a flat and spherical interface between two media.
• Provide examples of experiments that substantiate scientific ideas.
• Put forward, on the basis of observations and constructions, hypotheses about the relationship of the characteristics of images from the distance of the object to the lens.
• Know the purpose of the collecting lens.
• Draw conclusions based on experimental data.
• Explain the essence of the content of the supporting synopsis.
• To be able to conduct analogs of the ray path in a prism and a collecting lens.
• Illustrate the role of physics in the creation and improvement of the most important technical objects using lenses: planetariums, observatories, multimedia projectors, cameras, military equipment.
• Know the areas of application of the lenses.

Didactic means : presentation, attachments with handouts, cards with assignments, EOR.

Lesson plan.

Time	Lesson steps	Teacher activities	Students' activities
1’	I. Phase of knowledge update	Conversation of the teacher, preparation of the basic synopsis ( Annex 1) and worksheets ( Appendix 2).	Lesson preparation
5’	II. Basic repetition. Working with cards.	Organization of repetition for assimilation of new material in the form of a test. I will show on the screen a slide with questions and answer options ( Appendix 3).	Fill in item 1 ( Appendices 2 a) worksheet with correct answers (no corrections allowed);
7’	III.Knowledge Update Phase	The teacher informs about the forthcoming study of the use of light refraction at the spherical interface of two media - in the lens. The topic of the lesson is called. Defining the goals and objectives of the lesson	Listen to and find theory on the topic of the lesson in the supporting outline
		Frontal poll: What is a lens? What kind of lenses are there? Where are lenses used? Which lens is called a converging lens and which lens is called a diverging lens? What is the purpose of a collecting lens?	Answer questions using the reference notes (appendix)
		The problem is posed. How does the light inside the converging and diffusing lens behave? Show animation, and then a real physics experiment (movie1).	Hypotheses are put forward. Watching a film and commenting on the experiment. They independently make a conclusion about the direction of the beam displacement in the prism.
15’	IV. Learning new material	The concept of a thin lens is given (see the figure in the synopsis). The main characteristics of the lens are introduced. Shown alternately movies 2 and 3... with a parallel explanation. A detailed examination of the construction of the image in the lens using "comfortable" rays. Derivation of the formula for a thin lens. The concept of optical power of linear magnification is introduced.	They listen, watch a video, make notes in notebooks.
	Pinning a lesson stage	Frontal poll. (What is the name of the straight line passing through "O"? What is the main optical axis? What is the focus of the lens? Why is it called real? How many focuses does the lens have?)	Answer questions using notes.
10’	V. Consolidation of knowledge, abilities, skills.	Construction (on a board) of an image of an object in a collecting lens for the case when d> 2F (1). Shows the path of rays in a diffusing lens, draws attention to the legend, asks students to characterize the resulting image, writes on the board. 2 students are invited to the board to build images (case d< F и F>d> 0) A training task is given to everyone: in the card, build and characterize the image of the object in the collecting lens, if the object is between focus and double focus (2F	They listen, answer questions, draw conclusions, complete the task on the board, and the rest as an individual. the worksheet is filled in clause 2 ( App. 2) All independently carry out the construction in the card.
4’	Vi. Summing up the lesson.	Testing the assimilation of knowledge. Reflection. General discussion of the results of the work Conclusions. Worksheets are collected for verification.	Message from the teacher. Message from students. Check and submit the cards for verification.
3’	Vii. Homework. Homework information and instructions on how to complete it.	1. On the slide: G.Ya. Myakishev, B.B. Bukhovtsev, V.M. Charugin §§63 - 65, basic synopsis, homework on the card "Construction of the image in the lens"(Appendix 5); Preparation of presentations in the multimedia library. Approximate topics: 1. Achievements of physics in the creation of technical objects using lenses; 2. Optical devices (multimedia projectors, cameras, etc.). (For additional assessment) 2. An explanation of the homework.	Write down homework. They ask clarifying questions.

Appendix 4.

Answers to the test:	option	1	2	3	4	5
	I	WITH	WITH	V	A-2, B-3, C-1.	V
	II	A	A	WITH	B	WITH

Software: Various programs and applications of the integrated MsOffice package were used to create the slides. In preparing the lesson, films from the collection of the creators of the site "Association of Teachers of St. Petersburg" www.eduspb.com were used .

List of electronic educational resources:

Student worksheetAppendix 2.

1. Answers to test questions.

option	1	2	3	4	5
I
II

1. Construct an image of an object AB in a collecting lens for cases 1 - 4.

Appendix 3.

Test	Test
Option 1	Option 2
1. When is the angle of refraction equal to the angle of incidence? Only when the refractive indices of the two media are the same B. Only when the incident beam is perpendicular to the interface between the two media. C. When the refractive indices of the two media are the same; the incident beam is perpendicular to the interface. 2. If the angle of incidence of the beam on the interface between two media increases, then the relative refractive index of these media: A. Increases. B. Decreases. C. Will not change. 3. When the beam passes into an optically denser medium, the angle of incidence is: A. Less angle of refraction. B. More angle of refraction. C. Equal to the angle of refraction. 4. Compare basic laws and formulas. A. The law of reflection. B. Absolute refractive index. C. Relative refractive index. 1. 2 . γ = α 3 ... n = V / s 5. A ray of light falls on the surface of the mirror at an angle of 30º to the horizon. What is the angle of reflection? A. 30 ° B. 60 ° C. 90 °	1. How does the apparent size of an object change in water? A. Increase B. Decrease. C. Do not change. 2. How does the limiting angle of reflection change at the interface between two media? water - air " with an increase in the angle of incidence? A. Will not change. B. Increases. C. Decreases. 3. When the beam passes into an optically less dense medium, the angle of refraction is: A. Less angle of incidence. B. Equal to the angle of incidence. C. More angle of incidence. 4. For a certain value α of the angle of incidence of the light beam on the interface between the two media, the ratio of the sine of the angle of incidence to the sine of the angle of refraction is equal to n. What is this ratio when the angle of incidence is doubled? A. n / 2 B. n C. 2n 5. Determine the angle of incidence of the beam on the surface of the mirror, if the beam is reflected at an angle of 15º to the horizon. A. 15 ° B. 65 ° C. 75 °

Abstract.Annex 1.

Lens Is a transparent solid enclosed by two spherical surfaces. In some cases, one surface of the lens may be flat.

Lens characteristics... Depending on the shape, a distinction is made between collecting (positive) and diffusing (negative) lenses. Collecting lenses - lenses with a center thicker than their edges. Diffusion lenses - lenses with edges that are thicker than the middle. Lenses are characterized, as a rule, by their optical power D and are expressed in diopters (diopters), or focal length. The reciprocal of the focal length is called the optical power of the lens:

To obtain an image of an object, it is necessary to build its individual points, and then connect them.

To construct images obtained with a converging lens, the focus and optical center of which are set, we will mainly use three types of "convenient" rays:

• a ray parallel to the main optical axis, refracted in the lens, passes through its focus.
• the ray going to the lens through its focus, after refraction, will be directed parallel to the main optical axis.
• the ray passing through the optical center of the lens does not change its direction.

Two of the three "convenient" rays can be used to build an image.

A formula connecting three quantities: the distance d from the object to the lens, the distance f from the image to the lens, and the focal length F.

If the lens is converging, then F> 0, and in the case of a diverging lens - F< 0. И еще, знак «плюс» означает, что изображение действительное , а знак «минус» — мнимо е. Изображение, получаемое с помощью линзы, отличается своими размерами от предмета. Различие размеров предмета и изображения характеризуют увеличением. Линейное увеличение линзы

Homework "Building the image in the lens"Appendix 5.

1. Construct an image given by a thin converging lens (select a scale for drawing a drawing in a notebook).
2. Determine the magnitude of the linear magnification of the lens: G = H / h, where H is the magnification, h is the size of the object.

The table for each option shows the corresponding values F(focal length) and d(distance from object to lens). Select the option you need from the table (do not cross out the table).

Sample questions when defending a job:

1. According to the task, calculate the optical power of the lens.
2. Formulate the basic rules for the propagation of rays through a thin lens used in the construction of images.

1. Laws of reflection and refraction of light.

2. Total internal reflection. Fiber optics.

3. Lenses. Optical power of the lens.

4. Lens aberrations.

5. Basic concepts and formulas.

6. Tasks.

When solving many problems related to the propagation of light, one can use the laws of geometric optics, based on the concept of a light ray as a line along which the energy of a light wave propagates. In a homogeneous environment, light rays are rectilinear. Geometric optics is the limiting case of wave optics as the wavelength tends to zero (λ →0).

23.1. The laws of reflection and refraction of light. Total internal reflection, light guides

Reflection laws

Light reflection- a phenomenon that occurs at the interface between two media, as a result of which the light beam changes its direction of propagation, remaining in the first medium. The nature of the reflection depends on the ratio between the dimensions (h) of the irregularities of the reflecting surface and the wavelength (λ) incident radiation.

Diffuse reflection

When irregularities are located chaotically, and their sizes are of the order of the wavelength or exceed it, there is diffuse reflection- light scattering in all possible directions. It is due to diffuse reflection that non-self-luminous bodies become visible when light is reflected from their surfaces.

Mirror reflection

If the dimensions of the irregularities are small compared to the wavelength (h<< λ), то возникает направленное, или mirror, reflection of light (fig.23.1). In this case, the following laws are observed.

The incident ray, the reflected ray and the normal to the interface between the two media drawn through the point of incidence of the ray lie in the same plane.

The angle of reflection is equal to the angle of incidence:β = a.

Rice. 23.1. Beam path with specular reflection

Refraction laws

When a light beam hits the interface between two transparent media, it is divided into two beams: reflected and refracted(fig.23.2). The refracted ray propagates in the second medium, changing its direction. The optical characteristic of the medium is absolute

Rice. 23.2. Refracted rays

refractive index, which is equal to the ratio of the speed of light in a vacuum to the speed of light in this medium:

The direction of the refracted ray depends on the ratio of the refractive indices of the two media. The following laws of refraction are fulfilled.

The incident ray, the refracted ray and the normal to the interface between the two media drawn through the point of incidence of the ray lie in the same plane.

The ratio of the sine of the angle of incidence to the sine of the angle of refraction is a constant value equal to the ratio of the absolute refractive indices of the second and first media:

23.2. Full internal reflection. Fiber optics

Let us consider the transition of light from a medium with a large refractive index n 1 (optically denser) to a medium with a lower refractive index n 2 (optically less dense). Figure 23.3 shows the rays incident on the glass-air interface. For glass, the refractive index n 1 = 1.52; for air n 2 = 1.00.

Rice. 23.3. Total internal reflection (n 1> n 2)

Increasing the angle of incidence increases the angle of refraction until the angle of refraction is equal to 90 °. With a further increase in the angle of incidence, the incident beam is not refracted, but fully reflected from the interface. This phenomenon is called complete internal reflection. It is observed when light falls from a denser medium onto the border with a less dense medium and consists of the following.

If the angle of incidence exceeds the limiting angle for these media, then refraction at the interface does not occur and the incident light is completely reflected.

The limiting angle of incidence is determined by the ratio

The sum of the intensities of the reflected and refracted rays is equal to the intensity of the incident ray. With an increase in the angle of incidence, the intensity of the reflected beam increases, while the intensity of the refracted beam decreases and becomes equal to zero for the limiting angle of incidence.

Fiber optics

The phenomenon of total internal reflection is used in flexible optical fibers.

If the light is directed to the end of a thin glass fiber surrounded by a cladding with a lower refractive index of the angle, then the light will propagate along the fiber, experiencing full reflection at the glass-cladding interface. This fiber is called light guide. Bends in the light guide do not obstruct the passage of light

In modern optical fibers, the loss of light due to its absorption is very small (about 10% per km), which makes it possible to use them in fiber-optic communication systems. In medicine, bundles of thin light guides are used to make endoscopes, which are used for visual examination of hollow internal organs (Fig. 23.5). The number of fibers in an endoscope reaches a million.

With the help of a separate light guide channel, laid in a common harness, laser radiation is transmitted for the purpose of therapeutic effects on internal organs.

Rice. 23.4. Propagation of light beams along the light guide

Rice. 23.5. Endoscope

There are also natural light guides. For example, in herbaceous plants, the stem plays the role of a light guide, supplying light to the underground part of the plant. The stem cells form parallel columns, which resembles the design of industrial fibers. If

illuminate such a column, examining it through a microscope, it is clear that its walls remain dark, and the inside of each cell is brightly lit. The depth to which the light is delivered in this way does not exceed 4-5 cm. But even such a short light guide is enough to provide light to the underground part of the herbaceous plant.

23.3. Lenses. Optical power of the lens

Lens - a transparent body, usually bounded by two spherical surfaces, each of which can be convex or concave. The straight line passing through the centers of these spheres is called main optical axis of the lens(word home usually omitted).

A lens, the maximum thickness of which is significantly less than the radii of both spherical surfaces, is called thin.

Passing through the lens, the light beam changes direction - it deflects. If the deviation occurs to the side optical axis, then the lens is called collecting, otherwise the lens is called scattering.

Any ray incident on a converging lens parallel to the optical axis, after refraction, passes through a point on the optical axis (F), called main focus(fig. 23.6, a). For a diffusing lens through the focus passes continuation refracted ray (Fig. 23.6, b).

Each lens has two focuses located on either side. The distance from the focus to the center of the lens is called main focal length(f).

Rice. 23.6. Focus of collecting (a) and diffusing (b) lenses

In the calculation formulas, f is taken with the "+" sign for collecting lenses and with a "-" sign for scattering lenses.

The reciprocal of the focal length is called lens power: D = 1 / f. Optical power unit - diopter(diopters). 1 diopters is the optical power of a lens with a focal length of 1 m.

Optical power thin lens and its focal length depend on the radii of the spheres and the refractive index of the lens material relative to the environment:

where R 1, R 2 are the radii of curvature of the lens surfaces; n is the refractive index of the lens material relative to the environment; the "+" sign is taken for convex surface, and the "-" sign - for concave. One of the surfaces can be flat. In this case, take R = ∞ , 1 / R = 0.

Lenses are used to take images. Consider an object located perpendicular to the optical axis of the collecting lens, and construct an image of its upper point A. The image of the entire object will also be perpendicular to the lens axis. Depending on the position of the object relative to the lens, two cases of ray refraction are possible, shown in Fig. 23.7.

1. If the distance from the object to the lens exceeds the focal length f, then the rays emitted by point A, after passing through the lens intersect at point A ", which is called a valid image. The actual image is obtained inverted.

2. If the distance from the object to the lens is less than the focal length f, then the rays emitted by point A, after passing through the lens races

Rice. 23.7. Real (a) and imaginary (b) images given by the collecting lens

walk and at point A "their extensions intersect. This point is called imaginary image. The virtual image is obtained direct.

The scattering lens gives a virtual image of the object in all its positions (Fig. 23.8).

Rice. 23.8. Ghost image from a diffusing lens

To calculate the image, use lens formula, which establishes the link between the provisions points and her Images

where f is the focal length (for a diffusing lens it is negative), a 1 - the distance from the object to the lens; a 2 is the distance from the image to the lens (the “+” sign is taken for a real image, and the “-” sign - for a virtual image).

Rice. 23.9. Lens formula parameters

The ratio of the size of the image to the size of the object is called linear increase:

The linear increase is calculated by the formula k = a 2 / a 1. Lens (even thin) will give the "correct" image obeying lens formula, only under the following conditions:

The refractive index of the lens is independent of the wavelength of light or the light is sufficient monochromatic.

When taking images with the lens real subjects, these restrictions, as a rule, are not met: there is variance; some points of the object lie to the side of the optical axis; incident light beams are not paraxial, the lens is not thin. All this leads to distortion images. To reduce distortion, the lenses of optical devices are made from several lenses located close to each other. The optical power of such a lens is equal to the sum of the optical powers of the lenses:

23.4. Lens aberrations

Aberrations is a general name for image errors that occur when using lenses. Aberrations (from lat. "aberratio"- deviation), which appear only in non-monochromatic light, are called chromatic. All other types of aberrations are monochromatic, since their manifestation is not associated with the complex spectral composition of real light.

1. Spherical aberration- monochromatic aberration due to the fact that the extreme (peripheral) parts of the lens deflect the rays coming from a point source more strongly than its central part. As a result, the peripheral and central areas of the lens form different images (S 2 and S "2, respectively) of the point source S 1 (Fig. 23.10). Therefore, at any position of the screen, the image on it is obtained in the form of a light spot.

This type of aberration is eliminated by using concave and convex lens systems.

Rice. 23.10. Spherical aberration

2. Astigmatism- monochromatic aberration, consisting in the fact that the image of a point has the form of an elliptical spot, which at some positions of the image plane degenerates into a segment.

Astigmatism of oblique beams appears when the rays emanating from a point make significant angles with the optical axis. In Figure 23.11, and the point source is located on the secondary optical axis. In this case, two images appear in the form of straight line segments located perpendicular to each other in planes I and II. The source image can be obtained only in the form of a blurry spot between planes I and II.

Astigmatism due to asymmetry optical system. This type of astigmatism occurs when the symmetry of the optical system with respect to the light beam is broken due to the structure of the system itself. With this aberration, the lenses create an image in which contours and lines oriented in different directions have different sharpness. This is observed in cylindrical lenses (Fig. 23.11, b).

The cylindrical lens forms a line image of a point object.

Rice. 23.11. Astigmatism: oblique beams (a); due to the cylindricity of the lens (b)

In the eye, astigmatism is formed with asymmetry in the curvature of the lens and cornea systems. To correct astigmatism, glasses are used that have different curvatures in different directions.

3. Distortion(distortion). When the rays emitted by the object make a large angle with the optical axis, another species is detected. monochromatic aberrations - distortion. In this case, the geometric similarity between the object and the image is violated. The reason is that in reality the linear magnification given by the lens depends on the angle of incidence of the rays. As a result, the image of the square grid takes either pillow-, or barrel-shaped view (fig.23.12).

To combat distortion, a lens system with opposite distortion is selected.

Rice. 23.12. Distortion: a - pincushion, b - barrel-shaped

4. Chromatic aberration manifests itself in the fact that a beam of white light emanating from a point gives its image in the form of a rainbow circle, violet rays intersect closer to the lens than red ones (Fig. 23.13).

The cause of chromatic aberration is the dependence of the refractive index of a substance on the wavelength of the incident light (dispersion). To correct this aberration in optics, lenses made of glasses with different dispersions (achromats, apochromats) are used.

Rice. 23.13. Chromatic aberration

23.5. Basic concepts and formulas

Table continuation

End of the table

23.6. Tasks

1. Why do air bubbles in water shine?

Answer: due to the reflection of light at the water-air interface.

2. Why does the spoon appear enlarged in a thin-walled glass of water?

Answer: the water in the glass acts as a cylindrical collecting lens. We see a virtual enlarged image.

3. The optical power of the lens is 3 diopters. What is the focal length of the lens? Express answer in cm.

Solution

D = 1 / f, f = 1 / D = 1/3 = 0.33 m. Answer: f = 33 cm.

4. The focal lengths of the two lenses are equal, respectively: f = +40 cm, f 2 = -40 cm. Find their optical powers.

6. How can the focal length of a collecting lens be determined in clear weather?

Solution

The distance from the Sun to the Earth is so great that all the rays falling on the lens are parallel to each other. If you get an image of the Sun on the screen, then the distance from the lens to the screen will be equal to the focal length.

7. For a lens with a focal length of 20 cm, find the distance to the object at which the linear size of the actual image will be: a) twice the size of the object; b) equal to the size of the object; c) half the size of the object.

8. The optical power of the lens for a person with normal vision is 25 diopters. Refractive index 1.4. Calculate the radii of curvature of the lens if it is known that one radius of curvature is 2 times the other.

Construction of images obtained with the help of lenses Objectives: to form practical skills to apply knowledge about the properties of lenses to find images by graphic method; Learn to build the path of rays in lenses, analyze images obtained with lenses.

A lens is a transparent body bounded by two curved (most often spherical) or curved and flat surfaces. A lens is a transparent body bounded by two curved (most often spherical) or curved and flat surfaces. The first mention of lenses can be found in the ancient Greek play "Clouds" by Aristophanes (424 BC), where fire was made with the help of convex glass and sunlight. A lens (German Linse, from Latin lentils) is usually a disc made of a transparent homogeneous material, limited by two polished surfaces, spherical or flat .. What is a lens?

The main elements of the lens MAIN OPTICAL AXIS - a straight line passing through the centers of the spherical surface of the lens OPTICAL CENTER - intersection of the main optical axis with the lens Side optical axis - any straight line passing through the optical center Main optical axis Side optical axis О О - optical center

If a beam of rays parallel to the main optical axis is incident on the converging lens, then after refraction in the lens they are collected in one optical axis, then after refraction in the lens they are collected at one point F, which is called the main focus of the lens. which before refraction were parallel to its main optical axis. The focus of the diffusing lens is imaginary. There are two main focuses; they are located on the main optical axis at the same distance from the optical center of the lens on different sides. What is a focus lens? F- focus of the lens optical center of the lens main optical axis of the lens

Rule To obtain an image of any point on an object, you need to use TWO "wonderful" beams: 1. A beam passing through the center of the lens. It never refracts, always straight 2. A ray parallel to the main optical axis. After the lens, it will definitely go through the focus

Construction of the image Construction of the image F F We draw a lens, the main optical axis, Subject AB, The first ray is drawn from point A through the optical center of the lens, it is not refracted! The second ray is conducted from the same point A parallel to the main optical axis, it is refracted and always passes through the focus of the lens. At the intersection of these two rays we get an image of point A A B From point A1 we draw a perpendicular to the main optical axis. A1B1 is an image of the object AB A1 B1

The collecting lens of the object is in double focus The collecting lens of the object is behind the double focus A We draw two "wonderful" beams from point A and obtain its image. Using two beams we also obtain an image of point B. Connecting the obtained points, we obtain an image of the object.

Collecting lens Collecting lens A Conduct two "wonderful" beams from point A and obtain its image. Using two beams, we obtain an image of point B. Connecting the obtained points, we obtain an image of an object. between focus and double focus

Collecting lens A Conduct two "wonderful" rays from point A In the same way we obtain an image of point B Connecting the obtained points, we obtain an image of an object Image of an object: enlarged, direct, imaginary FF A В В В the object is between the focus and the lens What to do? and the rays are spreading! We continue the rays after the lens in the opposite direction At the intersection of the imaginary rays we get an image of point A

Diffusing lens A Conduct a ray from point A through the center of the lens, it will not be refracted Similarly, we obtain an image of point B Connecting the obtained points, we obtain an image of an object The image of an object is always imaginary, reduced, straight B FFA B Draw a ray from point A parallel to the axis, it will be refracted so, that its imaginary continuation will pass through the focus At the intersection of two rays we get an image of point A

The collecting lens used as a magnifying glass gives ... 1. real enlarged image actual enlarged image actual enlarged image 2. actual thumbnail image actual thumbnail image valid thumbnail image 3. virtual enlarged image virtual enlarged image image enlarged image 4. virtual thumbnail image virtual thumbnail image thumbnail image Question 1. Question 2

With the help of a lens, an inverted image of a candle flame is obtained on the screen. How will the image be resized if part of the lens is obscured by a sheet of paper? 1.part of the image will disappear; part of the image will disappear 2.the dimensions of the image will not change; the dimensions of the image will not change; 3.size will increase; dimensions will increase; 4.Size down. Down in size. Question 2. Question 3

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22

The use of lenses. The use of lenses. Lenses are a versatile optical element in most optical systems. Lenses are a versatile optical element in most optical systems. Biconvex lenses are used in most optical instruments, the same lens is the lens of the eye. Biconvex lenses are used in most optical instruments, the same lens is the lens of the eye. Lenses - menisci are widely used in glasses and contact lenses. In a converging beam behind a collecting lens, light energy is concentrated at the focus of the lens. Burning with a magnifying glass is based on this principle. Lenses - menisci are widely used in glasses and contact lenses. In a converging beam behind a collecting lens, light energy is concentrated at the focus of the lens. Burning with a magnifying glass is based on this principle.

The section of optics, in which the laws of propagation of light are considered based on the concept of light rays, is called geometric optics... Under light rays we mean the lines normal to the wave surfaces along which the flux of light energy propagates. Geometric optics, while remaining an approximate method for constructing images in optical systems, makes it possible to analyze the main phenomena associated with the passage of light through them, and therefore is the basis of the theory of optical devices.

Lenses are transparent bodies bounded by two surfaces (one of them is usually spherical, sometimes cylindrical, and the other is spherical or flat) that refract light rays capable of forming optical images of objects. The material for the lenses is glass, quartz, crystals, plastics, etc. According to their external shape (Fig. 232), lenses are divided into: 1) biconvex; 2) flat-convex; 3) biconcave; 4) flat-concave; 5) convex-concave; 6) concave-convex.

By optical properties lenses are divided into collecting and scattering.

The lens is called thin if its thickness (the distance between the bounding surfaces) is significantly less than the radii of the surfaces bounding the lens. The straight line passing through the centers of curvature of the lens surfaces is called main optical axis... For every lens, there is a point called optical center of the lens, lying on the main optical axis and having the property that the rays pass through it without refraction. For simplicity, the optical center O the lens will be assumed to coincide with the geometric center of the middle part of the lens (this is true only for biconvex and biconcave lenses with the same radii of curvature of both surfaces; for plano-convex and plano-concave lenses, the optical center O lies at the intersection of the main optical axis with the spherical surface).

To derive the formula for a thin lens - the relationship between the radii of curvature R 1 and R 2 lens surfaces with distances a and b from the lens to the object and its image, we will use Fermat's principle(P. Fermat (1601-1665) - French mathematician and physicist), or the principle of least time: the actual path of propagation of light (the path of the light beam) is the path for which the light takes the shortest time compared to any other conceivable path between the same points.

Consider two trajectories of a light beam (Fig. 233) - a straight line connecting the points A and V(Ray AOB), and the trajectory passing through the edge of the lens (ray ASV), - using the condition of equality of the transit time of light along these trajectories.

Light travel time along the trajectory AOB

where N = n/n 1 - relative refractive index ( n and n 1 - respectively, the absolute refractive indices of the lens and the environment). Light travel time along the trajectory ASV equals

Since =, then

Consider paraxial (paraxial) rays, i.e., rays forming small angles with the optical axis. Only for paraxial rays it turns out stigmatic portrayal, i.e., all rays of the paraxial beam emanating from the point A, intersect the optical axis at the same point V... Then<< (a+e), << (b+d) and

Likewise,

Substituting the found expressions into (166.1), we obtain

(166.2)

For thin lens e<< a and d << b, therefore (166.2) can be represented as

Taking into account that and, respectively, we obtain

(166.3)

Expression (166.3) is thin lens formula... The radius of curvature of the convex surface of the lens is considered positive, the concave - negative.

If a=, that is, the rays fall on the lens in a parallel beam (Fig. 234. a), then

The corresponding distance b= OF = f called focal length of the lens:

It depends on the relative refractive index and radii of curvature.

If b=, i.e. the image is at infinity and, therefore, the rays leave the lens in a parallel beam (Fig. 234, b), then a= OF = f... Thus, the focal lengths of a lens surrounded on both sides by the same environment. are equal. Points F lying on both sides of the lens at a distance equal to the focal distance are called focus lenses... The focus is the point at which, after refraction, all rays incident on the lens parallel to the main optical axis are collected. The magnitude

(166.4)

called lens power... Its unit is diopter (diopter). Diopter- the optical power of a lens with a focal length of 1 m: 1 diopters = 1 / m.

Lenses with positive optical power are collecting, With negative - scattering... The planes passing through the foci of the lens perpendicular to its main optical axis are called focal planes... Unlike a converging lens, a diverging lens has imaginary foci. In an imaginary focus (after refraction), imaginary extensions of rays incident on a scattering lens parallel to the main optical axis (Fig. 235) converge.

Taking into account (166.4), the lens formula (166.3) can be written in the form

For diffusing lens distances f and b should be considered negative.

The construction of an image of an object in lenses is carried out using the following rays:

1) a ray passing through the optical center of the lens and not changing its direction;

2) a beam running parallel to the main optical axis; after refraction in the lens, this ray (or its extension) passes through the second focus of the lens;

3) a ray (or its extension) passing through the first focus of the lens; after refraction in it, it leaves the lens parallel to its main optical axis.

For example, the construction of images in the collecting (Fig. 236) and in the scattering (Fig. 237) lenses are given: real (Fig. 236, a) and imaginary (Fig. 236, b) images - in a collecting lens, imaginary - in a scattering lens.

The ratio of the linear dimensions of the image and the object is called linear lens magnification... Negative ramp magnification values ​​correspond to the actual image (it is inverted), and positive values ​​correspond to the virtual image (it is upright). Combinations of collecting and diffusing lenses are used in optical devices used for solving various scientific and technical problems.

THIN LENSES

Objective: master the method of obtaining images using lenses, learn how to determine the focal length of lenses.

Questions you must know

for admission to work:

1. What is a lens?

2. What are Thin Lenses?

3. What are point source, optical center of lens, main and secondary optical axes, focus, focal plane and focal length?

4. Collecting and diffusing lenses.

5. Real and imaginary images of an object.

6. What rays are called paraxial?

7. Formula of a thin lens.

8. Lens magnification.

9. Optical power of lenses.

10. Basic laws of geometric optics.

11. Construction of images in collecting and diffusing lenses for various cases of the location of the object relative to the lens. For each case, answer the following questions:

a) Where will the image be?

b) Will the image be real or imaginary, how to observe it?

c) Will it be enlarged, reduced or in full size?

d) Will it be inverted or not?

INTRODUCTION

A lens is a transparent body bounded by two curved (usually spherical) surfaces or one curved and one flat surface. If the thickness of the lens itself is small compared to the radii of curvature of the refracting surfaces, then the lens is called thin .

The straight line passing through the centers of curvature O 1 and O 2 of the refracting surfaces is called the main optical axis of the lens (Fig. 1). In the case of thin lenses, it can be approximately assumed that the main optical axis intersects with the lens at one point, which is commonly called the optical center of the lens O.

All straight lines passing through the optical center are called secondary (auxiliary) optical axes. .

Distances measured from the center of the lens in the direction of the beam (to the right of the point O if the light source S is on the left), we will consider positive, and against the path of the light beam (to the left of the point O) - negative. So in fig. 1 radius R 1> 0, and R 2< 0.

If the source S 1 is far to the left of the collecting lens, i.e., the beam of rays falls on the lens parallel to the main optical axis (Fig. 2, a), then it is known from experience that the rays will cross the optical axis at a distance a 2 behind the lens. The corresponding distance a 2 = OF 2 = f 2 is called the focal length of the lens, and the point F 2- back focus .

If the parallel beam goes to the right, we get f 1 = –f 2 corresponding point F 1 called the front focus (Fig. 2, c). Please note that for a thin lens | f 1 | = | f 2 | ≡ f if the medium is the same on both sides of the lens.

If the beam after refraction turns out to be diverging, then the point where the imaginary extensions of rays incident parallel to the main optical axis converge (after refraction) is called an imaginary focus (Fig. 2, b).

Thus, the focus lens is the point at which, after refraction, all rays (or their imaginary extensions) are collected, incident on the lens parallel to the main optical axis.

Planes V 1 and V 2(Fig. 3), passing through the foci perpendicular to the main (main) optical axis, are called the focal planes of the lens.

If the light beam falls parallel to the main optical axis, then the rays are collected in the main foci, but if the light beam falls parallel to the side axis, then the rays are collected in the side foci located on the focal planes of the lens (Fig. 3).

Let's denote the distance from the light source S 1 to the optical center of the lens - a 1, a 2 Is the distance from the optical center of the lens to the source image (Fig. 4). In the drawing a 2> 0, a a 1 < 0 и R < 0, так как эти расстояния отсчитываются влево от линзы. Проводя аналитическое решение можно показать, что расстояния a 2 and a 1 are related to the radii of curvature of the lens in the air by the following relationship:

where f- the focal length of the lens, that is, the distance from the focus to the optical center of the lens; n l Is the refractive index of the lens material.

This ratio is called the thin lens formula. It follows from this formula that a 2 does not depend on angles β , i.e., all rays that emerged from S 1 at different angles, gather at the same distance a 2 from the interface (at point S 2).

This is true for rays emanating from a point S 1 at small angles β < 10° (such rays are called paraxial) to the optical axis, passing through the lens, the rays are refracted twice on spherical surfaces and are collected at one point S 2 also located on the optical axis and called the point image S 1(fig. 4).

Formula (1) can be written as:

The magnitude D is called the optical power of the lens and in the SI system is measured in diopters (or m –1 ). The diopter is equal to the optical power of a lens with a focal length of one meter. It can be positive or negative.

Lenses with meaning D> 0 are called collecting, since they collect a parallel beam to a point, and with D < 0 – рассеивающими.

For the convenience of constructing the path of rays in thin lenses in the drawings, the lenses themselves are depicted as follows: a- collecting lens, b- scattering (Fig. 5). The diffusing lens has imaginary focuses.

This means that for her back focus F 2 located on the left, and the front F 1- on right. It forms only a virtual thumbnail image.

The image of the object, given by the lens, can be obtained directly by geometric construction, using the property of the following rays (Fig. 6):

· The ray passing through the optical center of the lens is not refracted, ray (1);

· A ray falling on the lens parallel to the optical axis after refraction passes through the focus, ray (2);

· The ray passing through the front focus, after refraction, is parallel to the optical axis, ray (3).

If the beam from the source goes at a certain angle to the main optical axis, then it is necessary to construct a side axis and find a side focus, the refracted beam will pass through this focus (Fig. 7).

Consider the construction of an image in a thin converging lens (Fig. 6).

Moreover, if the image is formed directly by refracted rays, it is called real, and if their imaginary extensions of the rays, then imaginary.

The ratio of the linear dimensions of the image and the original object is called linear or transverse magnification β, is determined by the following relationship (Fig. 6):

Linear magnification is an algebraic quantity. It is positive if the image is direct, that is, it is oriented in the same way as the object itself, and negative if the image is reversed.