# Discrepancy between Power Radiated and the Power Loss Due to Radiation Reaction for an Accelerated Charge

## Abstract

**:**

## 1. Introduction

## 2. Two Discrepant Formulations for Radiation Losses from an Accelerated Charge

#### 2.1. An Inappropriate Usage of the Poynting Theorem

#### 2.2. Applicability of Larmor’s Formula to Compute Radiative Power Losses in Case of a Periodic Motion

#### 2.3. Discrepancy in Two Power Formulas is Due to the Difference in Power Going in Self-Fields at ‘Real’ and Retarded Times

## 3. A Uniformly Accelerated Charge

#### 3.1. The Contribution of Acceleration Fields to the Energy-Momentum of Self-Field

#### 3.2. Poynting Flux in the Case of a Uniformly Accelerated Charge

#### A Definition of Radiation at Infinity Incompatible with Green’s Theorem

#### 3.3. Far Fields and the Relative Location of the Uniformly Accelerated Charge

## 4. Conclusions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The self-force $\mathbf{f}$ on a charged spherical shell of a small radius $\u03f5$, moving non-relativistically with an acceleration $\mathbf{a}=\dot{\mathbf{v}}$. The net self-force on the charged shell at any instant is proportional to the acceleration it had at a time interval $\u03f5/c$ earlier [23]. This implies that for a uniform acceleration a, the self-force on the charge would only be an ‘inertial’ force $-{m}_{\mathrm{el}}a$, where ${m}_{\mathrm{el}}=2{e}^{2}/3\u03f5{c}^{2}$ is the electromagnetic mass of the charge [24], without any radiation reaction, whatsoever, consistent with the fact that there is no radiation emitted in this case.

**Figure 2.**Angular distribution of the electric field strength with respect to the time-retarded position ${z}_{\mathrm{r}}$ of the uniformly accelerated charge, moving along the z-axis with velocity $v\to c$ and the corresponding Lorentz factor $\gamma \gg 1$. Due to the relativistic beaming, the field strength is mostly appreciable only within a cone of angle $\theta \sim 1/\gamma $ about the direction of motion. When at time t, the fields from the retarded position ${z}_{\mathrm{r}}$ are at the spherical light-front of radius $r=ct$, the charge meanwhile has moved to ${z}_{\mathrm{o}}$, quite close to the spherical light-front. The circle represented by points P on the spherical light-front $r=ct$ where the field strength is maximum as a function of $\theta $, lies almost vertically above ${z}_{\mathrm{o}}$, the ‘present’ position of the charge, and thus are not very far from it, implying that the field at large r is still around the ‘present’ location of the charge.

**Figure 3.**The electric field distribution (

**a**) of a uniformly accelerated charge, with a ‘present’ velocity ${v}_{\mathrm{o}}=0.99995c$, corresponding to ${\gamma}_{\mathrm{o}}=100$ (

**b**) of a charge moving with a uniform velocity ${v}_{\mathrm{o}}=0.99995c$, corresponding to ${\gamma}_{\mathrm{o}}=100$. In both cases, the electric field lines are confined mostly within a small angle $\sim 1/\gamma $ with respect to the electric field lines that begin from the charge position ${z}_{\mathrm{o}}$, in plane perpendicular to the direction of motion.

**Figure 4.**The Poynting vector for a charge (

**a**) moving with a uniform proper acceleration, and is presently at ${z}_{\mathrm{o}}$ moving with a ‘present’ velocity ${v}_{\mathrm{o}}=0.99995c$, corresponding to ${\gamma}_{\mathrm{o}}=100$ (

**b**) moving with a uniform velocity ${v}_{\mathrm{o}}=0.99995c$, corresponding to ${\gamma}_{\mathrm{o}}=100$. The spherical light-front $r=ct$ is shown in the case of uniformly accelerated charge, which looks like a plane on this scale. The overall Poynting flow in both cases is along the direction of motion of the charge. Arrows show Poynting vector directions at different distances from the charge. The length of an arrow is not a direct indicator of the magnitude of the corresponding Poynting vector, the plot shows the trend only qualitatively. In fact, the magnitude of the Poynting vector, represented by larger arrows, is maximum at the plane normal to the direction of motion, passing through the charge at ${z}_{\mathrm{o}}$, and drops rapidly off the plane. At the positions of smaller arrows, shown in the figure, the magnitude of the Poynting vector falls as much as by a factor of $\sim {10}^{8}$.

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Singal, A.K.
Discrepancy between Power Radiated and the Power Loss Due to Radiation Reaction for an Accelerated Charge. *Symmetry* **2020**, *12*, 1833.
https://doi.org/10.3390/sym12111833

**AMA Style**

Singal AK.
Discrepancy between Power Radiated and the Power Loss Due to Radiation Reaction for an Accelerated Charge. *Symmetry*. 2020; 12(11):1833.
https://doi.org/10.3390/sym12111833

**Chicago/Turabian Style**

Singal, Ashok K.
2020. "Discrepancy between Power Radiated and the Power Loss Due to Radiation Reaction for an Accelerated Charge" *Symmetry* 12, no. 11: 1833.
https://doi.org/10.3390/sym12111833