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The paper describes the net momentum transported by the transient electromagnetic radiation field of a long transient dipole in free space. In the dipole a current is initiated at one end and propagates towards the other end where it is absorbed. The results show that the net momentum transported by the radiation is directed along the axis of the dipole where the currents are propagating. In general, the net momentum

Electromagnetic radiation fields are associated not only with energy but also with a momentum [

The literature on the radiation produced by Hertizian dipoles is numerous, and it is sufficed to refer to several textbooks dealing with the subject such as Jackson [

In this paper we will consider the momentum transported by the radiation fields of a long transient dipole. We will consider the general case where the characteristic wavelength of the current pulse propagating along the dipole can take any value with respect to the length of the dipole.

The features of radiation fields generated by Hertzian dipoles are described in any textbook on electromagnetic theory. Consider a dipole, of length

One can use this set of equations to evaluate the energy and momentum transported by Hertzian dipoles.

The geometry relevant to the problem under consideration is shown in

In the present study, the temporal variation of the current pulse propagation along the dipole is represented by a Gaussian curve, which can be described mathematically by the following analytical function:

Note that, for the ease of calculation, the Gaussian current pulse is shifted forward in time by

An expression for the radiation field produced by the propagation current pulse at a distant point (

Note that

The radiation field generated by a long dipole (with

The energy density generated by an electromagnetic field is given by the Poynting vector

This gives the power transmitted by the electromagnetic wave across a unit area located perpendicular to the direction of propagation of the wave. The momentum density associated with this electromagnetic field

This expression gives the linear momentum transferred per unit time by the wave across a unit area located perpendicular to the direction of propagation of the wave. In the above equations,

The Poynting vector and the momentum density vector associated with the Hertzian dipole is given by

The total energy radiated by the dipole is given by

One can easily show that, due to symmetry, net momentum transported by the radiation field in any given direction is zero. For example, the net momentum transported in the

Due to symmetry, the value of the above integral is zero. In other words, the ratio

Using the expressions for the electric radiation fields given in Equation (5), and noting that

The energy radiated across a unit area in any given direction

The total energy radiated by the system can be obtained by integrating the Poynting vector over a spherical region. That is, the total energy dissipated by the system is given by

The net momentum transported by the electromagnetic field through a unit area in the direction

Due to the angular symmetry of the emitted radiation, the

Since this is the only component of the net momentum transported by the radiation, we will drop the subscript z and write it simply as

Consider a transient electromagnetic field emitted in a direction

In the case of a normal dipole, the net momentum transported by the radiation field departs from this equation because the radiation is not directed in one particular direction. However, as we will show in this paper, as the effective wavelength

In order to take into account the radiation fields generated by long dipoles excited by current waveforms of different durations, let us define a parameter

In Equation (20),

The way in which the radiated energy of a dipole varies as a function of

In the above equation,

Since the time dependence of the current, which appeared in the expressions for both the energy and the momentum transfer, is the same except for a geometrical factor that arises due to the presence of

Thus, the ratio of the total momentum and the total energy (divided by the speed of light) transported by the dipole is given by

This ratio depends only on the parameter

Observe that, as given by Equation (5c), the delay between the two radiation pulses generated by the dipole during current initiation and termination is equal to

Substituting for

As the value of

The factor 2 in the above equation comes from the fact that there are two identical radiation pulses, each with a time signature identical to that of the current waveform. The expression for the total momentum transported by the radiation field is given similarly by the expression

The spatial integrals in Equations (27) and (28) can be evaluated without much difficulty. Furthermore, when one evaluates the ratio

Note in the above equation that the ratio becomes 1 as

Let us consider the ratio ^{5}–10^{6}, this ratio is almost equal to unity. This shows that as

The results presented here are obtained for a Gaussian current pulse. However, observe that the ratio

The results presented in this paper can be utilized in a hypothetical experiment to derive an interesting property associated with dipole radiation. Consider an electromagnetic dipole completely at rest. Its location is unknown. At a given time, the dipole emits a burst of electromagnetic radiation whose duration is

In the above equation,

In our case, the condition ^{5}. Thus, for dipoles working in time domain where the condition

In order to be treated as a Hertzian dipole, the effective wavelength associated with the radiation

In this paper, we have considered an idealized situation in which current pulses are assumed to propagate along the long dipole with speed of light and without attenuation. This idealization was warranted here because the goal of the study reported in this paper was to understand the connection between the energy and momentum transported by the long dipole radiation. However, the application of this procedure in more practical situations needs further consideration. For example, dipole antennas are constructed in practice by locating vertical conductors over perfectly conducting ground. In this case, the long dipole acts as a vertical transmission line. Pulses propagating along vertical conductors located over a ground plane are affected by the finite conductivity of the material of the conductor and by the effects of radiation damping [^{6}–10^{7}. Further research work is necessary to confirm this result.

Another natural phenomenon that acts as a dipole located over a ground plane is the lightning return stroke. Experimental data show that the return stroke can be treated as a current pulse propagating along a more or less vertical channel with an average speed of propagation of about (1–2) × 10^{8} m/s while undergoing attenuation. The actual front speed decreases with increasing height [

The paper describes the net momentum transported by the radiation emitted by a transient dipole working in time domain. In the dipole a current is initiated at one end and propagates towards the other end where it is absorbed. The net momentum associated with the radiation is directed along the axis of the dipole. It is shown that as the duration of the excitation current of the dipole decreases with respect to the travel time of the current along the dipole, ^{−5}, the net momentum and the radiated energy are connected by the relationship

The research presented in this paper is supported by the fund from B. John F. and Svea Andersson donation at Uppsala University.

The research problem was suggested by the first author. Both authors contributed equally in carrying out this research work and in writing the paper. Both the authors have read and approved the final manuscript.

The authors declare no conflict of interest.

Geometry relevant to the derivation of equations presented in this paper. (

The normalized Gaussian current pulse with standard deviation

Normalized radiation field generated along the direction

The variation of the ratio of