Electromagnetic wave square

If the motion of a particle in the double-slit experiment is to be represented by a wave function, then that wave function must determine the probability density P(x). For mechanical waves in matter and for electromagnetic waves, the intensity of a wave is proportional to the square of its amplitude. By analogy, the probability density P(x) is postulated to be the square of the absolute value of the wave function (x)... 

The X-ray detector measures the intensity of electromagnetic waves, i.e., the absolute square 2 of their amplitude. Thus, in combination, the upper path between density and intensity through the square is written as... 

The velocity of electromagnetic waves through any material other than the vacuum is (e ) 2 = v and the ratio n = c/v is called the index of refraction of that material. It follows that n = y /x/eoMo and, since the ratio n/fio 1, except for ferromagnetic materials, the index of refraction is commonly defined as the square root of the dielectric constant, e/e0- Since the frequency of the field is not affected by the medium, refraction can be described equally well as a change of the wavelength of light passing between different transparent media. 

The principal interaction of an electromagnetic wave, such as visible light, with a substance is that of the electric field of the wave and the electric charges of the substance. The dielectric constant of the substance determines the magnitude of this interaction in fact, it is equal to the square of the dielectric constant ... 

Signal power square of signal amplitude. For a transverse (e.g., electromagnetic) wave, the power is the energy deposited per unit time per unit cross-sectional area. 

Electromagnetic waves transmit energy as well. The average intensity I (in watts per square meter) is proportional to the average squared electric field ... 

This variation of AAm/Bm with separation comes from the "relativistic screening function" R (l), which is subsequently elaborated. This factor becomes important at large distances when we must be concerned with the finite velocity of the electromagnetic wave. At short distances, Rn(l) = 1 the energy of interaction between two flat surfaces varies with the square of separation. At large distances any effective power-law variation of the interaction depends on the particular separation of materials and wavelengths of the operative electromagnetic waves between them. 

It is sometimes useful to distinguish between conducting or dissipative media (crel > 0) and non-conducting media (cej = 0) For non-conducting media the velocity of an electromagnetic wave is proportional to (p.e)1/2, just as the velocity of a sound wave is proportional to the square root of the compressibility. 

The polarization factor arises from partial polarization of the electromagnetic wave after scattering. Considering the orientation of the electric vector, the partially polarized beam can be represented by two components one has its amplitude parallel (Ay) to the goniometer axis and another has the amplitude perpendicular (Ax) to the same axis. The diffracted intensity is proportional to the square of the amplitude and the two projections of the partially polarized beam on the diffracted wavevector are proportional to 1 for (A ) and cos 20 for (Ax). Thus, partial polarization after scattering yields the following overall factor (also see Thomson equation in the footnote on page 140) ... 

We will now study the characteristics of the electromagnetic wave based on Maxwell s equations, but from the point of view of energy. As the electromagnetic waves satisfy the general wave equation described above, the solution of Eqs. (1.54) and (1.55) will take the form of Eq. (1.7). The energy of the electromagnetic wave is proportional to the square of the wavefunction. Therefore we define the pointing vector S ... 

From Maxwell s theory of electromagnetic waves it follows that the relative permittivity of a material is equal to the square of its refractive index measured at the same frequency. Refractive index given by Table 1.2 is measured at the frequency of the D line of sodium. Thus it gives the proportion of (electronic) polarizability still effective at very high frequencies (optical frequencies) compared with polarizability at very low frequencies given by the dielectric constant. It can be seen from Table 1.2 that the dielectric constant is equal to the square of the refractive index for apolar molecules whereas for polar molecules the difference is mainly because of the permanent dipole. In the following discussion the Clausius-Mossoti equation will be used to define supplementary terms justifying the difference between the dielectric constant and the square of the refractive index (Eq. (29) The Debye model). 

At high frequencies co Ven), the absorption coefficient is proportional to the square of wavelength (/u., a co a therefore, short electromagnetic waves propagate in plasma more readily. 

The electrons enter the wiggler/undulator in unstructured packets (e.g. storage ring case) or in a quasi-continuous stream. As stated above, the electromagnetic waves, produced by oscillation of the various electrons traversing the device, add incoherently. There will be partial destruction since positive and negative amplitudes add algebraically. The emitted amplitude is proportional to the square root of the number of electrons in the packet, i.e. to so that the radiation intensity will be proportional to Ng. 

As in magnetic resonance with electromagnetic waves, the intensity of absorption is proportional to the energy quantum hv and the population difference hv/kT), giving in all cases a factor v /kT. Thus the intensity of the +2 transitions, involving the square of the hyperfine interaction, increases with the second power of the frequency. The position is dilferent for +1 transitions because of an additional frequency dependence their acoustic matrix elements involve the magnetic field, and their intensity thus rises with the square of the magnetic field. This introduces a further factor proportional to v, so that their intensity increases with the fourth power of the frequency. These frequency dependences have been confirmed in an experiment at liquid-helium temperatures where the frequency is varied from 800 to 1600 MHz, as shown in fig. 15. The measurements also verify that the absorption increases as T . 

Some properties of electromagnetic waves depend on frequency. For example, some wavelengths are visible. Some can penetrate and heat materials, including human tissue. Some properties apply across the spectmm. For example, energy from a radiation source diminishes with the square of the distance from the source. Compared to one unit from a source, the energy level at two units away will be one fourth. 

The behavior of electromagnetic waves in normal metals at ordinary temperatures and microwave frequencies is quite adequately described by the classical treatment based on MaxwelPs equations and Ohm s law. At low temperatures this is no longer true even though MaxwelPs equations are still valid, Ohm s law is inadequate to describe the relation between high frequency electric currents and fields in metals. According to classical theory, the surface resistance R is inversely proportional to the square root of the dc conductivity cr. Consequently, as the temperature is lowered and o- increases, the classical theory predicts that R cc. This is not borne out in practice, as will be seen by referring to Fig, 1. The ordinate is IR the observed surface conductance, and the abscissa is proportional to c T. Initially the behavior is classical and as the temperature is lowered. As the dc conductivity becomes larger, however, I does not increase proportionately and in the low temperature limit it becomes independent of a (and temperature). This phenomenon is known as the anomalous skin effect. The experimental data shown are due to Chambers [1]. The solid curve is the curve predicted from the theory of Reuter and Sondheimer [2],... 

The key properties of electromagnetic waves are velocity (V), wavelength (A), frequency (v), amplimde, polarisation, intensity and coherence. These are illustrated in Fig. 1.2. The relationship between the first three is given by V — Av. Polarisation can be either linear, or circular, and hnear polarised light can be represented as the sum of two equal amplitude, circularly polarised waves moving clockwise and anticlockwise in phase. The intensity of the wave is proportional to the square of the amplimde. 

We have already indicated that the square of the electromagnetic wave is interpreted as the probability density function for finding photons at various places in space. We now attribute an analogous meaning to for matter waves. Thus, in a one-dimensional problem (for example, a particle constrained to move on a line), the probability that the particle will be found in the interval dx around the point xi is taken to be ir x ) dx. If i/ is a complex fimction, then the absolute square, is used instead of... 

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