We show how to simulate a toy model of electrodynamics in dimensions and describe several numerical experiments. The toy model is much simpler than ordinary electrodynamics, but shares many of the same physical features. For example, there are analogs to the electric and magnetic fields, and these fields generate forces between charged particles and support freely propagating radiation. Unlike electrodynamics, however, the toy model is not Lorentz invariant, gives an attractive force between charges of the same sign, and yields a radiation reaction force that depends on the particle velocity.
REFERENCES
1.
The toy model that we will consider is not the same as electrodynamics in dimensions, which is discussed in Appendix A.
2.
The computer program used to perform the simulations will be provided upon request.
3.
A complete description of the toy model is given in
A. D.
Boozer
, “A toy model of electrodynamics in dimensions
,” Eur. J. Phys.
28
, 447
–464
(2007
). Derivations of the various results we cite regarding the toy model can be found there.4.
We have chosen a system of units such that the speed at which waves are propagated by the field is equal to one.
5.
Note that there is a minus sign on the right-hand side of Eq. (5), but there is no analogous minus sign for the Lorentz force law. Also, the Lorentz force law contains a term that involves the magnetic field, but there is no corresponding term in Eq. (5). The reason for choosing the force law given in Eq. (5) is explained in Ref. 3.
6.
This is not the only way of performing the decomposition, but it is the most useful for our present purposes. See
A. D.
Boozer
, “Retarded potentials and the radiative arrow of time
,” Eur. J. Phys.
28
, 1131
–1143
(2007
).7.
Here is the sign function, defined such that if , if , and if .
8.
The retarded fields of a charged point particle in ordinary electrodynamics are derived in
D. J.
Griffiths
, Introduction to Electrodynamics
, 2nd ed. (Prentice Hall, Englewood Cliffs
, NJ
, 1989
), Sec. 9.2.2.9.
Expressions for the energy and momentum density of the field are derived in Ref. 3. By using these expressions one can show that total energy and momentum of the coupled particle-field system are conserved, and that the energy lost by the particle to radiation damping is equal to the energy gained by the field.
10.
A detailed comparison of the toy model and ordinary electrodynamics is given in Ref. 3.
11.
For the toy model there is a preferred reference frame in which the equations of motion are valid, and a particle moving with respect to this preferred frame feels a drag force that is given by Eq. (19).
14.
J. D.
Jackson
, Classical Electrodynamics
, 2nd ed. (John Wiley & Sons
, New York
, 1975
), Sec. 17.7.15.
Pictures of the fields for a charge undergoing simple harmonic motion in ordinary electrodynamics are given in
R. Y.
Tsien
, “Pictures of dynamic electric fields
,” Am. J. Phys.
40
(1
), 46
–56
(1972
)and
R. H.
Good
, “Dipole radiation: A film
,” Am. J. Phys.
49
(2
), 185
–187
(1981
).An animation of the fields for dipole radiation in ordinary electrodynamics is discussed in
J. W.
Belcher
and S.
Olbert
, “Field line motion in classical electromagnetism
,” Am. J. Phys.
71
(3
), 220
–228
(2003
).17.
Pictures of the fields for braking radiation in ordinary electrodynamics are given in
J. C.
Hamilton
and J. L.
Schwartz
, “Electric fields of moving charges: A series of four film loops
,” Am. J. Phys.
39
(12
), 1540
–1542
(1971
)and
R. H.
Good
, “Dipole radiation: Simulation using a microcomputer
,” Am. J. Phys.
52
(12
), 1150
–1151
(1984
).18.
The analogous problem for ordinary electrodynamics is discussed in Ref. 14, Sec. 17.8.
19.
From Eqs. (61) and (62) we see that implies , so the dipole approximation implies the low-velocity approximation.
20.
We have used the fact that . This relation is the toy-model analog to the optical theorem of ordinary electrodynamics (see Ref. 14, Sec. 9.14).
21.
Note that in dimensions the scattering cross section is a dimensionless quantity ranging from 0 to 1 that gives the fraction of the incident power that is back-reflected from the scattering site.
22.
Note that although the particle radiates power to the right, on the right-hand side of the particle the radiated field exactly cancels the incident field , so there is no radiation in the total field .
23.
Because depends on , we must keep a record of the past trajectory of the particle to perform the numerical integration.
24.
Electrodynamics in dimensions is discussed in many places. See for example,
H.
Galić
, “Fun and frustration with hydrogen in a dimension
,” Am. J. Phys.
56
(4
), 312
–317
(1987
).© 2009 American Association of Physics Teachers.
2009
American Association of Physics Teachers
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