# On the Momentum Transported by the Radiation Field of a Long Transient Dipole and Time Energy Uncertainty Principle

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## Abstract

**:**

## 1. Introduction

## 2. Radiation Fields Produced by a Transient Dipole

#### 2.1. Transient Hertzian Dipole

#### 2.2. Transient Long Dipole

## 3. The Energy and Momentum of the Radiation Field

#### 3.1. Transient Hertzian Dipole

#### 3.2. Long Transient Dipole

## 4. The Total Electromagnetic Energy Radiated and the Total Electromagnetic Momentum Transported by a Long Dipole—General Case

## 5. The Total Electromagnetic Energy Dissipated by the Long Dipole and the Total Momentum Transported by the Radiation—Case When β ≪ 1

## 6. The Relationship between the Total Energy Dissipated and the Total Momentum Transported by the Long Dipole Field

^{5}–10

^{6}, this ratio is almost equal to unity. This shows that as $1/\beta $ increases, the radiation becomes more and more directional and it aligns along the z-axis (or along the axis of the dipole). When $1/\beta \ge {10}^{5}$, the total momentum transported by the field is almost equal to $U/c$, where $U$ is the total energy radiated. In other words, for $1/\beta \gg 1$ the radiation emitted by the dipole can be treated as almost unidirectional.

## 7. Discussion

^{5}. Thus, for dipoles working in time domain where the condition $1/\beta \ge {10}^{5}$ is satisfied, the emitted radiation should satisfy approximately the condition given by Equation (31). This is an interesting result, which shows that dipole fields, when excited by fast current pulses, satisfy a time–energy uncertainty relationship as given by Equation (31). Indeed, Cooray and Cooray [8] used this relationship to show that the smallest charge that can radiate in an antenna working in time domain is equal to the electronic charge.

^{6}–10

^{7}. Further research work is necessary to confirm this result.

^{8}m/s while undergoing attenuation. The actual front speed decreases with increasing height [13]. In reality, the ground is finitely conducting and the radiation fields are affected by the propagation losses. Research work is underway to investigate the momentum transferred by radiation fields of lightning return strokes and how it is related to the total energy radiated by them. Furthermore, how the magnitude of the radiated momentum and its relationship to total radiated energy vary as a function of return stroke parameters—such as current amplitude, current signature, speed of propagation, and current attenuation along the channel—are under investigation. We hope to present the results of these investigations in the near future.

## 8. Conclusions

^{−5}, the net momentum and the radiated energy are connected by the relationship $P\approx U/c$. Due to momentum conservation, the radiating dipole experiences a momentum of equal magnitude but in the opposite direction. The results show that under the conditions when $P\approx U/c$ is satisfied, the radiation fields satisfy the relationship $\Delta t\Delta U\ge h/4\pi $ where $\Delta t$ is the duration of the radiation, $\Delta U$ is the uncertainty in the radiated energy, and $h$ is the Plank constant.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Geometry relevant to the derivation of equations presented in this paper. (

**a**) Hertzian dipole; (

**b**) long dipole. Observe that in defining the unit vectors ${a}_{\mathit{r}}$, ${a}_{\mathit{\theta}}$, and ${a}_{\mathit{\phi}}$ we are using a spherical coordinate system with the centre of the dipole located at the origin of coordinate system. In this coordinate system $r$ is the radial distance, $\theta $ is the polar angle, and $\phi $ is the azimuthal angle. ${a}_{\mathit{r}}$, ${a}_{\mathit{\theta}}$, and ${a}_{\mathit{\phi}}$ are unit vectors in the direction of increasing radial distance, polar angle and azimuthal angle, respectively.

**Figure 3.**Normalized radiation field generated along the direction $\theta =20\xb0$ at a distant point by a long dipole. The dipole is excited by a Gaussian current pulse. In the calculation, $\sigma $ = 10 ns and $\tau /(L/c)=0.01$.

**Figure 4.**The variation of the ratio of z-momentum to $U/c$ as a function of $1/\beta $. The solid line is calculated using Equation (24) and the dashed line by using Equation (29).

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## Share and Cite

**MDPI and ACS Style**

Cooray, V.; Cooray, G.
On the Momentum Transported by the Radiation Field of a Long Transient Dipole and Time Energy Uncertainty Principle. *Atmosphere* **2016**, *7*, 151.
https://doi.org/10.3390/atmos7110151

**AMA Style**

Cooray V, Cooray G.
On the Momentum Transported by the Radiation Field of a Long Transient Dipole and Time Energy Uncertainty Principle. *Atmosphere*. 2016; 7(11):151.
https://doi.org/10.3390/atmos7110151

**Chicago/Turabian Style**

Cooray, Vernon, and Gerald Cooray.
2016. "On the Momentum Transported by the Radiation Field of a Long Transient Dipole and Time Energy Uncertainty Principle" *Atmosphere* 7, no. 11: 151.
https://doi.org/10.3390/atmos7110151