__Relativistic Reaction Forces for a Spherical Shell of
Charge__

__Abstract.__ The relativistic form of reaction forces
experienced by a vibrating spherical shell of charge is stated, and the work
that must be expended to drive the shell is shown to equal the flux of field
energy into space.

__1. Overview.__

In this article the relativistic forms of the inertial and radiation reaction forces are discussed. A tiny spherical shell of charge (radius r<<1 meter) oscillates along the x-axis. The relativistic version of the radiation reaction force is tested by computing the energy flux per cycle through an enclosing spherical surface and comparing that result with the work per cycle done by the driving agent (who counteracts both the inertial and radiation reaction forces).

__2. Review.__

In electromagnetic theory a spherical shell of charge cannot maintain its dimensions without the non-electromagnetic intervention of a constraining agent. Each increment of charge in the shell experiences an infinitesimal radial force outward in the electric field of the other increments. And an external agent must counteract each of these tiny forces if r, the shell radius, is to remain constant.

If the shell is permanently at rest in some inertial frame, then the internal electric forces sum to zero. In this case the net counteracting agent force also sums to zero, and the agent is usually not explicitly mentioned. Lorentz and others found, however, that the net internal forces (and hence the counteracting agent forces) do not sum to zero when a, da/dt, etc. are nonzero. If the shell is tiny (radius r<<1 meter) and if its speed, v <<c at all times, and if its motion is always along the x-axis, then the net internal force is found to be

(2-1)

where m_{em}, the so-called electromagnetic mass, is defined to be

. (2-2)

The counteracting agent force is thus

. (2-3)

All known charged particles are believed also to have mechanical mass. Their total mass can be expressed as

. (2-4)

The general equation of motion for a spherical shell of charge with nonzero mechanical mass is therefore

. (2-5)

Eq. 2-5 reduces to Newton’s second law when m_{em}=0
(uncharged particle).

Abraham
and Lorentz aptly dubbed the da_{x}/dt term in Eq. 2-1 the
"radiation reaction force" because of its correlation with radiated
energy. For example, if the spherical shell oscillates at angular frequency w,
then the net work expended per cycle by the agent’s counteraction equals the
field energy flux per cycle through an enclosing surface:

(2-6)

where S=e_{o}c^{2}E X B is
the Poynting vector. (The work per cycle expended to counteract –ma_{x} is
zero.) If the tiny shell’s E and B
fields at points on the enclosing surface are assumed to be the same as those of
a point charge at the shell’s center, then Eq. 2-6 can readily be corroborated
by a computer program. Table 2-1 lists the computed left and right sides of Eq.
2-6 for several values of wA<<c. The percent
difference, defined to be (Flux-Work) / Work, is also given. The small
discrepancies can be attributed to numerical error and to the fact that the
non-relativistic formula for the radiation reaction force was used.

**Table
2-1**

wA (m/sec) |
Work per Cycle (Joules) |
Flux per Cycle (Joules) |
Percent Difference |

2.9979E6 |
1.8832E4 |
1.8834E4 |
7.5007E-5 |

5.6960E6 |
1.2917E5 |
1.2921E5 |
2.7085E-4 |

8.3942E6 |
4.1341E5 |
4.1365E5 |
5.8846E-4 |

1.1092E7 |
9.5392E5 |
9.5490E5 |
1.0282E-3 |

1.3790E7 |
1.8331E6 |
1.8360E6 |
1.5904E-3 |

1.6489E7 |
3.1332E6 |
3.1404E6 |
2.2756E-3 |

1.9187E7 |
4.9368E6 |
4.9520E6 |
3.0846E-3 |

2.1885E7 |
7.3261E6 |
7.3556E6 |
4.0182E-3 |

2.4583E7 |
1.0384E7 |
1.0436E6 |
5.0771E-3 |

2.7281E7 |
1.4192E7 |
1.4280E7 |
6.2626E-3 |

**Work
per Cycle and Energy Flux per Cycle Compared**

In
view of Newton’s second law, the a_{x} term
in Eq. 2-1 might be dubbed the inertial reaction
force. As mentioned, the work per cycle, expended by the agent while
counteracting this force, is zero:

. (2-7)

Or more generally,

. (2-8)

If the same programs are run for relativistic wA … particularly if the program that computes the agent work per cycle is run for such relativistic oscillators … then it is clear that the reaction forces require relativistic adjustment. For example, Table 2-2 lists the computed non-relativistic agent work per cycle and the energy flux per cycle for a few relativistic values of wA.

**Table
2-2**

wA (m/sec) |
Work per Cycle (joules) |
Flux per Cycle (joules) |
Percent Difference |

2.7281e8 |
1.4192E10 |
7.5452E10 |
4.3167 |

2.7681E8 |
1.4825E10 |
9.4407E10 |
5.3683 |

2.8081E8 |
1.5476E10 |
1.2321E11 |
6.9613 |

**"Classical"
Work per Cycle and Energy Flux per Cycle Compared**

__3.
Relativistic Adjustments.__

The point charge field solutions are accurate at all source charge speeds. Thus they compute to the correct energy flux per cycle through a surrounding surface for all wA<c. However, the left half of Eq. 2-6 does not produce approximately equal works per cycle when wA is relativistic. Both the inertial and radiation reaction forces of Eq. 2-1 require relativistic adjustment. These adjustments are discussed individually below.

**3.1
The Relativistic Inertial Reaction Force.**

Before Special Relativity was introduced, Lorentz and others realized that electromagnetic mass cannot be a constant attribute of a charged particle. Rather it must be a function of the particle’s speed. In one dimension,

. (3.1-1)

In
Eq. 3.1-1 m_{em(o)} is
the rest mass. For a spherical shell of charge it is defined by Eq. 2-2.
Einstein concluded that this speed dependence must extend to mechanical mass as
well.

It turns out that Newton’s second law for uncharged particles is relativistically correct when expressed in the form

(3.1-2)

.

Or, since

, (3.1-3)

Eq. 3.1-2 becomes

(3.1-4)

.

This is the force that a driving agent must exert in order to counteract the relativistically correct inertial reaction force:

. (3.1-5)

As in the non-relativistic case, the work per cycle expended to counteract this force is zero.

**3.2
The Relativistic Radiation Reaction Force.**

The relativistically correct expression for the radiation reaction force is (in the case of a spherical shell of charge):

, (3.2-1)

where
m_{em} will henceforth
be understood to be the electromagnetic rest mass. In view of Eq. 3.1-3, Eq.
3.2-1 expands to:

. (3.2-2)

The power expended by the counteracting agent is thus

. (3.2-3)

As in the non-relativistic case (Eq. 2-6), when x = A sin(wt) this power does not integrate to zero over one cycle time. A program computes the agent work per cycle over a range of relativistic wA. Table 3.2-1 repeats Table 2-1, but with .85c<wA<.95c. Note the excellent agreement between the relativistically correct work per cycle and the energy flux per cycle.

**Table
3.2-1**

wA (m/sec) |
Work per Cycle (Joules) |
Flux per Cycle (Joules) |
Percent Difference |

2.5482E8 |
3.6245E10 |
3.6245E10 |
-1.3820E-11 |

2.5782E8 |
4.0141E10 |
4.0141E10 |
-9.4581E-11 |

2.6082E8 |
4.4730E10 |
4.4730E10 |
-5.0624E-11 |

2.6382E8 |
5.0207E10 |
5.0207E10 |
-1.4300E-10 |

2.6681E8 |
5.6851E10 |
5.6851E10 |
-1.5775E-10 |

2.6981E8 |
6.5064E10 |
6.5064E10 |
-5.7236E-11 |

2.7281E8 |
7.5452E10 |
7.5452E10 |
-1.4557E-10 |

2.7581E8 |
8.8964E10 |
8.8964E10 |
-2.0784E-10 |

2.7881E8 |
1.0717E11 |
1.0717E11 |
-1.7965E-8 |

2.8180E8 |
1.3285E11 |
1.3285E11 |
-1.2741E-6 |

**Work
per Cycle and Energy Flux per Cycle Compared**

__4.
Relativistic Electrodynamics.__

Given a spherical shell of charge, the one-dimensional relativistically correct equation of motion is:

. (4-1)

This is the force that a driving agent must apply in order to counteract the inertial and radiation reaction forces. The agent power expenditure is:

. (4-2)

If the motion is sinusoidal (say x=A sin(wt)), then the expended agent work per cycle equals the field energy flux per cycle through an enclosing surface:

. (4-3)

Or,
since the g^{3}m_{o}a_{x} (inertial)
term integrates to zero,

(4-4)

.

By Fourier analysis this result extends to all periodic motions. The program used to generate Table 3.2-1 can easily be modified to show that this is the case.

It
is noteworthy that the preceding discussion is predicated upon a shift from the
customary paradigm for dynamics. Usually it is the force experienced by a
particle that is presumed to be given, and the particle’s motion that is then
calculated. In the present discussion it is the motion that is given (e.g. x=A
sin(wt)) and the force required to maintain that
motion that is computed. Of course given a force, Eq. 4-1 can in
theory be solved to find the
motion. In practice this may not always be analytically feasible. Since the
radiation reaction force is often much smaller than the inertial reaction force,
the abbreviated equation "F_{x} = g^{3}m_{o}a_{x}"
is often solved in order to obtain an approximate motion. However, this
relativistically correct form for Newton’s second law obeys the Work Energy
Theorem … any work expended by the driving agent is manifest as changes in
kinetic energy and is completely recoverable. Rigorously speaking, Eq. 4-1 more
accurately describes the relationship between applied force and particle motion
when there is excess charge of one sign or another. The work expended by a
driving agent is not fully recoverable; some of that work bleeds away into
infinite space as radiant energy.