Poynting vectors electromagnetic fields

There is a close similarity with planar electromagnetic cavities (H.-J. Stockmann, 1999). The basic equations take the same form and, in particular, the Poynting vector is the analog of the quantum mechanical current. It is therefore possible to experimentally observe currents, nodal points and streamlines in microwave billiards (M. Barth et.al., 2002 Y.-H. Kim et.al., 2003). The microwave measurements have confirmed many of the predictions of the random Gaussian wave fields described above. For example wave function statistics, current flow and... 

Consider an electromagnetic field (E, H), which is not necessarily time harmonic. The Poynting vector S = E X H specifies the magnitude and direction of the rate of transfer of electromagnetic energy at all points of space it is... 

Once we have obtained the electromagnetic fields inside and scattered by the particle, we can determine the Poynting vector at any point. However, we are usually interested only in the Poynting vector at points outside the particle. The time-averaged Poynting vector S at any point in the medium surrounding the particle can be written as the sum of three terms ... 

S, is the Poynting vector of the incident field and S, that of the scattered field we may interpret Sext as the term that arises because of interaction between the incident and scattered fields. Of greater interest here, however, is the flow of electromagnetic energy exclusive of that scattered. Thus, the Poynting vector under consideration (normalized by the magnitude of S,) is... 

In physical terms, this can be understood in the following way. Take an electromagnetic field with Poynting vector S = E x B. By a suitable Lorentz transformation [with direction unit vector n and velocity parameter q given by ntanh2ri = 2S/(E2 +B2)], we can change to a frame in which S = 0 at any... 

To see this, let us introduce some definitions first. The Poynting vector G represents energy flux density (units esu2cm-3 s-1 = ergcm2s 1). It is a capability of the electromagnetic field to perform work defined as... 

Let us consider the propagation of electromagnetic waves with both fields nonzero E O and B 0. As usual, propagation is parallel to the Poynting vector G, defined in Eq. (17). Evidently, by definition, vector G is perpendicular to both fields E and B. Hence, there cannot exist components of the magnetic field B parallel to the instantaneous direction of propagation G. 

The energy flow of the electromagnetic field can be calculated using the Poynt-ing vector P, introduced by the following formula ... 

Thus we see that four of the conservation equations in (24) correspond to all four conservation equations of the standard theory one is the conservation of energy (25a) (Poynting s equation), and the other three are the conservation of the three components of momentum (25b) of the standard form of electromagnetic field theory. But since (24) are eight real-number valued equations rather than four, the spinor formalism predicts more facts than the standard vector Maxwell formalism—it is a true generalization. 

When the distance from the source exceeds the characteristic length Xo, the magnetic field is mainly defined by component be, and the electromagnetic field at the arrival moment corresponds to the wave zone components of the field are perpendicular to the direction of the Poynting vector, and the ratio of components of electric and magnetic field, does not depend on the distance to the dipole. 

Poynting vector (S) - Eor electromagnetic radiation, the vector product of the electric field strength and the magnetic field strength. [1]... 

We eonelude this seetion with a eaution. It is important to remember that whereas the Poynting veetor ean be defined for an arbitrary eleetromagnetie field, the Stokes parameters can only be defined for transverse fields such as plane waves discussed in the previous section or spherical waves discussed in Section 12. Quite often the electromagnetic field at an observation point is not a well-defined transverse electromagnetic wave, in which case the Stokes vector formalism cannot be applied directly. 

From the electromagnetic fields, the Poynting vector can be obtained. It can be shown that the real part of the complex Poynting... 

The flux density of an electromagnetic wave is described by the Poynting vector. For the case of the plane wave field one obtains for the time average in the direction of propagation... 

The time-varying Poynting vector, which represents the directional energy flux density of the electromagnetic field, is thus given by... 

The Poynting vector has a dual role for it can be shown that the electromagnetic radiation fields transport momentum as well as energy, and tliat the momentum density is given by R/c. This relationship is most easily derived by making use of the idea tliat radiation consists of photons of energy hu whose momentum in vacuo is tioi/c, which follows from the quantum theory of radiation (Problem 2.7). 

A long straight cylindrical conductor of radius a and resistivity p carries a steady current I. Find an expression for the Poynting vector at a radius r < a. Show that for a cylindrical shell in the conductor of inner and outer radii r and T2, the net flow of energy from the electromagnetic field into the shell is... 

---