What is the phenomenon of total internal reflection. Critical angle or limiting angle c and total internal reflection

When waves propagate in a medium, including electromagnetic ones, to find a new wave front at any time, use Huygens' principle.

Each point on the wave front is a source of secondary waves.

In a homogeneous isotropic medium, the wave surfaces of secondary waves have the form of spheres of radius v×Dt, where v is the speed of wave propagation in the medium. By drawing the envelope of the wave fronts of the secondary waves, we obtain a new wave front at a given moment in time (Fig. 7.1, a, b).

Law of Reflection

Using Huygens' principle, it is possible to prove the law of reflection of electromagnetic waves at the interface between two dielectrics.

The angle of incidence is equal to the angle of reflection. The incident and reflected rays, together with the perpendicular to the interface between the two dielectrics, lie in the same plane.Ð a = Ð b. (7.1)

Let a plane light wave (rays 1 and 2, Fig. 7.2) fall on a flat LED interface between two media. The angle a between the beam and the perpendicular to the LED is called the angle of incidence. If at a given moment in time the front of the incident OB wave reaches point O, then according to Huygens’ principle this point

Rice. 7.2

begins to emit a secondary wave. During the time Dt = VO 1 /v, the incident beam 2 reaches point O 1. During the same time, the front of the secondary wave, after reflection in point O, propagating in the same medium, reaches points of the hemisphere with radius OA = v Dt = BO 1. The new wave front is depicted by the plane AO ​​1, and the direction of propagation by the ray OA. Angle b is called the angle of reflection. From the equality of triangles OAO 1 and OBO 1, the law of reflection follows: the angle of incidence is equal to the angle of reflection.

Law of refraction

An optically homogeneous medium 1 is characterized by , (7.2)

Ratio n 2 / n 1 = n 21 (7.4)

called

(7.5)

For vacuum n = 1.

Due to dispersion (light frequency n » 10 14 Hz), for example, for water n = 1.33, and not n = 9 (e = 81), as follows from electrodynamics for low frequencies. If the speed of light propagation in the first medium is v 1, and in the second - v 2,

Rice. 7.3

then during the time Dt the incident plane wave travels the distance AO 1 in the first medium AO 1 = v 1 Dt. The front of the secondary wave, excited in the second medium (in accordance with Huygens' principle), reaches points of the hemisphere, the radius of which OB = v 2 Dt. The new front of the wave propagating in the second medium is represented by the BO 1 plane (Fig. 7.3), and the direction of its propagation by the rays OB and O 1 C (perpendicular to the wave front). Angle b between the ray OB and the normal to the interface between two dielectrics at point O called the angle of refraction. From the triangles OAO 1 and OBO 1 it follows that AO 1 = OO 1 sin a, OB = OO 1 sin b.

Their attitude expresses law of refraction(law Snell):

. (7.6)

The ratio of the sine of the angle of incidence to the sine of the angle of refraction is equal to the relative refractive index of the two media.

Total internal reflection

Rice. 7.4

According to the law of refraction, at the interface between two media one can observe total internal reflection, if n 1 > n 2, i.e. Ðb > Ða (Fig. 7.4). Consequently, there is a limiting angle of incidence Ða pr when Ðb = 90 0 . Then the law of refraction (7.6) takes the following form:

sin a pr = , (sin 90 0 =1) (7.7)

With a further increase in the angle of incidence Ða > Ða pr, the light is completely reflected from the interface between the two media.

This phenomenon is called total internal reflection and are widely used in optics, for example, to change the direction of light rays (Fig. 7.5, a, b).

It is used in telescopes, binoculars, fiber optics and other optical instruments.

In classical wave processes, such as the phenomenon of total internal reflection of electromagnetic waves, phenomena similar to the tunnel effect in quantum mechanics are observed, which is associated with the wave-corpuscular properties of particles.

Indeed, when light passes from one medium to another, refraction of light is observed, associated with a change in the speed of its propagation in different media. At the interface between two media, a light beam is divided into two: refracted and reflected.

A ray of light falls perpendicularly onto face 1 of a rectangular isosceles glass prism and, without refraction, falls on face 2, total internal reflection is observed, since the angle of incidence (Ða = 45 0) of the beam on face 2 is greater than the limiting angle of total internal reflection (for glass n 2 = 1.5; Ða pr = 42 0).

If the same prism is placed at a certain distance H ~ l/2 from face 2, then a ray of light will pass through face 2 * and exit the prism through face 1 * parallel to the ray incident on face 1. The intensity J of the transmitted light flux decreases exponentially with increasing the gap h between the prisms according to the law:

,

where w is a certain probability of the beam passing into the second medium; d is the coefficient depending on the refractive index of the substance; l is the wavelength of the incident light

Therefore, the penetration of light into the “forbidden” region is an optical analogue of the quantum tunneling effect.

The phenomenon of total internal reflection is truly complete, since in this case all the energy of the incident light is reflected at the interface between two media than when reflected, for example, from the surface of metal mirrors. Using this phenomenon, one can trace another analogy between the refraction and reflection of light, on the one hand, and Vavilov-Cherenkov radiation, on the other hand.

WAVE INTERFERENCE

7.2.1. The role of vectors and

In practice, several waves can propagate simultaneously in real media. As a result of the addition of waves, a number of interesting phenomena are observed: interference, diffraction, reflection and refraction of waves etc.

These wave phenomena are characteristic not only of mechanical waves, but also electric, magnetic, light, etc. All elementary particles also exhibit wave properties, which has been proven by quantum mechanics.

One of the most interesting wave phenomena, which is observed when two or more waves propagate in a medium, is called interference. An optically homogeneous medium 1 is characterized by absolute refractive index , (7.8)

where c is the speed of light in vacuum; v 1 - speed of light in the first medium.

Medium 2 is characterized by the absolute refractive index

where v 2 is the speed of light in the second medium.

Attitude (7.10)

called the relative refractive index of the second medium relative to the first. For transparent dielectrics in which m = 1, using Maxwell's theory, or

where e 1, e 2 are the dielectric constants of the first and second media.

For vacuum n = 1. Due to dispersion (light frequency n » 10 14 Hz), for example, for water n = 1.33, and not n = 9 (e = 81), as follows from electrodynamics for low frequencies. Light is electromagnetic waves. Therefore, the electromagnetic field is determined by the vectors and , which characterize the strengths of the electric and magnetic fields, respectively. However, in many processes of interaction of light with matter, for example, such as the effect of light on the organs of vision, photocells and other devices, the decisive role belongs to the vector, which in optics is called the light vector.

The limiting angle of total reflection is the angle of incidence of light at the interface between two media, corresponding to a refraction angle of 90 degrees.

Fiber optics is a branch of optics that studies the physical phenomena that arise and occur in optical fibers.

4. Wave propagation in an optically inhomogeneous medium. Explanation of ray bending. Mirages. Astronomical refraction. Inhomogeneous medium for radio waves.

Mirage is an optical phenomenon in the atmosphere: the reflection of light by a boundary between layers of air that are sharply different in density. For an observer, such a reflection means that together with a distant object (or part of the sky), its virtual image is visible, shifted relative to the object. Mirages are divided into lower ones, visible under the object, upper ones, above the object, and side ones.

Inferior Mirage

It is observed with a very large vertical temperature gradient (it decreases with height) over an overheated flat surface, often a desert or an asphalt road. The virtual image of the sky creates the illusion of water on the surface. So, the road stretching into the distance on a hot summer day seems wet.

Superior Mirage

Observed above the cold earth's surface with an inverted temperature distribution (increases with its height).

Fata Morgana

Complex mirage phenomena with a sharp distortion of the appearance of objects are called Fata Morgana.

Volume mirage

In the mountains, it is very rare, under certain conditions, to see the “distorted self” at a fairly close distance. This phenomenon is explained by the presence of “standing” water vapor in the air.

Astronomical refraction is the phenomenon of refraction of light rays from celestial bodies when passing through the atmosphere. Since the density of planetary atmospheres always decreases with altitude, the refraction of light occurs in such a way that the convexity of the curved ray in all cases is directed towards the zenith. In this regard, refraction always “raises” the images of celestial bodies above their true position

Refraction causes a number of optical-atmospheric effects on Earth: magnification day length due to the fact that the solar disk, due to refraction, rises above the horizon several minutes earlier than the moment at which the Sun should have risen based on geometric considerations; the oblateness of the visible disks of the Moon and the Sun near the horizon due to the fact that the lower edge of the disks rises higher by refraction than the upper; twinkling of stars, etc. Due to the difference in the magnitude of refraction of light rays with different wavelengths (blue and violet rays deviate more than red ones), an apparent coloring of celestial bodies occurs near the horizon.

5. The concept of a linearly polarized wave. Polarization of natural light. Unpolarized radiation. Dichroic polarizers. Polarizer and light analyzer. Malus's law.

Wave polarization- the phenomenon of breaking the symmetry of the distribution of disturbances in transverse wave (for example, the electric and magnetic field strengths in electromagnetic waves) relative to the direction of its propagation. IN longitudinal polarization cannot occur in a wave, since disturbances in this type of wave always coincide with the direction of propagation.

linear - disturbance oscillations occur in one plane. In this case they talk about “ plane-polarized wave";

circular - the end of the amplitude vector describes a circle in the plane of oscillation. Depending on the direction of rotation of the vector, there may be right or left.

Light polarization is the process of ordering the oscillations of the electric field strength vector of a light wave when light passes through certain substances (during refraction) or when the light flux is reflected.

A dichroic polarizer contains a film containing at least one dichroic organic substance, the molecules or fragments of molecules of which have a flat structure. At least part of the film has a crystalline structure. A dichroic substance has at least one maximum of the spectral absorption curve in the spectral ranges of 400 - 700 nm and/or 200 - 400 nm and 0.7 - 13 μm. When manufacturing a polarizer, a film containing a dichroic organic substance is applied to the substrate, an orienting effect is applied to it, and it is dried. In this case, the conditions for applying the film and the type and magnitude of the orienting influence are chosen so that the order parameter of the film, corresponding to at least one maximum on the spectral absorption curve in the spectral range 0.7 - 13 μm, has a value of at least 0.8. The crystal structure of at least part of the film is a three-dimensional crystal lattice formed by molecules of dichroic organic matter. The spectral range of the polarizer is expanded while simultaneously improving its polarization characteristics.

Malus's law is a physical law that expresses the dependence of the intensity of linearly polarized light after it passes through a polarizer on the angle between the polarization planes of the incident light and the polarizer.

Where I 0 - intensity of light incident on the polarizer, I- intensity of light emerging from the polarizer, k a- polarizer transparency coefficient.

6. Brewster phenomenon. Fresnel formulas for the reflection coefficient for waves whose electric vector lies in the plane of incidence, and for waves whose electric vector is perpendicular to the plane of incidence. Dependence of reflection coefficients on the angle of incidence. The degree of polarization of reflected waves.

Brewster's law is a law of optics that expresses the relationship of the refractive index with the angle at which light reflected from the interface will be completely polarized in a plane perpendicular to the plane of incidence, and the refracted beam is partially polarized in the plane of incidence, and the polarization of the refracted beam reaches its greatest value. It is easy to establish that in this case the reflected and refracted rays are mutually perpendicular. The corresponding angle is called the Brewster angle. Brewster's Law: , Where n 21 - refractive index of the second medium relative to the first, θ Br- angle of incidence (Brewster angle). The amplitudes of the incident (U inc) and reflected (U ref) waves in the KBB line are related by the relation:

K bv = (U pad - U neg) / (U pad + U neg)

Through the voltage reflection coefficient (K U), the KVV is expressed as follows:

K bv = (1 - K U) / (1 + K U) With a purely active load, the BV is equal to:

K bv = R / ρ at R< ρ или

K bv = ρ / R for R ≥ ρ

where R is the active load resistance, ρ is the characteristic impedance of the line

7. The concept of light interference. The addition of two incoherent and coherent waves whose polarization lines coincide. Dependence of the intensity of the resulting wave upon addition of two coherent waves on the difference in their phases. The concept of the geometric and optical difference in wave paths. General conditions for observing interference maxima and minima.

Light interference is the nonlinear addition of the intensities of two or more light waves. This phenomenon is accompanied by alternating maxima and minima of intensity in space. Its distribution is called an interference pattern. When light interferes, energy is redistributed in space.

Waves and the sources that excite them are called coherent if the phase difference between the waves does not depend on time. Waves and the sources that excite them are called incoherent if the phase difference between the waves changes over time. Formula for the difference:

, Where , ,

8. Laboratory methods for observing the interference of light: Young’s experiment, Fresnel biprism, Fresnel mirrors. Calculation of the position of interference maxima and minima.

Young's experiment - In the experiment, a beam of light is directed onto an opaque screen screen with two parallel slits, behind which a projection screen is installed. This experiment demonstrates the interference of light, which is proof of the wave theory. The peculiarity of the slits is that their width is approximately equal to the wavelength of the emitted light. The effect of slot width on interference is discussed below.

If we assume that light consists of particles ( corpuscular theory of light), then on the projection screen one could see only two parallel strips of light passing through the slits of the screen. Between them, the projection screen would remain virtually unlit.

Fresnel biprism - in physics - a double prism with very small angles at the vertices.
A Fresnel biprism is an optical device that allows the formation of two coherent waves from one light source, which make it possible to observe a stable interference pattern on the screen.
The Frenkel biprism serves as a means of experimentally proving the wave nature of light.

Fresnel mirrors are an optical device proposed in 1816 by O. J. Fresnel to observe the phenomenon of interference of coherent light beams. The device consists of two flat mirrors I and II, forming a dihedral angle that differs from 180° by only a few angular minutes (see Fig. 1 in the article Interference of Light). When mirrors are illuminated from a source S, beams of rays reflected from the mirrors can be considered as emanating from coherent sources S1 and S2, which are virtual images of S. In the space where the beams overlap, interference occurs. If the source S is linear (slit) and parallel to the edge of the photons, then when illuminated with monochromatic light, an interference pattern in the form of equally spaced dark and light stripes parallel to the slit is observed on the screen M, which can be installed anywhere in the area of ​​beam overlap. The distance between the stripes can be used to determine the wavelength of the light. Experiments conducted with photons were one of the decisive proofs of the wave nature of light.

9. Interference of light in thin films. Conditions for the formation of light and dark stripes in reflected and transmitted light.

10. Strips of equal slope and strips of equal thickness. Newton's interference rings. Radii of dark and light rings.

11. Interference of light in thin films at normal light incidence. Coating of optical instruments.

12. Optical interferometers of Michelson and Jamin. Determination of the refractive index of a substance using two-beam interferometers.

13. The concept of multi-beam interference of light. Fabry-Perot interferometer. The addition of a finite number of waves of equal amplitudes, the phases of which form an arithmetic progression. Dependence of the intensity of the resulting wave on the phase difference of the interfering waves. The condition for the formation of the main maxima and minima of interference. The nature of the multi-beam interference pattern.

14. The concept of wave diffraction. Wave parameter and limits of applicability of the laws of geometric optics. Huygens-Fresnel principle.

15. Fresnel zone method and proof of rectilinear propagation of light.

16. Fresnel diffraction by a round hole. Radii of Fresnel zones for a spherical and plane wave front.

17. Diffraction of light on an opaque disk. Calculation of the area of ​​Fresnel zones.

18. The problem of increasing the amplitude of a wave when passing through a round hole. Amplitude and phase zone plates. Focusing and zone plates. Focusing lens as a limiting case of a stepped phase zone plate. Lens zoning.

At a certain angle of incidence of light $(\alpha )_(pad)=(\alpha )_(pred)$, which is called limit angle, the angle of refraction is equal to $\frac(\pi )(2),\ $in this case the refracted ray slides along the interface between the media, therefore, there is no refracted ray. Then from the law of refraction we can write that:

Picture 1.

In the case of total reflection, the equation is:

has no solution in the region of real values ​​of the refraction angle ($(\alpha )_(pr)$). In this case, $cos((\alpha )_(pr))$ is a purely imaginary quantity. If we turn to the Fresnel Formulas, it is convenient to present them in the form:

where the angle of incidence is denoted $\alpha $ (for brevity), $n$ is the refractive index of the medium where the light propagates.

From the Fresnel formulas it is clear that the modules $\left|E_(otr\bot )\right|=\left|E_(otr\bot )\right|$, $\left|E_(otr//)\right|=\ left|E_(otr//)\right|$, which means the reflection is "full".

Note 1

It should be noted that the inhomogeneous wave does not disappear in the second medium. So, if $\alpha =(\alpha )_0=(arcsin \left(n\right),\ then\ )$ $E_(pr\bot )=2E_(pr\bot ).$ Violations of the law of conservation of energy in a given case no. Since Fresnel's formulas are valid for a monochromatic field, that is, for a steady-state process. In this case, the law of conservation of energy requires that the average change in energy over the period in the second medium be equal to zero. The wave and the corresponding fraction of energy penetrates through the interface into the second medium to a small depth of the order of the wavelength and moves in it parallel to the interface with a phase velocity that is less than the phase velocity of the wave in the second medium. It returns to the first medium at a point that is offset relative to the entry point.

The penetration of the wave into the second medium can be observed experimentally. The intensity of the light wave in the second medium is noticeable only at distances shorter than the wavelength. Near the interface on which the light wave falls and undergoes total reflection, the glow of a thin layer can be seen on the side of the second medium if there is a fluorescent substance in the second medium.

Total reflection causes mirages to occur when the earth's surface is hot. Thus, the complete reflection of light that comes from clouds leads to the impression that there are puddles on the surface of heated asphalt.

Under ordinary reflection, the relations $\frac(E_(otr\bot ))(E_(pad\bot ))$ and $\frac(E_(otr//))(E_(pad//))$ are always real. At full reflection they are complex. This means that in this case the phase of the wave undergoes a jump, while it is different from zero or $\pi $. If the wave is polarized perpendicular to the plane of incidence, then we can write:

where $(\delta )_(\bot )$ is the desired phase jump. Let us equate the real and imaginary parts, we have:

From expressions (5) we obtain:

Accordingly, for a wave that is polarized in the plane of incidence, one can obtain:

The phase jumps $(\delta )_(//)$ and $(\delta )_(\bot )$ are not the same. The reflected wave will be elliptically polarized.

Applying Total Reflection

Let us assume that two identical media are separated by a thin air gap. A light wave falls on it at an angle that is greater than the limiting one. It may happen that it penetrates the air gap as a non-uniform wave. If the thickness of the gap is small, then this wave will reach the second boundary of the substance and will not be very weakened. Having passed from the air gap into the substance, the wave will turn back into a homogeneous one. Such an experiment was carried out by Newton. The scientist pressed another prism, which was ground spherically, to the hypotenuse face of the rectangular prism. In this case, the light passed into the second prism not only where they touch, but also in a small ring around the contact, in a place where the thickness of the gap is comparable to the wavelength. If observations were carried out in white light, then the edge of the ring had a reddish color. This is as it should be, since the penetration depth is proportional to the wavelength (for red rays it is greater than for blue ones). By changing the thickness of the gap, you can change the intensity of the transmitted light. This phenomenon formed the basis of the light telephone, which was patented by Zeiss. In this device, one of the media is a transparent membrane, which vibrates under the influence of sound falling on it. Light that passes through an air gap changes intensity in time with changes in sound intensity. When it hits a photocell, it generates alternating current, which changes in accordance with changes in sound intensity. The resulting current is amplified and used further.

The phenomena of wave penetration through thin gaps are not specific to optics. This is possible for a wave of any nature if the phase velocity in the gap is higher than the phase velocity in the environment. This phenomenon is of great importance in nuclear and atomic physics.

The phenomenon of total internal reflection is used to change the direction of light propagation. Prisms are used for this purpose.

Example 1

Exercise: Give an example of the phenomenon of total reflection, which occurs frequently.

Solution:

We can give the following example. If the highway is very hot, then the air temperature is maximum near the asphalt surface and decreases with increasing distance from the road. This means that the refractive index of air is minimal at the surface and increases with increasing distance. As a result of this, rays that have a small angle relative to the highway surface are completely reflected. If you concentrate your attention, while driving in a car, on a suitable section of the highway surface, you can see a car driving quite far ahead upside down.

Example 2

Exercise: What is the Brewster angle for a beam of light that falls on the surface of a crystal if the limiting angle of total reflection for a given beam at the air-crystal interface is 400?

Solution:

\[(tg(\alpha )_b)=\frac(n)(n_v)=n\left(2.2\right).\]

From expression (2.1) we have:

Let's substitute the right side of expression (2.3) into formula (2.2) and express the desired angle:

\[(\alpha )_b=arctg\left(\frac(1)((sin \left((\alpha )_(pred)\right)\ ))\right).\]

Let's carry out the calculations:

\[(\alpha )_b=arctg\left(\frac(1)((sin \left(40()^\circ \right)\ ))\right)\approx 57()^\circ .\]

Answer:$(\alpha )_b=57()^\circ .$

We pointed out in § 81 that when light falls on the interface between two media, the light energy is divided into two parts: one part is reflected, the other part penetrates through the interface into the second medium. Using the example of the transition of light from air to glass, i.e. from a medium that is optically less dense to a medium that is optically denser, we saw that the proportion of reflected energy depends on the angle of incidence. In this case, the fraction of reflected energy increases greatly as the angle of incidence increases; however, even at very large angles of incidence, close to , when the light beam almost slides along the interface, some of the light energy still passes into the second medium (see §81, tables 4 and 5).

A new interesting phenomenon arises if light propagating in any medium falls on the interface between this medium and a medium that is optically less dense, that is, having a lower absolute refractive index. Here, too, the fraction of reflected energy increases with increasing angle of incidence, but the increase follows a different law: starting from a certain angle of incidence, all light energy is reflected from the interface. This phenomenon is called total internal reflection.

Let us consider again, as in §81, the incidence of light at the interface between glass and air. Let a light beam fall from the glass onto the interface at different angles of incidence (Fig. 186). If we measure the fraction of reflected light energy and the fraction of light energy passing through the interface, we obtain the values ​​given in Table. 7 (glass, like in Table 4, had a refractive index ).

Rice. 186. Total internal reflection: the thickness of the rays corresponds to the fraction of light energy charged or passed through the interface

The angle of incidence from which all light energy is reflected from the interface is called the limiting angle of total internal reflection. For the glass for which the table was compiled. 7 (), the limiting angle is approximately .

Table 7. Fractions of reflected energy for various angles of incidence when light passes from glass to air

Let us note that when light is incident on the interface at a limiting angle, the angle of refraction is equal to , i.e., in the formula expressing the law of refraction for this case,

when we have to put or . From here we find

At angles of incidence greater than that, there is no refracted ray. Formally, this follows from the fact that at angles of incidence large from the law of refraction for, values ​​larger than unity are obtained, which is obviously impossible.

In table Table 8 shows the limiting angles of total internal reflection for some substances, the refractive indices of which are given in table. 6. It is easy to verify the validity of relation (84.1).

Table 8. Limiting angle of total internal reflection at the boundary with air

Total internal reflection can be observed at the boundary of air bubbles in water. They shine because the sunlight falling on them is completely reflected without passing into the bubbles. This is especially noticeable in those air bubbles that are always present on the stems and leaves of underwater plants and which in the sun appear to be made of silver, that is, from a material that reflects light very well.

Total internal reflection finds application in the design of glass rotating and turning prisms, the action of which is clear from Fig. 187. The limiting angle for a prism is depending on the refractive index of a given type of glass; Therefore, the use of such prisms does not encounter any difficulties with regard to the selection of the angles of entry and exit of light rays. Rotating prisms successfully perform the functions of mirrors and are advantageous in that their reflective properties remain unchanged, whereas metal mirrors fade over time due to oxidation of the metal. It should be noted that the wrapping prism is simpler in design than the equivalent rotating system of mirrors. Rotating prisms are used, in particular, in periscopes.

Rice. 187. Path of rays in a glass rotating prism (a), a wrapping prism (b) and in a curved plastic tube - light guide (c)

On the image Ashows a normal ray that passes through the air-Plexiglas interface and exits the Plexiglas plate without undergoing any deflection as it passes through the two boundaries between the Plexiglas and the air. On the image b shows a ray of light entering a semicircular plate normally without deflection, but making an angle y with the normal at point O inside the plexiglass plate. When the beam leaves a denser medium (plexiglass), its speed of propagation in a less dense medium (air) increases. Therefore, it is refracted, making an angle x with respect to the normal in air, which is greater than y.



Based on the fact that n = sin (the angle that the beam makes with the normal in the air) / sin (the angle that the beam makes with the normal in the medium), plexiglass n n = sin x/sin y. If multiple measurements of x and y are made, the refractive index of the plexiglass can be calculated by averaging the results for each pair of values. Angle y can be increased by moving the light source in an arc of a circle centered at point O.

The effect of this is to increase the angle x until the position shown in the figure is reached V, i.e. until x becomes equal to 90 o. It is clear that the angle x cannot be greater. The angle that the ray now makes with the normal inside the plexiglass is called critical or limiting angle with(this is the angle of incidence on the boundary from a denser medium to a less dense one, when the angle of refraction in the less dense medium is 90°).

A weak reflected beam is usually observed, as is a bright beam that is refracted along the straight edge of the plate. This is a consequence of partial internal reflection. Note also that when white light is used, the light appearing along the straight edge is split into the colors of the spectrum. If the light source is moved further around the arc, as in the figure G, so that I inside the plexiglass becomes greater than the critical angle c and refraction does not occur at the boundary of the two media. Instead, the beam experiences total internal reflection at an angle r with respect to the normal, where r = i.

To make it happen total internal reflection, the angle of incidence i must be measured inside a denser medium (plexiglass) and it must be greater than the critical angle c. Note that the law of reflection is also valid for all angles of incidence greater than the critical angle.

Diamond critical angle is only 24°38". Its "flare" therefore depends on the ease with which multiple total internal reflection occurs when it is illuminated by light, which depends largely on the skillful cutting and polishing which enhances this effect. Previously it was it is determined that n = 1 /sin c, so an accurate measurement of the critical angle c will determine n.

Study 1. Determine n for plexiglass by finding the critical angle

Place a half-circle piece of plexiglass in the center of a large piece of white paper and carefully trace its outline. Find the midpoint O of the straight edge of the plate. Using a protractor, construct a normal NO perpendicular to this straight edge at point O. Place the plate again in its outline. Move the light source around the arc to the left of NO, all the time directing the incident ray to point O. When the refracted ray goes along the straight edge, as shown in the figure, mark the path of the incident ray with three points P 1, P 2, and P 3.



Temporarily remove the plate and connect these three points with a straight line that should pass through O. Using a protractor, measure the critical angle c between the drawn incident ray and the normal. Carefully place the plate again in its outline and repeat what was done before, but this time move the light source around the arc to the right of NO, continuously directing the beam to point O. Record the two measured values ​​of c in the results table and determine the average value of the critical angle c. Then determine the refractive index n n for plexiglass using the formula n n = 1 /sin s.


The apparatus for Study 1 can also be used to show that for light rays propagating in a denser medium (Plexiglas) and incident on the Plexiglas-air interface at angles greater than the critical angle c, the angle of incidence i is equal to the angle reflections r.

Study 2. Check the law of light reflection for angles of incidence greater than the critical angle

Place the semi-circular plexiglass plate on a large piece of white paper and carefully trace its outline. As in the first case, find the midpoint O and construct the normal NO. For plexiglass, the critical angle c = 42°, therefore, angles of incidence i > 42° are greater than the critical angle. Using a protractor, construct rays at angles of 45°, 50°, 60°, 70° and 80° to the normal NO.

Carefully place the plexiglass plate back into its outline and direct the light beam from the light source along the 45° line. The beam will go to point O, be reflected and appear on the arcuate side of the plate on the other side of the normal. Mark three points P 1, P 2 and P 3 on the reflected ray. Temporarily remove the plate and connect the three points with a straight line that should pass through point O.



Using a protractor, measure the angle of reflection r between and the reflected ray, recording the results in a table. Carefully place the plate into its outline and repeat for angles of 50°, 60°, 70° and 80° to the normal. Record the value of r in the appropriate space in the results table. Plot a graph of the angle of reflection r versus the angle of incidence i. A straight line graph drawn over the range of incidence angles from 45° to 80° will be sufficient to show that angle i is equal to angle r.