Converging and diverging lenses. lenses

Lesson development (lesson notes)

Line UMK A. V. Peryshkin. Physics (7-9)

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Lesson Objectives:

• find out what a lens is, classify them, introduce the concepts: focus, focal length, optical power, linear magnification;
• continue to develop skills to solve problems on the topic.

During the classes

I sing praise before you in delight
Not expensive stones, nor gold, but GLASS.

M.V. Lomonosov

Within the framework of this topic, we recall what a lens is; consider the general principles of imaging in a thin lens, and also derive a formula for a thin lens.

Previously, we got acquainted with the refraction of light, and also derived the law of refraction of light.

Checking homework

1) survey § 65

2) frontal survey (see presentation)

1. Which of the figures correctly shows the course of a beam passing through a glass plate in the air?

2. In which of the following figures is the image correctly constructed in a vertically positioned flat mirror?

3. A beam of light passes from glass into air, refracting at the interface between two media. Which of the directions 1-4 corresponds to the refracted beam?

4. A kitten runs towards a flat mirror at a speed V= 0.3 m/s. The mirror itself moves away from the kitten at a speed u= 0.05 m/s. With what speed does the kitten approach its image in the mirror?

Learning new material

In general, the word lens- This is a Latin word that translates as lentils. Lentils are a plant whose fruits are very similar to peas, but the peas are not round, but have the appearance of pot-bellied cakes. Therefore, all round glasses having such a shape began to be called lenses.

The first mention of lenses can be found in the ancient Greek play "Clouds" by Aristophanes (424 BC), where fire was made using convex glass and sunlight. And the age of the oldest of the discovered lenses is more than 3000 years. This so-called lens Nimrud. It was found during the excavations of one of the ancient capitals of Assyria in Nimrud by Austin Henry Layard in 1853. The lens has a shape close to an oval, roughly polished, one of the sides is convex and the other is flat. Currently, it is stored in the British Museum - the main historical and archaeological museum in Great Britain.

Lens of Nimrud

So, in the modern sense, lenses are transparent bodies bounded by two spherical surfaces . (write in notebook) Spherical lenses are most commonly used, in which the bounding surfaces are spheres or a sphere and a plane. Depending on the relative placement of spherical surfaces or spheres and planes, there are convex and concave lenses. (Children look at lenses from the Optics set)

In its turn convex lenses are divided into three types- flat convex, biconvex and concave-convex; a concave lenses are classified into flat-concave, biconcave and convex-concave.

(write down)

Any convex lens can be represented as a combination of a plane-parallel glass plate in the center of the lens and truncated prisms expanding towards the middle of the lens, and a concave lens can be represented as a combination of a plane-parallel glass plate in the center of the lens and truncated prisms expanding towards the edges.

It is known that if the prism is made of a material that is optically denser than the environment, then it will deflect the beam towards its base. Therefore, a parallel beam of light after refraction in a convex lens becomes convergent(these are called gathering), a in a concave lens conversely, a parallel beam of light after refraction becomes divergent(hence such lenses are called scattering).

For simplicity and convenience, we will consider lenses whose thickness is negligible compared to the radii of spherical surfaces. Such lenses are called thin lenses. And in the future, when we talk about a lens, we will always understand a thin lens.

The following technique is used to symbolize thin lenses: if the lens gathering, then it is denoted by a straight line with arrows at the ends directed from the center of the lens, and if the lens scattering, then the arrows are directed towards the center of the lens.

Conventional designation of a converging lens

Conventional designation of diverging lens

(write down)

Optical center of the lens is the point through which the rays do not experience refraction.

Any straight line passing through the optical center of the lens is called optical axis.

The optical axis, which passes through the centers of spherical surfaces that limit the lens, is called main optical axis.

The point at which the rays incident on the lens parallel to its main optical axis (or their continuation) intersect is called main focus of the lens. It should be remembered that any lens has two main focuses - front and rear, because. it refracts light falling on it from two directions. And both of these foci are located symmetrically with respect to the optical center of the lens.

converging lens

(draw)

diverging lens

(draw)

The distance from the optical center of a lens to its main focus is called focal length.

focal plane is a plane perpendicular to the main optical axis of the lens, passing through its main focus.
The value equal to the reciprocal focal length of the lens, expressed in meters, is called optical power of the lens. It is denoted by a capital Latin letter D and measured in diopters(abbreviated diopter).

(Record)

For the first time, the thin lens formula we obtained was derived by Johannes Kepler in 1604. He studied the refraction of light at small angles of incidence in lenses of various configurations.

Linear magnification of the lens is the ratio of the linear size of the image to the linear size of the object. It is denoted by a capital Greek letter G.

Problem solving(at the blackboard) :

• Str 165 exercise 33 (1.2)
• The candle is located at a distance of 8 cm from a converging lens, the optical power of which is 10 diopters. At what distance from the lens will the image be obtained and what will it look like?
• At what distance from a lens with a focal length of 12 cm must an object be placed so that its real image is three times larger than the object itself?

At home: §§ 66 nos. 1584, 1612-1615 (Lukasik collection)

1) Picture can be imaginary or valid. If the image is formed by the rays themselves (i.e., light energy enters a given point), then it is real, but if not by the rays themselves, but by their continuations, then they say that the image is imaginary (light energy does not enter the given point).

2) If the top and bottom of the image are oriented similarly to the object itself, then the image is called direct. If the image is upside down, then it is called reverse (inverted).

3) The image is characterized by the acquired dimensions: enlarged, reduced, equal.

Image in a flat mirror

The image in a flat mirror is imaginary, straight, equal in size to the object, located at the same distance behind the mirror as the object is in front of the mirror.

lenses

The lens is a transparent body bounded on both sides by curved surfaces.

There are six types of lenses.

Collecting: 1 - biconvex, 2 - flat-convex, 3 - convex-concave. Scattering: 4 - biconcave; 5 - plano-concave; 6 - concave-convex.

converging lens

diverging lens

Lens characteristics.

NN- the main optical axis - a straight line passing through the centers of spherical surfaces limiting the lens;

O- optical center - a point that, for biconvex or biconcave (with the same surface radii) lenses, is located on the optical axis inside the lens (in its center);

F- the main focus of the lens - the point at which a beam of light is collected, propagating parallel to the main optical axis;

OF- focal length;

N"N"- side axis of the lens;

F"- side focus;

Focal plane - a plane passing through the main focus perpendicular to the main optical axis.

The path of the rays in the lens.

The beam passing through the optical center of the lens (O) does not experience refraction.

A beam parallel to the main optical axis, after refraction, passes through the main focus (F).

The beam passing through the main focus (F), after refraction, goes parallel to the main optical axis.

A beam running parallel to the secondary optical axis (N"N") passes through the secondary focus (F").

lens formula.

When using the lens formula, you should correctly use the sign rule: +F- converging lens; -F- diverging lens; +d- the subject is valid; -d- an imaginary object; +f- the image of the subject is valid; -f- the image of the object is imaginary.

The reciprocal of the focal length of a lens is called optical power.

Transverse magnification- the ratio of the linear size of the image to the linear size of the object.

Modern optical devices use lens systems to improve image quality. The optical power of a system of lenses put together is equal to the sum of their optical powers.

1 - cornea; 2 - iris; 3 - albuginea (sclera); 4 - choroid; 5 - pigment layer; 6 - yellow spot; 7 - optic nerve; 8 - retina; 9 - muscle; 10 - ligaments of the lens; 11 - lens; 12 - pupil.

The lens is a lens-like body and adjusts our vision to different distances. In the optical system of the eye, focusing an image on the retina is called accommodation. In humans, accommodation occurs due to an increase in the convexity of the lens, carried out with the help of muscles. This changes the optical power of the eye.

The image of an object that falls on the retina is real, reduced, inverted.

The distance of best vision should be about 25 cm, and the limit of vision (far point) is at infinity.

Nearsightedness (myopia) A vision defect in which the eye sees blurry and the image is focused in front of the retina.

Farsightedness (hyperopia) A visual defect in which the image is focused behind the retina.

There are objects that are capable of changing the density of the electromagnetic radiation flux incident on them, that is, either increasing it by collecting it at one point, or decreasing it by scattering it. These objects are called lenses in physics. Let's consider this question in more detail.

What are lenses in physics?

This concept means absolutely any object that is capable of changing the direction of propagation of electromagnetic radiation. This is the general definition of lenses in physics, which includes optical glasses, magnetic and gravitational lenses.

In this article, the main attention will be paid to optical glasses, which are objects made of a transparent material and limited by two surfaces. One of these surfaces must necessarily have curvature (that is, be part of a sphere of finite radius), otherwise the object will not have the property of changing the direction of propagation of light rays.

The principle of the lens

The essence of this simple optical object is the phenomenon of refraction of sunlight. At the beginning of the 17th century, the famous Dutch physicist and astronomer Willebrord Snell van Rooyen published the law of refraction, which currently bears his last name. The formulation of this law is as follows: when sunlight passes through the interface between two optically transparent media, then the product of the sine between the beam and the normal to the surface and the refractive index of the medium in which it propagates is a constant value.

To clarify the above, we give an example: let the light fall on the surface of the water, while the angle between the normal to the surface and the beam is equal to θ 1 . Then, the light beam is refracted and begins its propagation in the water already at an angle θ 2 to the normal to the surface. According to Snell's law, we get: sin (θ 1) * n 1 \u003d sin (θ 2) * n 2, here n 1 and n 2 are the refractive indices for air and water, respectively. What is the refractive index? This is a value showing how many times the speed of propagation of electromagnetic waves in vacuum is greater than that for an optically transparent medium, that is, n = c/v, where c and v are the speeds of light in vacuum and in the medium, respectively.

The physics of refraction lies in the implementation of Fermat's principle, according to which light moves in such a way as to overcome the distance from one point to another in space in the shortest time.

The type of optical lens in physics is determined solely by the shape of the surfaces that form it. The direction of refraction of the beam incident on them depends on this shape. So, if the curvature of the surface is positive (convex), then, upon exiting the lens, the light beam will propagate closer to its optical axis (see below). Conversely, if the curvature of the surface is negative (concave), then passing through the optical glass, the beam will move away from its central axis.

We note again that the surface of any curvature refracts the rays in the same way (according to Stella's law), but the normals to them have a different slope relative to the optical axis, resulting in a different behavior of the refracted beam.

A lens bounded by two convex surfaces is called a converging lens. In turn, if it is formed by two surfaces with negative curvature, then it is called scattering. All other views are associated with a combination of the indicated surfaces, to which a plane is also added. What property the combined lens will have (diffusing or converging) depends on the total curvature of the radii of its surfaces.

Lens elements and ray properties

To build in lenses in imaging physics, it is necessary to get acquainted with the elements of this object. They are listed below:

• Main optical axis and center. In the first case, they mean a straight line passing perpendicular to the lens through its optical center. The latter, in turn, is a point inside the lens, passing through which the beam does not experience refraction.
• Focal length and focus - the distance between the center and a point on the optical axis, in which all rays incident on the lens parallel to this axis are collected. This definition is true for collecting optical glasses. In the case of divergent lenses, it is not the rays themselves that will converge to a point, but their imaginary continuation. This point is called the main focus.
• optical power. This is the name of the reciprocal of the focal length, that is, D \u003d 1 / f. It is measured in diopters (diopters), that is, 1 diopter. = 1 m -1.

The following are the main properties of rays that pass through a lens:

• the beam passing through the optical center does not change the direction of its movement;
• rays incident parallel to the main optical axis change their direction so that they pass through the main focus;
• rays falling on optical glass at any angle, but passing through its focus, change their direction of propagation in such a way that they become parallel to the main optical axis.

The above properties of rays for thin lenses in physics (as they are called, because it does not matter what spheres they are formed and how thick they have, only the optical properties of the object matter) are used to build images in them.

Images in optical glasses: how to build?

The figure below shows in detail the schemes for constructing images in the convex and concave lenses of an object (red arrow) depending on its position.

Important conclusions follow from the analysis of the circuits in the figure:

• Any image is built on only 2 rays (passing through the center and parallel to the main optical axis).
• Converging lenses (denoted with arrows at the ends pointing outward) can give both an enlarged and reduced image, which in turn can be real (real) or imaginary.
• If the object is in focus, then the lens does not form its image (see the lower diagram on the left in the figure).
• Scattering optical glasses (denoted by arrows at their ends pointing inward) always give a reduced and virtual image regardless of the position of the object.

Finding the distance to an image

To determine at what distance the image will appear, knowing the position of the object itself, we give the lens formula in physics: 1/f = 1/d o + 1/d i , where d o and d i are the distance to the object and to its image from the optical center, respectively, f is the main focus. If we are talking about a collecting optical glass, then the f-number will be positive. Conversely, for a diverging lens, f is negative.

Let's use this formula and solve a simple problem: let the object be at a distance d o = 2*f from the center of the collecting optical glass. Where will his image appear?

From the condition of the problem we have: 1/f = 1/(2*f)+1/d i . From: 1/d i = 1/f - 1/(2*f) = 1/(2*f), i.e. d i = 2*f. Thus, the image will appear at a distance of two foci from the lens, but on the other side than the object itself (this is indicated by the positive sign of the value d i).

Short story

It is curious to give the etymology of the word "lens". It comes from the Latin words lens and lentis, which means "lentil", since optical objects in their shape really look like the fruit of this plant.

The refractive power of spherical transparent bodies was known to the ancient Romans. For this purpose, they used round glass vessels filled with water. Glass lenses themselves began to be made only in the 13th century in Europe. They were used as a reading tool (modern glasses or a magnifying glass).

The active use of optical objects in the manufacture of telescopes and microscopes dates back to the 17th century (at the beginning of this century, Galileo invented the first telescope). Note that the mathematical formulation of Stella's law of refraction, without knowledge of which it is impossible to manufacture lenses with desired properties, was published by a Dutch scientist at the beginning of the same 17th century.

Other types of lenses

As noted above, in addition to optical refractive objects, there are also magnetic and gravitational objects. An example of the former are magnetic lenses in an electron microscope, a vivid example of the latter is the distortion of the direction of the light flux when it passes near massive cosmic bodies (stars, planets).

Definition 1

Lens is a transparent body having 2 spherical surfaces. It is thin if its thickness is less than the radii of curvature of spherical surfaces.

The lens is an integral part of almost every optical device. Lenses are, by their definition, collecting and scattering (Fig. 3.3.1).

Definition 2

converging lens is a lens that is thicker in the middle than at the edges.

Definition 3

A lens that is thicker at the edges is called scattering.

Figure 3. 3. one . Collecting (a) and diverging (b) lenses and their symbols.

Definition 4

Main optical axis is a straight line that passes through the centers of curvature O 1 and O 2 of spherical surfaces.

In a thin lens, the main optical axis intersects at one point - the optical center of the lens O. The light beam passes through the optical center of the lens without deviating from its original direction.

Definition 5

Side optical axes are straight lines passing through the optical center.

Definition 6

If a beam of rays is directed to the lens, which are parallel to the main optical axis, then after passing through the lens the rays (or their continuation) will be concentrated at one point F.

This point is called main focus of the lens.

A thin lens has two main foci, which are located symmetrically on the main optical axis with respect to the lens.

Definition 7

Focus of a converging lens valid, and for the scattering imaginary.

Beams of rays parallel to one of the entire set of secondary optical axes, after passing through the lens, are also aimed at the point F ", located at the intersection of the secondary axis with the focal plane Ф.

Definition 8

focal plane- this is a plane perpendicular to the main optical axis and passing through the main focus (Fig. 3.3.2).

Definition 9

The distance between the main focus F and the optical center of the lens O is called focal(F).

Figure 3. 3. 2. Refraction of a parallel beam of rays in a converging (a) and diverging (b) lens. O 1 and O 2 are the centers of spherical surfaces, O 1 O 2 is the main optical axis, O – optical center, F is the main focus, F" is the focus, O F" is the secondary optical axis, Ф is the focal plane.

The main property of lenses is the ability to transmit images of objects. They, in turn, are:

• Real and imaginary;
• Straight and inverted;
• Enlarged and reduced.

Geometric constructions help determine the position of the image, as well as its nature. For this purpose, the properties of standard rays are used, the direction of which is defined. These are rays that pass through the optical center or one of the foci of the lens, and rays that are parallel to the main or one of the side optical axes. Drawings 3 . 3. 3 and 3. 3. 4 show construction data.

Figure 3. 3. 3. Building an image in a converging lens.

Figure 3. 3. four . Building an image in a diverging lens.

It is worth highlighting that the standard beams used in Figures 3 . 3. 3 and 3. 3. 4 for imaging, do not pass through the lens. These rays are not used in imaging, but can be used in this process.

Definition 10

The thin lens formula is used to calculate image position and character. If we write the distance from the object to the lens as d, and from the lens to the image as f, then thin lens formula looks like:

1d + 1f + 1F = D.

Definition 11

Value D is the optical power of the lens, equal to the reciprocal focal length.

Definition 12

Diopter(d p t r) is a unit of optical power, the focal length of which is equal to 1 m: 1 d p t r = m - 1 .

The formula for a thin lens is similar to that for a spherical mirror. It can be derived for paraxial rays from the similarity of triangles in figures 3 . 3. 3 or 3 . 3. four .

The focal length of the lenses is written with certain signs: a converging lens F > 0, a diverging lens F< 0 .

The value of d and f also obey certain signs:

• d > 0 and f > 0 - in relation to real objects (that is, real light sources) and images;
• d< 0 и f < 0 – применительно к мнимым источникам и изображениям.

For the case in Figure 3 . 3. 3 F > 0 (converging lens), d = 3 F > 0 (real object).

From the thin lens formula we get: f = 3 2 F > 0 , means that the image is real.

For the case in Figure 3 . 3. 4F< 0 (линза рассеивающая), d = 2 | F | >0 (real object), the formula f = - 2 3 F< 0 , следовательно, изображение мнимое.

The linear dimensions of the image depend on the position of the object in relation to the lens.

Definition 13

Linear magnification of the lens G is the ratio of the linear dimensions of the image h "and the object h.

It is convenient to write the value h "with plus or minus signs, depending on whether it is direct or inverted. It is always positive. Therefore, for direct images, the condition Γ\u003e 0 is applied, for inverted Γ< 0 . Из подобия треугольников на рисунках 3 . 3 . 3 и 3 . 3 . 4 нетрудно вывести формулу для расчета линейного увеличения тонкой линзы:

G \u003d h "h \u003d - f d.

In the example with a converging lens in Figure 3. 3. 3 for d = 3 F > 0 , f = 3 2 F > 0 .

Hence, Г = - 1 2< 0 – изображение перевернутое и уменьшенное в два раза.

In the diverging lens example in Figure 3. 3. 4 for d = 2 | F | > 0 , the formula f = - 2 3 F< 0 ; значит, Г = 1 3 >0 - the image is straight and reduced by a factor of three.

The optical power D of the lens depends on the radii of curvature R 1 and R 2 , its spherical surfaces, and also on the refractive index n of the lens material. In the theory of optics, the following expression takes place:

D \u003d 1 F \u003d (n - 1) 1 R 1 + 1 R 2.

A convex surface has a positive radius of curvature, while a concave surface has a negative radius. This formula is applicable in the manufacture of lenses with a given optical power.

Many optical instruments are designed in such a way that light passes through 2 or more lenses in succession. The image of the object from the 1st lens serves as an object (real or imaginary) for the 2nd lens, which, in turn, builds the 2nd image of the object, which can also be real or imaginary. The calculation of the optical system of 2 thin lenses consists in
2-fold application of the lens formula, and the distance d 2 from the 1st image to the 2nd lens should be proposed equal to the value l - f 1, where l is the distance between the lenses.

The value f 2 calculated by the lens formula predetermines the position of the 2nd image, as well as its character (f 2 > 0 is a real image, f 2< 0 – мнимое). Общее линейное увеличение Γ системы из 2 -х линз равняется произведению линейных увеличений 2 -х линз, то есть Γ = Γ 1 · Γ 2 . Если предмет либо его изображение находятся в бесконечности, тогда линейное увеличение не имеет смысла.

Kepler's astronomical tube and Galileo's terrestrial tube

Let's consider a special case - a telescopic path of rays in a system of 2 lenses, when both the object and the 2nd image are located at infinitely large distances from each other. The telescopic path of the rays is carried out in the telescopes: Galileo's earthly tube and Kepler's astronomical tube.

A thin lens has some drawbacks that do not allow high resolution images to be obtained.

Definition 14

Aberration is the distortion that occurs during the imaging process. Depending on the distance at which the observation is made, aberrations can be spherical or chromatic.

The meaning of spherical aberration is that with wide light beams, rays that are at a far distance from the optical axis do not cross it at the focus. The thin lens formula only works for rays that are close to the optical axis. The image of a distant source, which is created by a wide beam of rays refracted by a lens, is blurry.

The meaning of chromatic aberration is that the refractive index of the lens material is affected by the light wavelength λ. This property of transparent media is called dispersion. The focal length of a lens is different for light with different wavelengths. This fact leads to blurring of the image when non-monochromatic light is emitted.

Modern optical devices are equipped not with thin lenses, but with complex lens systems in which it is possible to eliminate some distortion.

In devices such as cameras, projectors, etc., converging lenses are used to form real images of objects.

Camera- this is a closed light-tight camera in which the image of the captured objects is created on the film by a system of lenses - lens. During the exposure, the lens is opened and closed using a special shutter.

The peculiarity of the camera is that on a flat film, rather sharp images of objects that are at different distances are obtained. Sharpness changes as the lens moves relative to the film. Images of points that do not lie in the plane of sharp pointing come out blurry in the images in the form of scattered circles. The size d of these circles can be reduced by lens aperture, that is, by reducing the relative aperture a F , as shown in Figure 3. 3. 5 . This results in increased depth of field.

Figure 3. 3. 5 . Camera.

With the help of a projection device, it is possible to shoot large-scale images. The lens O of the projector focuses the image of a flat object (diapositive D) on the remote screen E (Figure 3.3.6). The lens system K (condenser) is used to concentrate the light source S on the slide. An enlarged inverted image is recreated on the screen. The scale of the projection device can be changed by zooming in or out of the screen and at the same time changing the distance between the aperture D and the lens O.

Figure 3. 3. 6 . projection apparatus.

Figure 3. 3. 7. thin lens model.

Figure 3. 3. eight . Model of a system of two lenses.

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"Lenses. Building an image in lenses"

Lesson Objectives:

Educational: we will continue the study of light rays and their distribution, introduce the concept of a lens, study the action of a converging and scattering lens; learn to build images given by the lens.

Developing: contribute to the development of logical thinking, the ability to see, hear, collect and comprehend information, independently draw conclusions.

Educational: cultivate attentiveness, perseverance and accuracy in work; learn to use the acquired knowledge to solve practical and cognitive problems.

Lesson type: combined, including the development of new knowledge, skills, consolidation and systematization of previously acquired knowledge.

During the classes

Organizing time(2 minutes):

greeting students;

checking the readiness of students for the lesson;

familiarization with the objectives of the lesson (the educational goal is set as a general one, without naming the topic of the lesson);

creation of psychological mood:

The universe, comprehending,
Know everything without taking away
What's inside - in the outside you will find,
What's outside, you'll find inside
So accept it without looking back
The world's intelligible riddles ...

I. Goethe

Repetition of previously studied material occurs in several stages.(26 min):

1. Blitz - poll(the answer to the question can only be yes or no, for a better overview of the students' answers, you can use signal cards, "yes" - red, "no" - green, it is necessary to specify the correct answer):

Does light travel in a straight line in a homogeneous medium? (Yes)

The angle of reflection is indicated by the Latin letter betta? (No)

Is reflection specular or diffuse? (Yes)

Is the angle of incidence always greater than the angle of reflection? (No)

At the boundary of two transparent media, does the light beam change its direction? (Yes)

Is the angle of refraction always greater than the angle of incidence? (No)

The speed of light in any medium is the same and equal to 3*10 8 m/s? (No)

Is the speed of light in water less than the speed of light in vacuum? (Yes)

Consider slide 9: “Building an image in a converging lens” ( ), using the reference abstract to consider the rays used.

Perform the construction of an image in a converging lens on the board, give its characteristics (performed by a teacher or student).

Consider slide 10: “Building an image in a diverging lens” ( ).

Perform the construction of an image in a diverging lens on the board, give its characteristics (performed by a teacher or student).

5. Checking the understanding of the new material, its consolidation(19 min):

Student work at the blackboard:

Construct an image of an object in a converging lens:

Advance task:

Independent work with a choice of tasks.

6. Summing up the lesson(5 minutes):

What did you learn in the lesson, what should you pay attention to?

Why is it not advised to water plants from above on a hot summer day?

Grades for work in the classroom.

7. Homework(2 minutes):

Construct an image of an object in a divergent lens:

If the object is beyond the focus of the lens.

If the object is between the focus and the lens.

Attached to the lesson , , and .