Applied to the case of a ray falling from a medium in which light propagates at a speed of ν 1 into a medium where light propagates at a speed of ν 2 >ν 1 it follows that the angle of refraction is greater than the angle of incidence:
But if the angle of incidence satisfies the condition:
(5.5)
then the angle of refraction turns to 90°, i.e., the refracted ray slides along the interface. This angle of incidence is called extreme(α pr.). With a further increase in the angle of incidence, the penetration of the beam into the depths of the second medium stops and complete reflection occurs (Fig. 5.6). A strict consideration of the issue from the wave point of view shows that in reality the wave penetrates into the second medium to a depth of the order of the wavelength.
Total reflection has various practical applications. Since for the glass-air system the limiting angle α is less than 45°, the prisms shown in Figure 5.7 make it possible to change the beam path, and reflection at the working boundary occurs practically without loss.
If you introduce light into a thin glass tube from its end, then, experiencing complete reflection on the walls, the beam will follow along the tube even with complex bends of the latter. Light guides operate on this principle - thin transparent fibers that allow a light beam to be guided along a curved path.
Figure 5.8 shows a section of light guide. The beam entering the light guide from the end at an angle of incidence a, meets the surface of the light guide at an angle γ=90°-β, where β is the angle of refraction. For complete reflection to occur, the following condition must be met:
where n is the refractive index of the fiber material. Since triangle ABC is right-angled, it turns out:
Hence,
Assuming a→90°, we find:
Thus, even with an almost grazing incidence, the beam experiences complete reflection in the light guide if the following condition is met:
In reality, the light guide is made up of thin flexible fibers with a refractive index n 1 surrounded by a cladding with a refractive index n 2 While studying the phenomenon of refraction, Newton performed an experiment that became a classic: a narrow beam of white light directed at a glass prism produced a series of color images of the beam's cross section - a spectrum. Then the spectrum fell on a second similar prism, rotated 180° around the horizontal axis. After passing through this prism, the spectrum reassembled into a single white cross-sectional image of the light beam. Thus, the complex composition of white light was proven. From this experiment it follows that the refractive index depends on the wavelength (dispersion). Let us consider the operation of a prism for monochromatic light incident at an angle α 1 on one of the refractive faces of a transparent prism (Fig. 5.9) with a refractive angle A. From the construction it is clear that the beam deflection angle δ is related to the refractive angle of the prism by a complex relationship: Let's rewrite it in the form and examine the beam deflection to the extreme. Taking the derivative and equating it to zero, we find: It follows that the extreme value of the deflection angle is obtained when the beam moves symmetrically inside the prism: It is easy to see that this results in a minimum deflection angle equal to: Equation (5.7) is used to determine the refractive index from the angle of minimum deviation. If the prism has a small refractive angle, such that sines can be replaced by angles, a visual relationship is obtained: Experience shows that glass prisms refract the short-wave part of the spectrum (blue rays) more strongly, but that there is no direct simple connection between λ and δ min. We will consider the theory of dispersion in Chapter 8. For now, it is important for us to introduce a measure of dispersion - the difference in the refractive indices of two specific wavelengths (one of them is taken in the red, the other in the blue part of the spectrum): The measure of dispersion is different for different types of glass. Figure 5.10 shows the course of the refractive index for two common types of glass: light - crown and heavy - flint. It can be seen from the figure that the dispersion measures differ significantly. This makes it possible to create a very convenient direct vision prism, where light is decomposed into a spectrum, almost without changing the direction of propagation. This prism is made from several (up to seven) prisms of different glass with slightly different refractive angles (Fig. 5.10, below). Due to different measures of dispersion, the beam path approximately shown in the figure is achieved. In conclusion, we note that passing light through a plane-parallel plate (Fig. 5.11) allows the beam to be displaced parallel to itself. Offset value depends on the properties of the plate and on the angle of incidence of the primary beam on it. Of course, in all the cases considered, along with refraction, there is also reflection of light. But we do not take it into account, since refraction in these matters is considered the main phenomenon. This remark also applies to the refraction of light on the curved surfaces of various lenses. Let the beam fall on one of the faces of the prism. Having refracted at point , the ray will go in the direction and, having refracted a second time at point, will exit the prism into the air (Fig. 189). Let's find the angle by which the ray, passing through the prism, will deviate from the original direction. We will call this angle the deflection angle. The angle between the refractive faces, called the refractive angle of the prism, will be denoted by . Rice. 189. Refraction in a prism From a quadrilateral in which the angles at and are right, we find that the angle is equal to . Using this, from the quadrilateral we find An angle, like an exterior angle in a triangle, is equal to where is the angle of refraction at point , and is the angle of incidence at the point of the ray emerging from the prism. Further, using the law of refraction, we have Using the resulting equations, knowing the refractive angle of the prism and the refractive index, we can calculate the deflection angle for any angle of incidence. The expression for the angle of deflection takes a particularly simple form when the refractive angle of the prism is small, that is, the prism is thin and the angle of incidence is small; then the angle is also small. Approximately replacing the sines of angles in formulas (86.3) and (86.4) with the angles themselves (in radians), we have Substituting these expressions into formula (86.1) and using (86.2), we find We will use this formula, which is valid for a thin prism when rays fall on it at a small angle. Please note that the angle of deflection of the beam in the prism depends on the refractive index of the substance from which the prism is made. As we indicated above, the refractive index for different colors of light is different (dispersion). For transparent bodies, the refractive index of violet rays is the highest, followed by blue, cyan, green, yellow, orange, and finally red rays, which have the lowest refractive index. In accordance with this, the angle of deflection for violet rays is the greatest, for red rays the smallest, and a white ray incident on the prism, upon exiting it, will be decomposed into a series of colored rays (Fig. 190 and Fig. I on the colored flyleaf), i.e. e. a spectrum of rays is formed. Rice. 190. Decomposition of white light during refraction in a prism. An incident beam of white light is depicted as a front with the direction of wave propagation perpendicular to it. For refracted beams, only the directions of wave propagation are shown 18.
By placing a screen behind a piece of cardboard with a small hole made in it, you can image sources on this screen. Under what conditions will the image on the screen be clear? Explain why the image appears upside down? 19.
Prove that a beam of parallel rays remains the same after reflection from a plane mirror Rice. 191. For exercise 27. If the cup is empty, the eye does not see the coin (a), but if the cup is filled with water, then the coin is visible (b). A stick immersed at one end in water appears to be broken (c). Mirage in the desert (d). How a fish sees a tree and a diver (d) 20.
What is the angle of incidence of the beam if the incident beam and the reflected beam form an angle? 21.
What is the angle of incidence of the beam if the reflected beam and the refracted beam form an angle? The refractive index of the second medium relative to the first is equal to . 22.
Prove the reversibility of the direction of light rays for the case of light reflection. 23.
Is it possible to invent a system of mirrors and prisms (lenses) through which one observer would see a second observer, but the second observer would not see the first? 24.
The refractive index of glass relative to water is 1.182: the refractive index of glycerin relative to water is 1.105. Find the refractive index of glass relative to glycerol. 25.
Find the limiting angle of total internal reflection for diamond at the interface with water. 26.
find the displacement of the ray when passing through a plane-parallel glass plate with a refractive index of 1.55, if the angle of incidence is , and the thickness of the plate is 27.
Using the laws of refraction and reflection, explain the phenomena shown in Fig. 191 Video tutorial 2: Geometric optics: Laws of refraction
Lecture: Laws of light refraction. Path of rays in a prism
At the moment when a ray falls on some other medium, it is not only reflected, but also passes through it. However, due to the difference in densities, it changes its path. That is, the beam, hitting the boundary, changes its propagation trajectory and moves with a displacement by a certain angle. Refraction will occur when the beam falls at a certain angle to the perpendicular. If it coincides with the perpendicular, then refraction does not occur and the beam penetrates the medium at the same angle. Air-Media The most common situation when light passes from one medium to another is the transition from air. So, in the picture JSC- ray incident on the interface, CO And OD- perpendiculars (normals) to the sections of the media, lowered from the point of incidence of the beam. OB- a ray that has been refracted and passed into another medium. The angle between the normal and the incident ray is called the angle of incidence (AOC). The angle between the refracted ray and the normal is called the angle of refraction (BOD). To find out the refractive intensity of a particular medium, a PV is introduced, which is called the refractive index. This value is tabular and for basic substances the value is a constant value that can be found in the table. Most often, problems use the refractive indices of air, water and glass. Laws of refraction for air-medium
1.
When considering the incident and refracted ray, as well as the normal to the sections of the media, all of the listed quantities are in the same plane. 2.
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 refractive index of the medium. From this relationship it is clear that the value of the refractive index is greater than unity, which means that the sine of the angle of incidence is always greater than the sine of the angle of refraction. That is, if the beam leaves the air into a denser medium, then the angle decreases. The refractive index also shows how the speed of propagation of light changes in a particular medium, relative to propagation in a vacuum: From this we can obtain the following relationship: When we consider air, we can make some neglects - we will assume that the refractive index of this medium is equal to unity, then the speed of light propagation in the air will be equal to 3 * 10 8 m/s. Ray reversibility
These laws also apply in cases where the direction of the rays occurs in the opposite direction, that is, from the medium into the air. That is, the path of light propagation is not affected by the direction in which the rays move. Law of refraction for arbitrary media
Let's consider a method for determining the refractive index, applicable for transparent substances. The method consists of measuring the angle of deflection of rays when light passes through a prism made of the material under study. A parallel beam of rays is directed at the prism, so it is enough to consider the path of one of them (S 1) in a plane perpendicular to the line of intersection of the beam of the refractive faces of the prism (Fig. 6). A 1
─direction of the normal to the face on which the ray S 1 falls, A 2
─ direction of the normal to the face from which the ray S 2 emerges, i 1
,
i 2
- angles of incidence, r 1
,
r 2
- angles of refraction at the interfaces AC and AB respectively, φ
- refractive angle of the prism, δ
- the angle of deflection of the beam emerging from the prism relative to the original direction. The path of the ray through the prism is calculated based on the laws of light refraction. During refraction at the first face of the prism AC we get where n is the refractive index of the prism material for a given wavelength of light. For edge AB the law of refraction will be written as Relations 12 and 13 allow us to find expressions for determining n. However, to experimentally determine the angles r
1
And i
1
It's hard enough. In practice, it is more convenient to measure the beam deflection angle with a prism δ
and the refracting angle of the prism φ.
We obtain the formula for determining the refractive index n
through the corners
δ
And φ
. First, we use the well-known theorem in geometry that the external angle of a triangle is equal to the sum of the internal angles not adjacent to it. Then from the triangle EDF we get φ
=
r
1
+
i
2
.
(14) From triangle EHF and using (14), we obtain: δ
=(i 1
-r 1
)+(r 2
– i 2
)= i 1
+r 2
–(r 1
+i 2
)= i 1
+r 2
+
φ
.
(15) Then we express the angle δ
through the corner r 1
, using the laws of refraction (12), (13) and (14), and determine the conditions for minimality δ
: i 1
= arcsin(n sin r 1); r 2 = arcsin(n sin i 2
) = arcsin(n sin ( φ-r 1
)); δ
= arcsin(n sin r 1
) +arcsin(n sin ( φ-r 1
)). Addiction δ
from r 1
has a minimum, the condition of which can be found by equating the derivative δ
from r 1
zero: Expression (16) is satisfied if r 1
=
φ
-
r 1.
In accordance with (14) we have φ
-
r 1
=
i 2
,
That's why
r 1
=
i 2
.
Then from the laws of refraction (12) and (13) it follows that the angles
i 1
,
r 2
must also be equal:
i 1
=
r 2
.
Taking into account (14) and (15), we obtain: φ
= 2
r 1
;
δ
min =2
i 1
–
φ
.
Taking these equalities into account, we finally obtain: Consequently, at the smallest beam deflection angle by the prism δ
min The refractive index of the prism substance can be determined by the formula Thus, determining the refractive index of a substance comes down to measuring refractive angle of the prism
And angle of smallest deviation
rays
.
Angle of smallest deviation δ
formed by two directions: the direction of the ray incident on the prism S 1
and the direction of the beam emerging from the prism S 2
. If the radiation source is not monochromatic, then due to the dispersion of the prism substance, the direction of the refracted ray EF, and, consequently, the direction of the emerging ray S 2
will be different for different wavelengths, i.e. S 2 =f( λ
). This leads to δ
And n for different λ,
will be different. Refracting angle of a prism φ
formed by the face of a prism SA, on which the beam falls and the face AB, from which the radiation comes out, or perpendicular to these faces A 1
And A 2
respectively. The source of radiation in the work is a mercury lamp. Monochromatic light falls on the edge AB glass prism (Fig. 16.28) located in the air, S 1 O 1 - incident ray, \(~\alpha_1\) - angle of incidence, O 1 O 2 - refracted ray, \(~\beta_1\) - angle of refraction. Since light passes from an optically less dense medium to an optically more dense one, then \(~\beta_1<\alpha_1.\) Пройдя через призму, свет падает на ее грань AC. Here it is refracted again \[~\alpha_2\] is the angle of incidence, \(~\beta_2\) is the angle of refraction. On this face, light passes from an optically more dense medium to an optically less dense one. therefore \(~\beta_2>\alpha_2.\) Edges VA And SA, at which light refraction occurs are called refractive edges. The angle \(\varphi\) between the refractive faces is called refractive angle prisms. The angle \(~\delta\) formed by the direction of the ray entering the prism and the direction of the ray leaving it is called deflection angle. The face lying opposite the refractive angle is called base of the prism. The following relations hold true for a prism: 1) For the first refractive face, the law of light refraction will be written as follows: \(\frac(\sin \alpha_1)(\sin \beta_1)=n,\) where n is the relative refractive index of the substance from which the prism is made. 2) For the second face: \(\frac(\sin \alpha_1)(\sin \beta_1)=\frac(1)(n).\) 3) Refractive angle of the prism: \(\varphi=\alpha_2 + \beta_1.\) Deviation angle of the prism beam from the original direction: \(\delta = \alpha_1 + \beta_2 - \varphi.\) Consequently, if the optical density of the prism substance is greater than that of the surrounding medium, then a ray of light passing through the prism is deflected towards its base. It is easy to show that if the optical density of the prism substance is less than that of the surrounding medium, then the light ray, after passing through the prism, will be deflected towards its top. Aksenovich L. A. Physics in secondary school: Theory. Tasks. Tests: Textbook. allowance for institutions providing general education. environment, education / L. A. Aksenovich, N. N. Rakina, K. S. Farino; Ed. K. S. Farino. - Mn.: Adukatsiya i vyakhavanne, 2004. - P. 469-470.
(5.7)
(5.8)
.
(12)
.
(13)
And 
.
.
(17)Literature