The formula for a thin converging lens is a conclusion. Converging and diverging lenses
"Lenses. Building an image in lenses"
Lesson Objectives:
Educational: we will continue the study of light rays and their propagation, 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 .
1. Types of lenses. Main optical axis of the lens
A lens is a body transparent to light, bounded by two spherical surfaces (one of the surfaces may be flat). Lenses with a thicker center than
the edges are called convex, and those whose edges are thicker than the middle are called concave. A convex lens made from a substance with an optical density greater than that of the medium in which the lens
is located, is converging, and a concave lens under the same conditions is divergent. Various types of lenses are shown in fig. 1: 1 - biconvex, 2 - biconcave, 3 - plano-convex, 4 - plano-concave, 3.4 - convex-concave and concave-convex.
Rice. 1. Lenses
The straight line O 1 O 2 passing through the centers of the spherical surfaces limiting the lens is called the main optical axis of the lens.
2. Thin lens, its optical center.
Side optical axes
A lens whose thickness l=|С 1 С 2 | (see Fig. 1) is negligible compared to the radii of curvature R 1 and R 2 of the lens surfaces and the distance d from the object to the lens, is called thin. In a thin lens, the points C 1 and C 2 , which are the vertices of the spherical segments, are located so close to each other that they can be taken as one point. This point O, lying on the main optical axis, through which light rays pass without changing their direction, is called the optical center of a thin lens. Any straight line passing through the optical center of the lens is called its optical axis. All optical axes, except for the main one, are called secondary optical axes.
Light rays traveling near the main optical axis are called paraxial (paraxial).
3. Main tricks and focal
lens distance
The point F on the main optical axis, at which the paraxial rays intersect after refraction, incident on the lens parallel to the main optical axis (or the continuation of these refracted rays), is called the main focus of the lens (Fig. 2 and 3). Any lens has two main foci, which are located on either side of it symmetrically to its optical center.
Rice. 2 Fig. 3
The converging lens (Fig. 2) has real foci, while the diverging lens (Fig. 3) has imaginary foci. Distance |OP| = F from the optical center of the lens to its main focus is called focal. A converging lens has a positive focal length, while a diverging lens has a negative focal length.
4. Focal planes of the lens, their properties
The plane passing through the main focus of a thin lens perpendicular to the main optical axis is called the focal plane. Each lens has two focal planes (M 1 M 2 and M 3 M 4 in Fig. 2 and 3), which are located on both sides of the lens.
Rays of light incident on a converging lens parallel to any of its secondary optical axis, after refraction in the lens, converge at the point of intersection of this axis with the focal plane (at point F' in Fig. 2). This point is called the side focus.
Lens formulas
5. Optical power of the lens
The value D, the reciprocal of the focal length of the lens, is called the optical power of the lens:
D=1/F(1)
For a converging lens F>0, therefore, D>0, and for a diverging lens F<0, следовательно, D<0, т.е. оптическая сила собирающей линзы положительна, а рассеивающей - отрицательна.
The unit of optical power is taken as the optical power of such a lens, the focal length of which is 1 m; This unit is called a diopter (dptr):
1 diopter = = 1 m -1
6. Derivation of the thin lens formula based on
geometric construction of the path of rays
Let there be a luminous object AB in front of the converging lens (Fig. 4). To construct an image of this object, it is necessary to construct images of its extreme points, and it is convenient to choose such rays, the construction of which will be the simplest. In general, there can be three such rays:
a) beam AC, parallel to the main optical axis, after refraction passes through the main focus of the lens, i.e. goes in a straight line CFA 1 ;
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b) the AO beam passing through the optical center of the lens is not refracted and also comes to point A 1 ;
c) the beam AB passing through the front focus of the lens, after refraction, goes parallel to the main optical axis along the straight line DA 1.
All three indicated beams where a real image of point A is obtained. Dropping the perpendicular from point A 1 to the main optical axis, we find point B 1, which is the image of point B. To build an image of a luminous point, it is enough to use two of the three listed beams.
Let us introduce the following notation |OB| = d is the distance of the object from the lens, |OB 1 | = f is the distance from the lens to the object image, |OF| = F is the focal length of the lens.
Using fig. 4, we derive the thin lens formula. From the similarity of triangles AOB and A 1 OB 1 it follows that
(2)
It follows from the similarity of triangles COF and A 1 FB 1 that
and since |AB| = |CO|, then
(4)
From formulas (2) and (3) it follows that
(5)
Since |OB1|= f, |OB| = d, |FB1| = f – F and |OF| = F, formula (5) takes the form f/d = (f – F)/F, whence
FF = df – dF (6)
Dividing formula (6) term by term by the product dfF, we obtain
(7)
where
(8)
Taking into account (1), we obtain
(9)
Relations (8) and (9) are called the thin converging lens formula.
At the diverging lens F<0, поэтому формула тонкой рассеивающей линзы имеет вид
(10)
7. Dependence of the optical power of a lens on the curvature of its surfaces
and refractive index
The focal length F and the optical power D of a thin lens depend on the radii of curvature R 1 and R 2 of its surfaces and the relative refractive index n 12 of the lens substance relative to the environment. This dependence is expressed by the formula
(11)
(12)
If one of the lens surfaces is flat (for it R= ∞), then the corresponding term 1/R in formula (12) is equal to zero. If the surface is concave, then the term 1/R corresponding to it enters this formula with a minus sign.
The sign of the right side of formula m (12) determines the optical properties of the lens. If it is positive, then the lens is converging, and if it is negative, it is diverging. For example, for a biconvex glass lens in air, (n 12 - 1) > 0 and
those. the right side of formula (12) is positive. Therefore, such a lens in air is converging. If the same lens is placed in a transparent medium with an optical density
larger than that of glass (for example, in carbon disulfide), then it will become scattering, because in this case it has (n 12 - 1)<0 и, хотя
, the sign at the right side of the formula/(17.44) will become
negative.
8. Linear magnification of the lens
The size of the image created by the lens changes depending on the position of the object relative to the lens. The ratio of the size of the image to the size of the depicted object is called linear magnification and is denoted by G.
Let's denote h the size of the object AB and H - the size of A 1 B 2 - its image. Then it follows from formula (2) that
(13)
10. Building images in a converging lens
Depending on the distance d of the object from the lens, there can be six different cases of constructing an image of this object:
a) d =∞. In this case, the light rays from the object fall on the lens parallel to either the main or some secondary optical axis. Such a case is shown in Fig. 2, from which it can be seen that if the object is infinitely removed from the lens, then the image of the object is real, in the form of a point, is in the focus of the lens (main or secondary);
b) 2F< d <∞. Предмет находится на конечном расстоянии от линзы большем, чем ее удвоенное фокусное расстояние (см. рис. 3). Изображение предмета действительное, перевернутое, уменьшенное находится между фокусом и точкой, отстоящей от линзы на двойное фокусное расстояние. Проверить правильность построения данного изображения можно
by calculation. Let d= 3F, h = 2 cm. It follows from formula (8) that
(14)
Since f > 0, the image is real. It is located behind the lens at a distance OB1=1.5F. Every real image is inverted. From the formula
(13) it follows that
; H=1 cm
i.e. the image is reduced. Similarly, using the calculation based on formulas (8), (10) and (13), one can check the correctness of the construction of any image in the lens;
c) d=2F. The object is at double the focal length from the lens (Fig. 5). The image of the object is real, inverted, equal to the object, located behind the lens on
twice the focal length from it;
Rice. 5
d) F
Rice. 6
e) d= F. The object is in the focus of the lens (Fig. 7). In this case, the image of the object does not exist (it is at infinity), since the rays from each point of the object, after refraction in the lens, go in a parallel beam;
Rice. 7
e) d
Rice. eight
11. Construction of images in a diverging lens
Let's build an image of an object at two different distances from the lens (Fig. 9). It can be seen from the figure that no matter how far the object is from the diverging lens, the image of the object is imaginary, direct, reduced, located between the lens and its focus
from the depicted object.
Rice. 9
Building images in lenses using side axes and the focal plane
(Building an image of a point lying on the main optical axis)
Rice. ten
Let the luminous point S be on the main optical axis of the converging lens (Fig. 10). To find where its image S’ is formed, we draw two beams from point S: a beam SO along the main optical axis (it passes through the optical center of the lens without being refracted) and a beam SВ incident on the lens at an arbitrary point B.
Let's draw the focal plane MM 1 of the lens and draw the side axis ОF', parallel to the beam SB (shown by a dashed line). It intersects with the focal plane at point S'.
As noted in paragraph 4, a ray must pass through this point F after refraction at point B. This ray BF'S' intersects with the ray SOS' at point S', which is the image of the luminous point S.
Constructing an image of an object whose size is larger than the lens
Let the object AB be located at a finite distance from the lens (Fig. 11). To find where the image of this object will turn out, we draw two rays from point A: the AOA 1 beam passing through the optical center of the lens without refraction, and the AC beam incident on the lens at an arbitrary point C. Draw the focal plane MM 1 of the lens and draw the side axis OF', parallel to beam AC (shown by dashed line). It intersects with the focal plane at point F'.
Rice. eleven
A beam refracted at point C will pass through this point F'. This beam CF'A 1 intersects with the beam AOA 1 at point A 1, which is the image of the luminous point A. To get the entire image A 1 B 1 of the object AB, we lower the perpendicular from point A 1 to the main optical axis.
magnifying glass
It is known that in order to see small details on an object, they must be viewed from a large angle of view, but an increase in this angle is limited by the limit of the accommodative capabilities of the eye. It is possible to increase the angle of view (keeping the distance of the best view d o) using optical devices (loupes, microscopes).
A magnifying glass is a short-focus biconvex lens or a system of lenses that act as a single converging lens, usually the focal length of a magnifying glass does not exceed 10 cm).
Rice. 12
The path of the rays in the magnifying glass is shown in Fig. 12. The magnifying glass is placed close to the eye,
and the object under consideration AB \u003d A 1 B 1 is placed between the magnifying glass and its front focus, a little closer to the latter. Select the position of the magnifying glass between the eye and the object so as to see a sharp image of the object. This image A 2 B 2 turns out to be imaginary, straight, enlarged and is located at the distance of the best view |OB|=d o from the eye.
As can be seen from fig. 12, the use of a magnifying glass results in an increase in the angle of view from which the eye views the object. Indeed, when the object was in position AB and viewed with the naked eye, the angle of view was φ 1 . The object was placed between the focus and the optical center of the magnifying glass in position A 1 B 1 and the angle of view became φ 2 . Since φ 2 > φ 1, this
means that with a magnifying glass you can see finer details on an object than with the naked eye.
From fig. 12 also shows that the linear magnification of the magnifying glass
Since |OB 2 |=d o , and |OB|≈F (focal length of the magnifying glass), then
G \u003d d about / F,
therefore, the magnification given by a loupe is equal to the ratio of the distance of the best view to the focal length of the loupe.
Microscope
A microscope is an optical instrument used to examine very small objects (including those invisible to the naked eye) from a large angle of view.
The microscope consists of two converging lenses - a short-focus lens and a long-focus eyepiece, the distance between which can be changed. Therefore, F 1<
The path of the rays in the microscope is shown in Fig. 13. The lens creates a real, inverted, enlarged intermediate image A 1 B 2 of the object AB.
Rice. 13
282.
Linear zoom
With the help of a micrometric
screw, the eyepiece is placed
with respect to the lens
so that it is intermediate
exact image A\B\ eye-
stuck between the front focus
som RF and optical center
Ocular eyepiece. Then the eyepiece
becomes a magnifying glass and creates an imaginary
mine, direct (relative to
intermediate) and increased
LHF image of the subject av.
Its position can be found
using the properties of the focal
plane and side axes (axis
O ^ P 'is carried out in parallel with the lu-
chu 1, and the axis OchR "- parallel-
but beam 2). As seen from
rice. 282, the use of micro
osprey leads to significantly
mu increase the angle of view,
under which the eye is viewed
there is an object (fa ^> fO, which pos-
wants to see the details, not vi-
visible to the naked eye.
microscope
\AM 1L2J2 I|d||
G=
\AB\ |L,5,| \AB\
Since \A^Vch\/\A\B\\== Gok is the linear magnification of the eyepiece and
\A\B\\/\AB\== Gob - linear magnification of the lens, then linear
microscope magnification
(17.62)
G == Gob Gok.
From fig. 282 shows that
» |L1Y,1 |0,R||
\ AB \ 150.1 '
where 10.5, | = |0/7, | +1/^21+1ad1.
Let 6 denote the distance between the back focus of the lens
and the front focus of the eyepiece, i.e. 6 = \P\P'r\. Since 6 ^> \OP\\
and 6 » \P2B\, then |0|5|1 ^ 6. Since |05|| ^ Rob, we get
b
Rob
(17.63)
The linear magnification of the eyepiece is determined by the same formula
(17.61), which is the magnification of the magnifying glass, i.e.
384
Gok=
a"
Gok
(17.64)
(17.65)
Substituting (17.63) and (17.64) into formula (17.62), we obtain
bio
G==
/^rev/m
Formula (17.65) determines the linear magnification of the microscope.
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, let's 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 propagation speed 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 the occurrence of refraction lies in the implementation of Fermat's principle, according to which light moves in such a way as to cover 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 ray.
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 imaginary 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).
The most important application of light refraction is the use of lenses, which are usually made of glass. In the figure you see cross sections of various lenses. Lens called a transparent body bounded by spherical or flat-spherical surfaces. Any lens that is thinner in the middle than at the edges will, in a vacuum or gas, diverging lens. Conversely, any lens that is thicker in the middle than at the edges will converging lens.
For clarification, refer to the drawings. On the left, it is shown that the rays traveling parallel to the main optical axis of the converging lens, after it "converge", passing through the point F - valid main focus converging lens. On the right, the passage of light rays through a diverging lens is shown parallel to its main optical axis. The rays after the lens "diverge" and seem to come from the point F ', called imaginary main focus diverging lens. It is not real, but imaginary because the rays of light do not pass through it: only their imaginary (imaginary) extensions intersect there.
In school physics, only the so-called thin lenses, which, regardless of their "sectional" symmetry, always have two main foci located at equal distances from the lens. If the rays are directed at an angle to the main optical axis, then we will find many other foci in the converging and / or diverging lens. These, side tricks, will be located away from the main optical axis, but still in pairs at equal distances from the lens.
A lens can not only collect or scatter rays. Using lenses, you can get enlarged and reduced images of objects. For example, thanks to a converging lens, an enlarged and inverted image of a golden figurine is obtained on the screen (see figure).
Experiments show: a distinct image appears, if the object, lens and screen are located at certain distances from each other. Depending on them, images can be inverted or straight, enlarged or reduced, real or imaginary.
The situation when the distance d from the object to the lens is greater than its focal length F, but less than the double focal length 2F, is described in the second row of the table. This is exactly what we observe with the figurine: its image is real, inverted and enlarged.
If the image is real, it can be projected onto a screen. In this case, the image will be visible from any place in the room from which the screen is visible. If the image is imaginary, then it cannot be projected onto the screen, but can only be seen with the eye, positioning it in a certain way in relation to the lens (you need to look “into it”).
Experiences show that diverging lenses give a reduced direct virtual image at any distance from the object to the lens.
In this lesson, we will repeat the features of the propagation of light rays in homogeneous transparent media, as well as the behavior of the rays when they cross the border between the light separation of two homogeneous transparent media, which you already know. Based on the knowledge already gained, we will be able to understand what useful information about a luminous or light-absorbing object we can get.
Also, applying the laws of refraction and reflection of light already familiar to us, we will learn how to solve the main problems of geometric optics, the purpose of which is to build an image of the object in question, formed by rays falling into the human eye.
Let's get acquainted with one of the main optical devices - a lens - and the formulas of a thin lens.
2. Internet portal "CJSC "Opto-Technological Laboratory"" ()
3. Internet portal "GEOMETRIC OPTICS" ()
Homework
1. Using a lens on a vertical screen, a real image of a light bulb is obtained. How will the image change if the upper half of the lens is closed?
2. Construct an image of an object placed in front of a converging lens in the following cases: 1. ; 2.; 3.; four. .