EntranceMethods for applying the speed of light. Experimental methods for determining the speed of light
There are various methods for measuring the speed of light, including astronomical ones and using various experimental techniques. Quantity measurement accuracy WITH is constantly increasing. The table provides an incomplete list of experimental works to determine the speed of light.
|
date |
Experiment |
Experimental methods |
Measurement results, km/sec |
|
1676 1725 1849 1850 1857 1868 1875 1880 1883 1883 1901 1907 1928 1932 1941 1952 |
Roemer Bradley Fizeau Foucault Weber-Kohlrausch Maxwell Cornu Michelson Thomson Newcomb Perrotine Rose and Dorsey Mittelyptedt Pease and Pearson Anderson Froome |
Eclipse of Jupiter's moon Light aberration Moving bodies Rotating mirrors Electromagnetic constants Electromagnetic constants Rotating mirrors Rotating mirrors Electromagnetic constants Rotating mirrors Rotating mirrors Electromagnetic constants Kerr gate cell Rotating mirrors Kerr gate cell Microwave interferometry |
214 459 308 000 313 290 298 000 310 000 288 000 299 990 299 910 282 000 299 880 299 777 299 784 299 778 299 774 299 782 299 792.45 |
The pictures show a reproduction of the drawing of the Roemer, as well as a schematic interpretation.
Römer's astronomical method is based on the measurement speed light from observations from Earth of eclipses of Jupiter's satellites. Jupiter has severalabout satellites that are either visible from Earth near Jupiter, or
Let at a certain moment in time the EarthZ1 and Jupiter J1 are in opposition, and at this moment in time one of Jupiter’s satellites, observed from Earth, disappears in the shadow of Jupiter (the satellite is not shown in the figure). Then, if we denote by R and r the radii of the orbits of Jupiter and Earth and by c the speed of light eta in in the coordinate system associated with the Sun C, on Earth, the departure of the satellite into the shadow of Jupiter will be recorded (R-r)/s seconds later than it occurs in the time reporting system associated with Jupiter.
After 0.545 years, Earth Z2 and Jupiter J2 are in conjunction. If at this time the nth eclipse of the same satellite of Jupiter occurs, then on Earth it will be registered with a delay of (R+r)/s seconds. Therefore, if the satellite’s orbital period around Jupiter is t, then the time interval T1 occurring between the first and nth eclipses observed from Earth is equal to
After another 0.545 year, Earth 33 and Jupiter 3 will again be in opposition. During this time, (n-1) revolutions of the satellite around Jupiter and (n-1) eclipses occurred, the first of which took place when the Earth and Jupiter occupied positions Z2 and Yu2, and the last when they occupied the positions Z3 and Yu3. The first eclipse was observed on Earth with a delay (R+r)/s, and the last with a delay (R-r)/s in relation to the moments of the satellite leaving the shadow of the planet Jupiter. Therefore, in this case we have
Roemer measured the time intervals T1 and T2 and found that T1-T2 = 1980 s. But from the formulas written above it follows that T1-T2 = 4r/s, therefore c = 4r/1980 m/s. Taking r, the average distance from the Earth to the Sun, equal to 1500000000 km, we find the value for the speed of light to be 3.01 * 10 6 m/s.
This result was the first measurement of the speed of light.
In 1725 James Bradley discovered that the star Draco, located at the zenith (i.e. directly overhead), makes an apparent motion with a period of one year in an almost circular orbit with a diameter equal to 40.5 arc seconds. For stars visible elsewhere in the sky, Bradley also observed a similar apparent motion - generally elliptical.
The phenomenon observed by Bradley is called aberration. It has nothing to do with the star's own motion. The reason for the aberration is that the speed of light is finite, and observation is carried out from the Earth moving in orbit at a certain speed v.
The angle of the cone at which the apparent trajectory of the star is visible from the Earth is determined by the expression: tgα=ν/c
Knowing the angle α and the speed of the Earth's orbit v, we can determine the speed of light c.
He obtained a value for the speed of light equal to 308,000 km/s.
In 1849, you were the first to determine the speed of light laboratory conditions A. Fizeau. His method was called the cogwheel method. A characteristic feature of his method is the automatic recording of the start and return moments of the signal, carried out by regularly interrupting the light flux (gear wheel).
Figure shows a diagram of an experiment to determine the speed of light using the gear wheel method.
The light from the source passed through the chopper (the teeth of the rotating wheel) and, reflected from the mirror, returned again to the gear wheel. Knowing the distance between the wheel and the mirror, the number of teeth of the wheel, and the speed of rotation, you can calculate the speed of light.
Knowing the distance D, the number of teeth z, the angular speed of rotation (rpm) v, you can determine the speed of light. He got it to be equal to 313,000 km/s.
Throughout his life, the American physicist Albert Abraham Michelson(1852–1931) improved the technique for measuring the speed of light. Creating increasingly complex installations, he tried to obtain results with minimal error. In 1924–1927, Michelson developed an experimental design in which a beam of light was sent from the top of Mount Wilson to the top of San Antonio (a distance of about 35 km). The rotating shutter was a rotating mirror, manufactured with extreme precision and driven by a specially designed high-speed rotor that makes up to 528 revolutions per second.
By changing the rotation frequency of the rotor, the observer achieved the appearance of a stable image of the light source in the eyepiece. Knowing the distance between the installations and the frequency of rotation of the mirror made it possible to calculate the speed of light.
From 1924 to the beginning of 1927, five independent series of observations were carried out, increasing the accuracy of measuring distance and rotor speed. The average measurement result was 299,798 km per second.
The results of all Michelson’s measurements can be written as c = (299796 ± 4) km/s.
The top figure shows a diagram of Michelson's experiment. The figure below shows a simplified diagram of the experiment. The user can change the rotation frequency of the octagonal prism, observing the movement of the light pulse and ensuring that it hits the observer's eyepiece.
The frequency can be changed from 0 to 1100 rpm in steps of 2 s –1. To make it easier to set the frequency in the experiment, a coarse speed control knob has been made; more precise settings can be set using additional keys to the right of the frequency window. The optimal result is achieved at 528 and 1056 rpm. At 0 revolutions, a static beam of light is drawn from the source to the observer.
An example of calculating the speed of light for an experiment in which the observer detects the appearance of light at a mirror rotation frequency of 528 s –1.
Here ν and T are the frequency and period of rotation of the octagonal prism, τ 1 is the time during which the light beam manages to travel the distance L from one installation to another and return back, it is also the time of rotation of one face of the mirror.
Based on materials from www.school-collection.edu.ru
Long before scientists measured the speed of light, they had to work hard to define the very concept of “light.” Aristotle was one of the first to think about this, who considered light to be a kind of mobile substance spreading in space. His ancient Roman colleague and follower Lucretius Carus insisted on the atomic structure of light.
TO XVII century Two main theories of the nature of light were formed - corpuscular and wave. Newton was one of the adherents of the first. In his opinion, all light sources emit tiny particles. During the “flight” they form luminous lines - rays. His opponent, the Dutch scientist Christiaan Huygens, insisted that light is a type of wave motion.
As a result of centuries-old disputes, scientists have come to a consensus: both theories have the right to life, and light is a spectrum of electromagnetic waves visible to the eye.
A little history. How was the speed of light measured?
Most ancient scientists were convinced that the speed of light is infinite. However, the results of research by Galileo and Hooke allowed for its extreme nature, which was clearly confirmed in the 17th century by the outstanding Danish astronomer and mathematician Olaf Roemer.
He made his first measurements by observing the eclipses of Io, the satellite of Jupiter, at the moment when Jupiter and the Earth were located with opposite sides relative to the Sun. Roemer recorded that as the Earth moved away from Jupiter by a distance equal to the diameter of the Earth's orbit, the delay time changed. The maximum value was 22 minutes. As a result of calculations, he received a speed of 220,000 km/sec.
50 years later in 1728, thanks to the discovery of aberration, the English astronomer J. Bradley “refined” this figure to 308,000 km/sec. Later, the speed of light was measured by French astrophysicists François Argot and Leon Foucault, obtaining an output of 298,000 km/sec. An even more accurate measurement technique was proposed by the creator of the interferometer, the famous American physicist Albert Michelson.
Michelson's experiment to determine the speed of light
The experiments lasted from 1924 to 1927 and consisted of 5 series of observations. The essence of the experiment was as follows. A light source, a mirror and a rotating octagonal prism were installed on Mount Wilson in the vicinity of Los Angeles, and a reflecting mirror was installed 35 km later on Mount San Antonio. First, light through a lens and a slit hit a prism rotating with a high-speed rotor (at a speed of 528 rps).
Participants in the experiments could adjust the rotation speed so that the image of the light source was clearly visible in the eyepiece. Since the distance between the vertices and the rotation frequency were known, Michelson determined the speed of light - 299,796 km/sec.
Scientists finally decided on the speed of light in the second half of the 20th century, when masers and lasers were created, characterized by the highest stability of the radiation frequency. By the beginning of the 70s, the error in measurements had dropped to 1 km/sec. As a result, on the recommendation of the XV General Conference on Weights and Measures, held in 1975, it was decided to assume that the speed of light in a vacuum is now equal to 299792.458 km/sec.
Is the speed of light achievable for us?
Obviously, exploration of the far corners of the Universe is unthinkable without spaceships flying at enormous speed. Preferably at the speed of light. But is this possible?
The speed of light barrier is one of the consequences of the theory of relativity. As you know, increasing speed requires increasing energy. The speed of light would require virtually infinite energy.
Alas, the laws of physics are categorically against this. At a spaceship speed of 300,000 km/sec, particles flying towards it, for example, hydrogen atoms, turn into a deadly source of powerful radiation equal to 10,000 sieverts/sec. This is about the same as being inside the Large Hadron Collider.
According to scientists at Johns Hopkins University, there is no adequate protection in nature from such monstrous cosmic radiation. The destruction of the ship will be completed by erosion from the effects of interstellar dust.
Another problem with light speed is time dilation. Old age will become much longer. The visual field will also be distorted, as a result of which the ship’s trajectory will pass as if inside a tunnel, at the end of which the crew will see a shining flash. Behind the ship there will be absolute pitch darkness.
So in the near future, humanity will have to limit its speed “appetites” to 10% of the speed of light. This means that it will take about 40 years to fly to the closest star to Earth, Proxima Centauri (4.22 light years).
One of important properties, is the speed of light propagation in vacuum and other optical media. The enormous value of the speed of light compared to the speed of propagation of various moving objects observed by humans in practical life posed many difficulties both in explaining many optical phenomena and in the practical determination of the speed of light. To show how difficult it was for a person to perceive the possibility of moving matter, in this case light, at enormous speeds, we can give an example of determining the speed of light undertaken by the Italian scientist Galileo Galilei, who, together with his collaborator, positioned themselves on two neighboring mountain peaks and signaled each other with the light of lanterns . One participant in this experiment opened the lid of the lantern and turned on the clock at the same time. The second participant, having received a light signal, also opened the lantern and sent light in the direction of the first experimenter, who, having received a response signal, stopped the clock. Knowing the distance between the tops of the mountains and the time it takes light to travel this distance back and forth, you can get the speed of light. It is, of course, clear to us why this attempt to determine the speed of light did not give the desired results.
It soon became clear that in order to measure the speed of propagation of light with the required accuracy, it was necessary to have large distances for the light to travel, firstly, and it was necessary to measure time with very high accuracy, secondly.
To obtain accurate time readings, light modulation is used, and three main modulation methods are used:
- Gear method,
- Rotating mirror method
- Electric shutter method.
In all these methods, the propagation time is determined from a measurement of the modulation frequency.
Let us briefly consider these three options for light modulation using examples.
Fizeau's method. Figure 1.3.1 shows a schematic diagram of the installation used in the Fizeau method, where the light flux is modulated by a rotating gear wheel. Light from a light source 1 condenser system is directed to a translucent mirror 2 , reflected from which passes between the teeth of a rotating gear wheel 5 . Next, the collimator system 3 directs a beam of rays onto a concave mirror 4 , reflected from which the light travels back along the same path to the translucent mirror 2 . Observation is made by the human eye through an eyepiece 6 .
If the gear wheel is stationary, then the light will pass through the gap between the teeth and return back through the same gap. By setting the gear wheel in rotation and increasing the rotation speed, it is possible to achieve that during the time the light comes from the wheel 5 to the mirror 4 and back the wheel will turn the width of the tooth and the tooth will take the place of the gap. In this case, the light will not enter the eyepiece 6 . By further increasing the speed of rotation of the wheel, you can obtain the passage of light back through the adjacent gap, etc.
Fizeau had a wheel with 720 teeth and a double path length of the light beam of about 17 km. From his experiments, the speed of light turned out to be 3.15. 10 10 cm/With. The main mistake here is related to the difficulty of recording the moment of darkening. Further improvements to this method led to more accurate measurements of the speed of light.
Rotating mirror method. This method, proposed by Wheatstone, was used by Foucault in 1960. The installation diagram is shown in Fig. 1.3.2. From the radiation source 1 light passing through a translucent mirror 2 and lens 3 guided by a rotating mirror 4 to a spherical mirror 5 . Reflected from the mirror 5 , the light flux went back and was focused by the observation system, including A(with a fixed mirror 4 ). With a rotating mirror, during the time the light travels twice the path L, the mirror had time to rotate through a certain angle and the light flux reflected from it in the reverse direction was focused at a point B. Measuring the distance between A And B, we get the angle at which the mirror rotates 4 and, therefore, knowing the speed of rotation of the mirror, the time it takes light to travel the distance. At , the found value of the speed of light propagation turned out to be equal to 2.98. 10 10 cm/With. Distance between A And B was equal to only 0.7 mm, and the main source of errors lay in the inaccuracy of measuring this distance.
Kerr electric shutter method. In this method, a Kerr cell acts as a modulating device (a Kerr cell filled with a polar liquid and placed between crossed nicols transmits light only when an electric field is applied). The installation diagram is shown in Fig. 1.3.3. Light from a mercury lamp 1 passes through a Kerr gate onto a translucent mirror 2 , is reflected from it to the right and hits the mirror 3 . After reflection from mirror 3, the light in the reverse path of the rays hits the energy receiver 8 .
Part of the light energy passes through a translucent mirror and overcomes the path determined by the mirrors 4 , 5 , 6 , 7 and back, also hits the receiver 8 .
The accuracy of this method is determined by the high frequency modulation of the light flux created by the Kerr cell, which is exposed to a high-frequency electric field, and the ability to accurately measure the phase shift of the two light streams coming from the mirror 3 and from the mirror 7 .
The value obtained for the speed of light is . The modern generally accepted value for the speed of light in a vacuum.
For optical media with a refractive index, the speed of light is determined by the expression: .
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Speed of light and methods for determining it
Plan
Introduction
1. Astronomical methods for measuring the speed of light
1.1 Roemer's method
1.2 Light aberration method
1.3 Interruption method (Fizeau method)
1.4 Rotating mirror method (Foucault method)
1.5 Michelson method
Introduction
The speed of light is one of the most important physical constants, which are called fundamental. This constant is of particular importance in both theoretical and experimental physics and related sciences. The exact value of the speed of light is required to be known in radio and light location, when measuring distances from the Earth to other planets, controlling satellites and spaceships. Determining the speed of light is most important for optics, in particular for the optics of moving media, and physics in general. Let's get acquainted with methods for determining the speed of light.
1. Astronomical methods for measuring the speed of light
1.1 Roemer's method
The first measurements of the speed of light were based on astronomical observations. A reliable value for the speed of light, close to its modern value, was first obtained by Roemer in 1676 while observing eclipses of the satellites of the planet Jupiter.
The time it takes for a light signal to travel from a celestial body to the Earth depends on the distance L location of the luminary. A phenomenon that occurs on some celestial body, is observed with a delay equal to the time of passage of light from the luminary to the Earth:
Where With- speed of light.
If we observe any periodic process occurring in a system distant from the Earth, then, with a constant distance between the Earth and the system, the presence of this delay will not affect the period of the observed process. If during the period the Earth moves away from the system or approaches it, then in the first case the end of the period will be recorded with a greater delay than its beginning, which will lead to an apparent increase in the period. In the second case, on the contrary, the end of the period will be recorded with less delay than its beginning, which will lead to an apparent decrease in the period. In both cases, the apparent change in period is equal to the ratio of the difference in distances between the earth and the system at the beginning and end of the period to the speed of light.
The above considerations form the basis of Römer's method.
Roemer conducted observations of the satellite Io, whose orbital period is 42 hours 27 minutes 33 seconds.
When the Earth moves along a portion of its orbit E 1 E 2 E 3 it is moving away from Jupiter and an increase in period should be observed. When moving around the area E 3 E 4 E 1 the observed period will be less than the true one. Since the change in one period is small (about 15 s), the effect is detected only when large number observations carried out over a long period of time. If, for example, you observe eclipses for six months, starting from the moment of opposition to the Earth (point E 1 ) until the moment of “connection” (point E 3 ), then the time interval between the first and last eclipses will be 1320 s longer than theoretically calculated. Theoretical calculation of the eclipse period was carried out at orbital points close to opposition. Where the distance between Earth and Jupiter practically does not change over time.
The resulting discrepancy can only be explained by the fact that within six months the Earth moved from the point E 1 exactly E 3 and the light at the end of the half-year has to travel a path greater than at the beginning, by the size of the segment E 1 E 3 , equal to the diameter of the earth's orbit. Thus, delays that are imperceptible for a particular period accumulate and form the resulting delay. The delay value determined by Roemer was 22 minutes. Taking the diameter of the Earth's orbit equal to km, we can obtain a value for the speed of light of 226,000 km/s.
The speed of light determined based on Roemer's measurements turned out to be less than the modern value. Later, more accurate observations of eclipses were made, in which the delay time turned out to be 16.5 minutes, which corresponds to the speed of light 301000 km/s.
1.2 Light aberration method
light speed measurement astronomical
For an observer on earth, the direction of the line of sight to the star will be different if this direction is determined at different times of the year, that is, depending on the position of the Earth in its orbit. If the direction to any star is determined at six-month intervals, that is, when the Earth is at opposite ends of the diameter of the Earth’s orbit, then the angle between the resulting two directions is called annual parallax (Fig. 2). The farther away a star is, the smaller its parallax angle. By measuring the parallax angles of various stars, it is possible to determine the distance of these stars from our planet.
In 1725-1728 Bradley James, an English astronomer, measured the annual parallax of the fixed stars. While observing one of the stars in the constellation Draco, he discovered that its position changed throughout the year. During this time, she described a small circle, the angular dimensions of which were equal to 40.9”. In the general case, as a result of the Earth's orbital motion, the star describes an ellipse, the major axis of which has the same angular dimensions. For stars lying in the ecliptic plane, the ellipse degenerates into a straight line, and for stars lying near the pole - into a circle. (The ecliptic is the great circle celestial sphere, along which the apparent annual movement of the Sun occurs.)
The amount of displacement measured by Bradley was significantly greater than the expected parallactic displacement. Bradley called this phenomenon an aberration of light and explained it by the finite speed of light. During the short time during which the light falling on the telescope lens spreads from the lens to the eyepiece, the eyepiece shifts by a very small segment as a result of the Earth’s orbital movement (Fig. 3). As a result, the image of the star will shift by a segment A. When pointing the telescope again at the star, it will have to be tilted slightly in the direction of the Earth's movement so that the image of the star again coincides with the center of the crosshairs in the eyepiece.
Let the angle of inclination of the telescope be equal to b. Let us denote the time required for light to travel a segment V, equal to the distance from the telescope lens to its eyepiece, is equal to f. Then the segment, and
From Bradley's measurements it was known that at two positions of the Earth lying on the same orbital diameter, the star appears displaced from its true position by the same angle. The angle between these observation directions, from where, knowing the speed of the Earth in orbit, the speed of light can be found. Bradley got With= 306000 km/s.
It should be noted that the phenomenon of light aberration is associated with a change in the direction of the Earth's speed throughout the year. The explanation of this phenomenon is based on corpuscular concepts of light. Consideration of light aberration from the standpoint of wave theory is more complex and is associated with the question of the influence of the Earth's motion on the propagation of light.
Roemer and Bradley showed that the speed of light is finite, although it is of great importance. For the further development of the theory of light, it was important to establish on what parameters the speed of light depends and how it changes when light passes from one medium to another. To do this, it was necessary to develop methods for measuring the speed of light from terrestrial sources. The first attempts at such experiments were made in early XIX century.
1.3 Interruption method (Fizeau method)
The first experimental method for determining the speed of light from terrestrial sources was developed in 1449 by the French physicist Armand Hippolyte Louis Fizeau. The experimental scheme is shown in Fig. .4.
Light spreading from a source s, partially reflected from a translucent plate R and goes to the mirror M. In the path of the beam there is a light breaker - a gear wheel TO, whose axis OO" parallel to the beam. Rays of light pass through the gaps between the teeth and are reflected by the mirror M and are sent back through the gear and plate R to the observer.
When the wheel rotates slowly TO the light, having passed through the gap between the teeth, manages to return through the same gap and enters the eye of the observer. At those moments when the path of the rays is crossed by a tooth, the light does not reach the observer. Thus, at low angular velocity, the observer perceives flickering light. If you increase the speed of rotation of the wheel, then at a certain value the light passing through one gap between the teeth, reaching the mirror and returning back, will not fall into the same gap d, but will be blocked by a tooth that has taken the position of the gap at this moment d. Consequently, at angular velocity, no light will enter the observer’s eye at all from the gap d, nor from all subsequent ones (first darkening). If we take the number of teeth P, then the time for turning the wheel on the slider is equal to
Time it takes light to travel the distance from the wheel to the mirror M and vice versa is equal
Where l- distance to the wheel from the mirror (base). Equating these two time intervals, we obtain the condition under which the first darkening occurs:
where can you determine the speed of light:
where is the number of revolutions per second.
In the Fizeau installation, the base was 8.63 km, the number of teeth in the wheel was 720, and the first darkening occurred at a frequency of 12.6 rps. If you double the speed of the wheel, a brightened field of view will be observed; at triple the rotation speed, darkness will occur again, etc. The speed of light calculated by Fizeau is 313300 km/s.
The main difficulty of such measurements is to accurately determine the moment of darkening. Accuracy increases both with larger bases and with interruption rates that allow higher order obscurations to be observed. Thus, Perrotin in 1902 carried out measurements with a base length of 46 km and obtained a value for the speed of light of 29987050 km/s. The work was carried out in extremely clean sea air using high-quality optics.
Instead of a rotating wheel, other, more advanced light interruption methods can be used, for example, a Kerr cell, which can be used to interrupt a light beam 107 times per second. In this case, you can significantly reduce the base. Thus, in Anderson’s setup (1941) with a Kerr cell and photoelectric recording, the base was only 3 m. He obtained the value With= 29977614 km/s.
1.4 Rotating mirror method (Foucault method)
The method for determining the speed of light, developed in 1862 by Foucault, can be attributed to the first laboratory methods. Using this method, Foucault measured the speed of light in media for which the refractive index n>1 .
The diagram of the Foucault installation is shown in Fig. 5.
Light from source S passes through a translucent plate R, lens L and falls on a flat mirror M1, which can rotate around its axis ABOUT, perpendicular to the drawing plane. After reflection from the mirror M1 a beam of light is directed onto a fixed concave mirror M 2, located so that this ray always falls perpendicular to its surface and is reflected along the same path onto the mirror M1 . If the mirror M1 motionless, then the beam reflected from it will return along its original path to the plate R, partially reflected from which it will give an image of the source S at the point S1 .
When the mirror rotates M1 during the time it takes the light to travel 2 l between both mirrors and returns back (), a mirror rotating with angular velocity M1 will turn to an angle
and will take the position shown in Fig. .5 dotted line. The beam reflected from the mirror will be rotated at an angle relative to the original one and will give an image of the source at the point S2 . Measuring the distance S1 S2 and knowing the geometry of the installation, you can determine the angle and calculate the speed of light:
Thus, the essence of the Foucault method is to accurately measure the time it takes light to travel a distance 2 l. This time is estimated by the angle of rotation of the mirror M1 , the rotation speed of which is known. Rotation angle is determined based on displacement measurements S1 S2 . In Foucault's experiments, the rotation speed was 800 rps, the base l varied from 4 to 20 km. The value was found With= 298000500 km/s.
Foucault was the first to measure the speed of light in water using his installation. Having placed a pipe filled with water between the mirrors, Foucault discovered that the shift angle increased by * times, and therefore, the speed of light propagation in water calculated using the formula written above turned out to be equal to (3/4) With. The refractive index of light in water, calculated using the formulas of the wave theory, turned out to be equal, which is fully consistent with Snell’s law. Thus, based on the results of this experiment, the validity of the wave theory of light was confirmed, and a century and a half dispute in its favor was ended.
1.5 Michelson method
In 1926, a Michelson installation was made between two mountain peaks, so that the distance traveled by a ray from a source to its image after reflections from the first face of an octagonal mirror prism, mirrors M 2 - M 7 and the fifth face was about 35.4 km. The rotation speed of the prism (approximately 528 rps) was chosen so that during the time of light propagation from the first facet to the fifth, the prism had time to rotate 1/8 of a revolution. The possible displacement of the bunny at an inaccurately selected speed played the role of a correction. The speed of light determined in this experiment turned out to be equal to 2997964 km/s.
Among other methods, we note the measurement of the speed of light performed in 1972 by independently determining the wavelength and frequency of light. The light source was a helium-neon laser generating radiation at 3.39 μm. In this case, the wavelength was measured using interferometric comparison with the standard length of krypton's orange radiation, and the frequency was measured using radio engineering methods. Speed of light
determined by this method was 299792.45620.001 km/s. The authors of the method believe that the achieved accuracy can be increased by improving the reproducibility of measurements of length and time standards.
In conclusion, we note that when determining the speed of light, the group velocity is measured And, which coincides with the phase one only for vacuum.
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Laboratory methods for determining the speed of light are essentially improvements on Galileo's method.
a) Interrupt method.
Fizeau (1849) was the first to determine the speed of light in laboratory conditions. A characteristic feature of his method is the automatic recording of the start and return moments of the signal, carried out by regularly interrupting the light flux (gear wheel). The scheme of Fizeau's experiment is shown in Fig. 9.3. Light from source S goes between the teeth of a rotating wheel W to the mirror M and, having been reflected back, must again pass between the teeth to the observer. For convenience, eyepiece E, serving for observation, is placed opposite A, and the light turns from S To W using a translucent mirror N. If the wheel rotates, and at such an angular speed that during the movement of light from A To M and back in place of the teeth there will be slits, and vice versa, then the returning light will not be transmitted to the eyepiece and the observer will not see the light (the first eclipse). As the angular velocity increases, the light will partially reach the observer. If the width of the teeth and gaps are the same, then at double speed there will be a maximum of light, at triple speed there will be a second eclipse, etc. Knowing the distance aM=D, number of teeth z, angular speed of rotation (number of revolutions per second) n, you can calculate the speed of light.
| Rice. 9.3. Scheme of the interruption method experiment. |
Or With=2Dzn.
The main difficulty of determination lies in the exact moment of the eclipse. Accuracy increases with increasing distance D and at interruption speeds that allow observation of higher order eclipses. Thus, Perrotin made his observations at D=46 km and observed a 32nd order eclipse. Under these conditions, high-aperture installations are required, fresh air(observations in the mountains), good optics, strong light source.
IN Lately Instead of a rotating wheel, other, more advanced methods of interrupting light are successfully used.
b) Rotating mirror method.
Foucault (1862) successfully implemented the second method, the principle of which had been proposed even earlier (1838) by Arago for the purpose of comparing the speed of light in air with the speed of light in other media (water). The method is based on very careful measurements of short periods of time using a rotating mirror. The experimental design is clear from Fig. 9.4. Light from source S guided by the lens L on a rotating mirror R, is reflected from it in the direction of the second mirror WITH and goes back, passing path 2 CR=2D during t. This time is estimated by the angle of rotation of the mirror R, the rotation speed of which is precisely known; the angle of rotation is determined from measuring the displacement of the bunny given by the returning light. Measurements are made using an eyepiece E and translucent plate M, playing the same role as in the previous method; S 1 – position of the bunny with a stationary mirror R, S" 1 – when the mirror rotates. Important Feature Foucault's installation was used as a mirror WITH concave spherical mirror, with the center of curvature lying on the axis of rotation R. Due to this, the light reflected from R To WITH, always ended up back on R; in the case of using a flat mirror WITH this would happen only with a certain mutual orientation R And WITH, when the axis of the reflected cone of rays is located normal to WITH.
Foucault, in accordance with Arago's original plan, also used his device to determine the speed of light in water, because he managed to reduce the distance RС up to 4 m, giving the mirror 800 revolutions per second. Foucault's measurements showed that the speed of light in water is less than in air, in accordance with the ideas of the wave theory of light.
Michelson's last (1926) installation was made between two mountain peaks, so the resulting distance D» 35.4 km (more precisely, 35,373.21 m). The mirror was an octagonal steel prism rotating at a speed of 528 rps.
The time it took for the light to travel the full way was 0.00023 s, so the mirror had time to rotate 1/8 of a revolution and the light fell on the edge of the prism. Thus, the bunny’s displacement was relatively insignificant, and the determination of its position played the role of a correction, and not the main measured value, as in Foucault’s first experiments, where the entire displacement reached only 0.7 mm.
Very accurate measurements of the speed of propagation of radio waves were also made. In this case, radiogeodetic measurements were used, i.e. determining the distance between two points using radio signals in parallel with precise triangulation measurements. The best value obtained by this method, reduced to vacuum, is c = 299,792 ± 2.4 km/s. Finally, the speed of radio waves was determined using the method of standing waves generated in a cylindrical resonator. The theory allows us to relate data on the dimensions of the resonator and its resonant frequency with the speed of the waves. The experiments were done with an evacuated resonator, so reduction to a vacuum was not required. Best value, obtained using this method, c = 299,792.5 ± 3.4 km/s.
c) Phase and group speeds of light.
Laboratory methods for determining the speed of light, which allow these measurements to be made on a short basis, make it possible to determine the speed of light in various media and, therefore, test the relationships of the theory of light refraction. As has already been mentioned several times, the refractive index of light in Newton’s theory is equal to n=sin i/sin r=υ 2 /υ 1, and in the wave theory n=sin i/sin r=υ 1 /υ 2 where υ 1 is the speed of light in the first medium, and υ 2 – speed of light in the second medium. Arago also saw in this difference the possibility of experimentum crucis and proposed the idea of an experiment, which was carried out later by Foucault, who found for the ratio of the speeds of light in air and water a value close to , as follows from Huygens’ theory, and not, as follows from Newton’s theory.
Conventional determination of refractive index n=sin i/sin r=υ 1 /υ 2 from the change in the direction of the wave normal at the boundary of two media gives the ratio of the phase velocities of the wave in these two media. However, the concept of phase velocity is applicable only to strictly monochromatic waves, which are not realistically feasible, since they would have to exist indefinitely in time and howl infinitely extended in space.
In reality, we always have a more or less complex impulse, limited in time and space. When observing such a pulse, we can highlight some specific place, for example, the place of maximum extension of that electrical or magnetic field, which is an electromagnetic pulse. The speed of the pulse can be identified with the speed of propagation of any point, for example, the point of maximum field strength.
However, the medium (with the exception of vacuum) is usually characterized by dispersion, i.e. monochromatic waves propagate with different phase velocities depending on their length, and the pulse begins to deform. In this case, the question of the speed of the impulse becomes more complex. If the dispersion is not very large, then the pulse deformation occurs slowly and we can monitor the movement of a certain field amplitude in the wave pulse, for example, the maximum field amplitude. However, the speed of movement of the pulse, called by Rayleigh group velocity, will differ from the phase velocity of any of its constituent monochromatic waves.
For simplicity of calculations, we will imagine a pulse as a set of two sinusoids of equal amplitude that are close in frequency, and not as a set of an infinite number of close sinusoids. With this simplification, the main features of the phenomenon are preserved. So, our impulse, or, as they say, a group of waves, is composed of two waves.
where the amplitudes are assumed to be equal, and the frequencies and wavelengths differ little from each other, i.e.
where and are small quantities. Impulse (wave group) at there is a sum at 1 and at 2, i.e.
Introducing the notation, let us represent our momentum in the form where A not constantly, but changes in time and space, but changes slowly, because δω And δk– small (compared to ω 0 and κ 0) quantities. Therefore, allowing for a certain carelessness in speech, we can consider our impulse to be a sinusoid with a slowly changing amplitude.
Thus, the speed of the impulse (group), which, according to Rayleigh, is called group velocity, is the speed of movement amplitudes, and, consequently, energy, carried by a moving impulse.
So, a monochromatic wave is characterized by a phase velocity υ=ω /κ , indicating the speed of movement phases, and the impulse is characterized by the group velocity u=dω/dκ, corresponding to the speed of propagation of the field energy of this pulse.
It is not difficult to find a connection between u And υ . Indeed,
or, since and therefore,
those. finally
(Rayleigh formula).
Difference between u And υ the more significant the greater the dispersion dυ/dλ. In the absence of dispersion ( dυ/dλ=0) we have u=υ. This case, as already said, occurs only for vacuum.
Rayleigh showed that in the known methods for determining the speed of light, by the very essence of the method, we are not dealing with a continuously lasting wave, but breaking it into small segments. The gear wheel and other interrupters in the interruption method provide weakening and increasing light excitation, i.e. group of waves. The same thing happens in Roemer's method, where the light is interrupted by periodic darkening. In the rotating mirror method, light also stops reaching the observer when the mirror is rotated sufficiently. In all these cases, we measure the group velocity in a dispersive medium, not the phase velocity.
Rayleigh believed that in the light aberration method we measure the direct phase velocity, because there the light is not interrupted artificially. However, Ehrenfest (1910) showed that the observation of light aberration is in principle indistinguishable from Fizeau’s method, i.e. also gives group velocity. Indeed, the aberration experience can be reduced to the following. Two disks with holes are rigidly fixed on a common axis. Light is sent along a line connecting these holes and reaches the observer. Let's put the whole apparatus into rapid rotation. Since the speed of light is finite, light will not pass through the second hole. To transmit light, it is necessary to rotate one disk relative to the other by an angle determined by the ratio of the speeds of the disks and light. This is a typical aberration experience; however, it is no different from Fizeau’s experiment, in which instead of two rotating disks with holes, there is one disk and a mirror for turning the rays, i.e. essentially two disks: the real one and its reflection in a fixed mirror. So, the aberration method gives the same as the interruption method, i.e. group speed.
Thus, in Michelson's experiments with both water and carbon disulfide, the ratio of group rather than phase velocities was measured.