One of the important laws of the propagation of light waves in transparent substances is the law of refraction, formulated at the beginning of the XVII century by the Dutchman Snell. The parameters that appear in the mathematical formulation of the phenomenon of refraction are the refractive indices and angles. This article discusses how light rays behave when passing through the surface of different media.
What is the phenomenon of refraction?
The main property of any electromagnetic wave is its rectilinear motion in homogeneous (homogeneous) space. If any inhomogeneity occurs, the wave experiences a deviation from a rectilinear trajectory to a greater or lesser extent. This heterogeneity may be the presence of a strong gravitational or electromagnetic field in a certain area of space. In this article, these cases will not be considered, and attention will be paid specifically to inhomogeneities associated with the substance.
The refraction effect of a ray of light in its classical formulation means a sharp change in one rectilinear direction of motion of this ray to another when passing through a surface that delimits two different transparent media.
The following examples satisfy the definition given above:
- beam transition from air to water;
- from glass to water;
- from water to diamond, etc.
Why does this occur?
The only reason for the described effect is the difference in the speeds of electromagnetic waves in two different media. If there is no such difference, or it will be insignificant, then when passing through the interface, the beam will retain its original direction of propagation.
Different transparent media have different physical density, chemical composition, temperature. All these factors affect the speed of light. For example, the phenomenon of a mirage is a direct consequence of the refraction of light in layers of air heated to different temperatures near the earth's surface.
The main laws of refraction
There are two of these laws, and everyone can check them if he is armed with a protractor, a laser pointer and a thick piece of glass.
Before formulating them, it is worth introducing some notation. The refractive index is written with the symbol n i , where i - identifies the corresponding medium. The angle of incidence is denoted by θ 1 (theta one), and the angle of refraction θ 2 (theta two). Both angles are measured relative to the normal to it rather than the section plane.
Law No. 1. The normal and two rays (θ 1 and θ 2 ) lie in the same plane. This law is completely analogous to the 1st law for reflection.
Law number 2. For the phenomenon of refraction, the equality is always true:
n 1 * sin (θ 1 ) = n 2 * sin (θ 2 ).
In the given form, this relation is the easiest to remember. In other forms, it looks less convenient. Below are two more options for recording Law No. 2:
sin (θ 1 ) / sin (θ 2 ) = n 2 / n 1 ;
sin (θ 1 ) / sin (θ 2 ) = v 1 / v 2 .
Where v i is the wave velocity in the i-th medium. The second formula is easily obtained from the first direct substitution of the expression for n i :
n i = c / v i .
Both of these laws are the result of numerous experiments and generalizations. However, they can be mathematically obtained using the so-called principle of least time or the Fermat principle. In turn, the Fermat principle is derived from the Huygens-Fresnel principle on secondary wave sources.
Features of Law No. 2
n 1 * sin (θ 1 ) = n 2 * sin (θ 2 ).
It can be seen that the larger the exponent n 1 (a dense optical medium in which the speed of light decreases strongly), the closer θ 1 will be to the normal (the function sin (θ) monotonically increases in the interval [0 o , 90 o ]).
The refractive indices and the velocities of electromagnetic waves in media are tabular values measured experimentally. For example, for air n is 1,00029, for water 1.33, for quartz 1.46, and for glass about 1.52. The light slows down its movement in the diamond (almost 2.5 times), its refractive index is 2.42.
The above figures indicate that any transition of the beam from the marked media into the air will be accompanied by an increase in the angle (θ 2 > θ 1 ). When changing the direction of the beam, the opposite conclusion is true.
The refractive index depends on the frequency of the wave. The above figures for different media correspond to a wavelength of 589 nm in vacuum (yellow). For blue light, these indicators will be slightly larger, and for red - less.
It is worth noting that the angle of incidence is equal to the angle of refraction of the beam in only one single case, when the indices n 1 and n 2 are the same.
Next, we consider two different cases of applying this law with the example of media: glass, air and water.
A beam moves from air to glass or water.
It is worth considering two cases for each environment. You can take for example the angles of incidence of 15 o and 55 o at the border of glass and water with air. The angle of refraction in water or glass can be calculated by the formula:
θ 2 = arcsin (n 1 / n 2 * sin (θ 1 )).
The first medium in this case is air, that is, n 1 = 1,00029.
Substituting the known angles of incidence into the expression above, we get:
(n 2 = 1.33): θ 2 = 11.22 o (θ 1 = 15 o ) and θ 2 = 38.03 o (θ 1 = 55 o );
(n 2 = 1.52): θ 2 = 9.81 o (θ 1 = 15 o ) and θ 2 = 32.62 o (θ 1 = 55 o ).
The data obtained allow us to draw two important conclusions:
- Since the angle of refraction from air to glass is smaller than for water, the glass changes the direction of the rays somewhat more strongly.
- The larger the angle of incidence, the stronger the beam deviates from the original direction.
Light moves from water or glass to air
It is interesting to calculate what the angle of refraction is for such a reverse case. The calculation formula remains the same as in the previous paragraph, only now the indicator n 2 = 1,00029, that is, corresponds to air. It will turn out
- when the beam moves out of the water:
(n 1 = 1.33): θ 2 = 20.13 o (θ 1 = 15 o ) and θ 2 = does not exist (θ 1 = 55 o );
- when a ray of glass moves:
(n 1 = 1.52): θ 2 = 23.16 o (θ 1 = 15 o ) and θ 2 = does not exist (θ 1 = 55 o ).
For the angle θ 1 = 55 o it is not possible to determine the corresponding θ 2 . This is due to the fact that it turned out to be more than 90 o . This situation is called total reflection inside an optically dense medium.
This effect is characterized by critical angles of incidence. They can be calculated by equating in law No. 2 sin (θ 2 ) to one:
θ 1c = arcsin (n 2 / n 1 ).
Substituting the indicators for glass and water in this expression, we get:
(n 1 = 1.33): θ 1c = 48.77 ° ;
(n 1 = 1.52): θ 1c = 41.15 o .
Any angle of incidence, which will be greater than the obtained values for the corresponding transparent media, will lead to the effect of complete reflection from the interface, that is, the refracted beam will not exist.