It has come to our attention that Figs. 3 and 4 in our recent article1 are incorrect. The field lines should connect up at the spherical surface. This can be shown by calculating the radial component of the field just inside the sphere [the hyperbolic field, Eq. (11)] and just outside the sphere [the Heaviside field, Eq. (17)]

$Ein·r̂=q4πϵ01(αz)2=Eout·r̂.$
(1)

Here, $r=s+(z−b1+α2)ẑ$ is the radial vector from the center of the sphere, and r = αb. Because these components are equal, the same number of field lines must exit any patch of area as enter it, and hence the field lines are continuous across the surface (though they do suffer a kink, since the tangential component is different on the two sides). As a consequence, there is no “connecting field”—or, more precisely, the connecting field carries no delta function and is irrelevant to the calculation.2

A second error occurred when we quoted the field of a charge moving at the speed of light [Eq. (18)] and noted that it does not have the correct dependence on s—which we took to confirm the role of the connecting field. In actual fact what we have here is not the field of a charge moving at speed c, but the field of a particle moving at close to the speed of light adjoined to the field of a particle in hyperbolic motion (Fig. 4). In the limit, the former approaches a delta function, but to capture the strength of that delta function we must integrate the s component of the Heaviside field [Eq. (17)] not from large negative z to large positive z, but only up to the surface of the sphere, $zmax=b1+α2−r2−s2$; beyond that the field is hyperbolic and does not contribute to the delta function. As shown by Jerrold Franklin3

$∫−∞zmaxEs dz=q4πϵ0∫−∞zmax(1−v2/c2)s[ R2−(v/c)2R2 sin2θ ]3/2 dz=q4πϵ0s1+α2∫−∞zmax1[ s2+(z1+α2−b)2 ]3/2 dz=q4πϵ01sz1+α2−bs2+(z1+α2−b)2|−∞zmax.$
(2)

In the limit α,

$zmax1+α2 → 12b(s2+b2),$
(3)

so

$∫−∞zmaxEs dz → q4πϵ01s[ s2−b2s2+b2+1 ]=q2πϵ01s2+b2,$
(4)

which is the right coefficient [Eq. (20)], with no contribution from the connecting field.

The authors thank Jerrold Franklin for calling these errors to our attention,3 and Daniel Cross for very illuminating correspondence.4 We apologize for any confusion these blunders may have caused, but we are pleased to note that correcting them makes the model much nicer.

1.
J.
Franklin
and
D. J.
Griffiths
, “
The fields of a charged particle in hyperbolic motion
,”
Am. J. Phys.
82
,
755
763
(
2014
). All figure and equation numbers refer to this paper.
2.
The connecting field comes from the infinitesimally brief moment when the motion transitions from uniform velocity to hyperbolic. In J. J. Thomson's classic model of a charge that starts or stops (
E. M.
Purcell
and
D. J.
Morin
,
Electricity and Magnetism
, 3rd ed. (
Cambridge U.P.
,
Cambridge
,
2013
), Sect. 5.7),
the velocity is discontinuous
,
so the acceleration is a delta function
. In our case, the acceleration is discontinuous, and the delta function is in the jerk. But the Liénard–Weichert field [Eq. (2)] does not explicitly involve the jerk, and it does not pick up a delta function. We thank D. Cross for this observation.
3.
For further details, see
J.
Franklin
, “
Electric field of a point charge in truncated hyperbolic motion
,” e-print arXiv:1411.0640v3.
4.
Cross has also shown how one can obtain the “extra” delta-function term directly from the Liénard–Weichert construction, without truncating the hyperbolic motion. See
D. J.
Cross
, “
Completing the Liénard–Weichert potentials: The origin of the delta function for a charged particle in hyperbolic motion
,”
Am. J. Phys.
(accepted); e-print arXiv:1409.1569.