Energy density and intensity of an electromagnetic wave. Light intensity and how to measure it

Intensity- scalar physical quantity, which quantitatively characterizes the power carried by the wave in the direction of propagation. Numerically, the intensity is equal to the radiation power averaged over the oscillation period of the wave passing through a single area located perpendicular to the direction of energy propagation. In mathematical form, this can be expressed as follows:

where is the period of the wave, is the power carried by the wave through the area.

The intensity of the wave is related to the average energy density in the wave and the speed of wave propagation by the following relation:

The unit of intensity in international system units (SI) is W / m², in the CGS system - erg / s cm².

Volumetric energy density electro magnetic field in a linear isotropic medium, as is known from electrodynamics, is given by the expression (here we also took into account the relationship between the vectors E and H in an electromagnetic wave):

Energy flux density vector electromagnetic wave(what in the theory of elastic waves is called the Umov vector) is called the Umov-Poynting vector, or more often just the Poynting vector R :

The mean value modulus of the Poynting vector is called intensity electromagnetic wave:

In the case of a sinusoidal monochromatic plane (when the planes of oscillations of the vectors E and H do not change with time) of an electromagnetic wave propagating in the direction X:

for intensity it turns out:

It should be noted that the intensity of an electromagnetic wave depends on the amplitude (either electric or magnetic field; they are related), but does not depend on the frequency of the wave - in contrast to the intensity of elastic mechanical waves.

The concept of coherence.

In physics, coherence is the correlation (consistency) of several oscillatory or wave processes in time, which manifests itself when they are added. Oscillations are coherent if the difference between their phases is constant in time, and when the oscillations are added, an oscillation of the same frequency is obtained.

A classic example of two coherent oscillations is two sinusoidal oscillations of the same frequency.

Wave coherence means that at different spatial points the waves oscillate synchronously, that is, the phase difference between two points does not depend on time. Lack of coherence, therefore - a situation where the phase difference between two points is not constant, but changes with time. Such a situation can take place if the wave was generated not by a single emitter, but by a set of identical, but independent (that is, uncorrelated) emitters.

The study of the coherence of light waves leads to the concepts of temporal and spatial coherence. When electromagnetic waves propagate in waveguides, phase singularities can occur. In the case of waves on water, the coherence of the wave is determined by the so-called second periodicity.

Without coherence, it is impossible to observe such a phenomenon as interference.

Wave interference- mutual increase or decrease in the resulting amplitude of two or more coherent waves when they are superimposed on each other. It is accompanied by alternation of maxima (antinodes) and minima (nodes) of intensity in space. The result of the interference (interference pattern) depends on the phase difference of the superimposed waves.

All waves can interfere, but a stable interference pattern will be observed only if the waves have the same frequency and the oscillations in them are not orthogonal. Interference can be stationary or non-stationary. A stationary interference pattern can only be given completely coherent waves. For example, two spherical waves on the surface of water propagating from two coherent point sources, when interfered, will give the resulting wave, the front of which will be a sphere.

During interference, the wave energy is redistributed in space. This does not contradict the law of conservation of energy because, on average, for a large region of space, the energy of the resulting wave is equal to the sum of the energies of the interfering waves.

When incoherent waves are superimposed, the average value of the squared amplitude (that is, the intensity of the resulting wave) is equal to the sum of the squared amplitudes (intensities) of the superimposed waves. The energy of the resulting oscillations of each point of the medium is equal to the sum of the energies of its oscillations, due to all incoherent waves separately. It is the difference between the resulting intensity of the wave process and the sum of the intensities of its components that is the sign of interference.

It can vary greatly, and visually we are not able to determine the degree of illumination, since the human eye is endowed with the ability to adapt to different lighting conditions. Meanwhile, the lighting intensity is extremely importance in a wide variety of fields of activity. For example, we can take the process of film or video filming, as well as, for example, growing indoor plants.

The human eye perceives light from 380 nm ( purple) up to 780 nm (red). Best of all, we perceive waves with a wavelength that is just not the most suitable for plants. Lighting that is bright and pleasant to our eyes may not be suitable for plants in a greenhouse, which may not receive the waves important for photosynthesis.

Light intensity is measured in lux. On a bright sunny afternoon in our middle lane it reaches about 100,000 lux, in the evening it drops to 25,000 lux. In a dense shadow, its value is tenths of these values. Indoors, the intensity of sunlight is much less, because the light is weakened by trees and window panes. The brightest lighting (on south window in the summer right behind the glass) at best 3-5 thousand lux, in the middle of the room (2-3 meters from the window) - only 500 lux. This is the minimum light necessary for the survival of plants. For normal growth, even unpretentious ones require at least 800 lux.

We cannot determine the intensity of light by eye. To do this, there is a device, the name of which is a luxmeter. When buying it, you need to clarify the wave range measured by it, because. The capabilities of the device, although wider than the capabilities of the human eye, are still limited.

Light intensity can also be measured with a camera or a photometer. True, you will have to recalculate the received units into suites. To carry out the measurement, you need to put in the place of measurement White list paper and point a camera at it, the sensitivity of which is set to 100, and the aperture to 4. Having determined the shutter speed, its denominator should be multiplied by 10, the resulting value will approximately correspond to the illumination in lux. For example, with a shutter speed of 1/60 sec. lighting around 600 lux.

If you are fond of growing flowers and caring for them, then, of course, you know that light energy is vital for plants for normal photosynthesis. Light affects the growth rate, direction, development of the flower, size and shape of its leaves. With a decrease in light intensity, all processes in plants slow down proportionally. Its amount depends on how far the light source is, on the side of the horizon to which the window is facing, on the degree of shading by street trees, on the presence of curtains or blinds. The brighter the room, the more actively the plants grow and the more they need water, heat and fertilizer. If the plants grow in the shade, then they require less maintenance.

When shooting a movie or TV show, lighting is very important. High-quality shooting is possible with illumination of about 1000 lux, achieved in a television studio with the help of special lamps. But acceptable image quality can be obtained with less lighting.

The intensity of lighting in the studio before and during shooting is measured using exposure meters or high-quality color monitors that are connected to the video camera. Before shooting, it is best to walk with the exposure meter around film set in order to determine its dark or overly lit areas in order to avoid negative phenomena when viewing the footage. In addition, by properly adjusting the lighting, you can achieve additional expressiveness of the scene being shot and the desired directorial effects.

Thus, in geometric optics light wave can be considered as a beam of rays. The rays, however, by themselves determine only the direction of propagation of light at each point; the question remains about the distribution of light intensity in space.

Let us single out an infinitesimal element on one of the wave surfaces of the considered beam. It is known from differential geometry that every surface has at each of its points two, generally speaking, different principal radii of curvature.

Let (Fig. 7) be the elements of the principal circles of curvature drawn on a given element of the wave surface. Then the rays passing through the points a and c will intersect each other at the corresponding center of curvature and the rays passing through b and d will intersect at the other center of curvature.

At given angles of opening, the rays emanating from the length of the segments are proportional to the corresponding radii of curvature (i.e., the lengths and); the area of ​​the surface element is proportional to the product of the lengths, i.e., proportional. In other words, if we consider an element of the wave surface, limited by a certain number of rays, then when moving along them, the area of ​​this element will change proportionally.

On the other hand, the intensity, i.e., the density of the energy flux, is inversely proportional to the surface area through which a given amount of light energy passes. Thus, we conclude that the intensity

This formula should be understood as follows. On each given ray (AB in Fig. 7) there are certain points and , which are the centers of curvature of all wave surfaces that intersect this ray. The distances u from the point O of the intersection of the wave surface with the beam to the points are the radii of curvature of the wave surface at the point O. Thus, formula (54.1) determines the light intensity at the point O on a given beam as a function of the distances to certain points on this beam. We emphasize that this formula is unsuitable for comparing intensities in different points the same wave surface.

Since the intensity is determined by the square of the field modulus, to change the field itself along the beam, we can write:

where in the phase factor, R can be understood as both and the quantities differ from each other only by a constant (for a given ray) factor, since the difference , the distance between both centers of curvature, is constant.

If both radii of curvature of the wave surface coincide, then (54.1) and (54.2) have the form

This is the case, in particular, always in cases where light is emitted from a point source (the wave surfaces are then concentric spheres, and R is the distance to the light source).

From (54.1) we see that the intensity becomes infinite at points, i.e., at the centers of curvature of the wave surfaces. Applying this to all rays in the beam, we find that the intensity of light in a given beam goes to infinity, generally speaking, on two surfaces - the locus of all centers of curvature of the wave surfaces. These surfaces are called caustics. In the special case of a beam of rays with spherical wave surfaces, both caustics merge into one point (focus).

Note that, according to the properties of the locus of centers of curvature of a family of surfaces known from differential geometry, the rays touch the caustics.

It must be borne in mind that (for convex wave surfaces) the centers of curvature of the wave surfaces may turn out to lie not on the rays themselves, but on their extensions beyond optical system from which they originate. In such cases one speaks of imaginary caustics (or imaginary foci). In this case, the intensity of light does not go to infinity anywhere.

As regards the intensification of the intensity to infinity, in reality, of course, the intensity at the points of the caustic becomes large, but remains finite (see the problem in § 59). The formal conversion to infinity means that the approximation geometric optics becomes in any case inapplicable near caustics. Related to the same circumstance is the fact that the phase change along the ray can be determined by formula (54.2) only in sections of the ray that do not include points of contact with the caustics. Below (in § 59) it will be shown that, in fact, when passing by the caustic, the phase of the field decreases by . This means that if in the section of the beam before it touches the first caustic, the field is proportional to the factor - the coordinate along the beam), then after passing by the caustic, the field will be proportional. The same will happen near the point of contact of the second caustic, and beyond this point the field will be proportional

Let us now calculate the total energy emitted by the charge during acceleration. For generality, let us take the case of arbitrary acceleration, however, assuming that the motion is nonrelativistic. When the acceleration is directed, say, vertically, electric field radiation is equal to the product of the charge and the projection of the retarded acceleration, divided by the distance. Thus, we know the electric field at any point, and from here we know the energy passing through the unit area for.

The value is often found in formulas for the propagation of radio waves. Its reciprocal can be called vacuum impedance (or vacuum resistance); it is equal to . Hence the power (in watts per square meter) is the mean square of the field divided by 377.

Using formula (29.1) for electric field we get

, (32.2)

where is the power at , radiated at an angle . As already noted, inversely proportional to the distance. By integrating, we obtain from here the total power radiated in all directions. To do this, first multiply by the area of ​​the strip of the sphere, then we get the energy flow in the interval of the angle (Fig. 32.1). The area of ​​the strip is calculated as follows: if the radius is , then the thickness of the strip is , and the length is , since the radius of the annular strip is . So the area of ​​the strip is

(32.3)

Figure 32.1. The area of ​​a ring on a sphere is .

Multiplying the flux [power by , according to formula (32.2)] by the area of ​​the strip, we find the energy radiated in the interval of angles and ; then you need to integrate over all angles from to :

(32.4)

When calculating, we use the equality and as a result we get . Hence finally

Several remarks need to be made about this expression. First of all, since there is a vector, then in the formula (32.5) means , i.e., the square of the length of the vector. Secondly, the formula (32.2) for the flow includes an acceleration taken with the delay taken into account, i.e., the acceleration at the moment of time when the energy passing now through the surface of the sphere was radiated. The thought may arise that the energy was actually radiated at exactly the indicated moment in time. But this is not entirely correct. The moment of emission cannot be determined exactly. It is possible to calculate the result of only such a movement, for example, oscillations, etc., where the acceleration finally disappears. Consequently, we can only find the total energy flux over the entire period of oscillation, which is proportional to the average square of the acceleration over the period. Therefore, in (32.5) should mean the time average of the squared acceleration. For such a movement, when the acceleration at the beginning and at the end vanishes, the total radiated energy is equal to the time integral of expression (32.5).

Let's see what formula (32.5) gives for an oscillating system, for which the acceleration has the form . The average for the period from the square of the acceleration is (when squaring, you must remember that in fact, instead of the exponent, its real part, the cosine, should be included, and the average from gives):

Consequently,

These formulas were obtained relatively recently - at the beginning of the 20th century. These are wonderful formulas, they had a huge historical meaning, and it would be worth reading about them in old books on physics. True, a different system of units was used there, and not the SI system. However, in the final results related to electrons, these complications can be eliminated using the following correspondence rule: the value where is the electron charge (in coulombs), previously written as . It is easy to verify that in the SI system the value is numerically equal to , since we know that and . In what follows, we will often use the convenient notation (32.7)

If this numerical value is substituted into the old formulas, then all other quantities in them can be considered defined in the SI system. For example, formula (32.5) previously had the form . And the potential energy of a proton and an electron at a distance is or , where SI.