We investigate the electrostatic energy of one-dimensional line charges, focusing on the energy difference between lines of different shapes. The self-energy of a strictly one-dimensional charge is infinite, but one can quantify the energy by considering geometries that approach a one-dimensional curve, for example, thin wires, thin strips, or chains of close point charges. In each model, the energy diverges logarithmically as the geometry approaches a perfect one-dimensional curve, but the energy also contains a finite term depending on the shape of the line—the “shape energy.” The difference in shape energy between a straight line and a circle is checked to be the same using a range of models. To calculate the shape energy of more complex shapes numerically, we propose a line integral where the singularity in the integrand is canceled. This integral is used to calculate the shape energy of a helix.

1.
N.
Korolev
,
O. V.
Vorontsova
, and
L.
Nordenskiöld
, “
Physicochemical analysis of electrostatic foundation for DNA-protein interactions in chromatin transformations
,”
Prog. Biophys. Mol. Biol.
95
(
1–3
),
23
–49 (
2007
).
2.
K.-K.
Kunze
and
R.
Netz
, “
Complexes of semiflexible polyelectrolytes and charged spheres as models for salt-modulated nucleosomal structures
,”
Phys. Rev. E
66
(
1
),
011918
(
2002
).
3.
J. K.
Nunes
,
J.
Li
,
I. M.
Griffiths
,
B.
Rallabandi
,
J.
Man
, and
H. A.
Stone
, “
Electrostatic wrapping of a microfiber around a curved particle
,”
Soft Matter
17
(
13
),
3609
3618
(
2021
).
4.
J.
Batle
,
O.
Ciftja
,
S.
Abdalla
,
M.
Elhoseny
,
M.
Alkhambashi
, and
A.
Farouk
, “
Equilibrium charge distribution on a finite straight one-dimensional wire
,”
Eur. J. Phys.
38
(
5
),
055202
(
2017
).
5.
D. J.
Griffiths
and
Y.
Li
, “
Charge density on a conducting needle
,”
Am. J. Phys.
64
(
6
),
706
714
(
1996
).
6.
M. H.
Partovi
and
D. J.
Griffiths
, “
Equilibrium charge density on a thin curved wire
,”
Am. J. Phys.
77
(
12
),
1173
1182
(
2009
).
7.
G. L.
Ferreira
, “
The electrostatic energy of thin charged straight threads and coils and the work to bend straight threads into coils
,”
J. Electrostat.
49
(
1–2
),
23
30
(
2000
).
8.
J. D.
Jackson
, “
Charge density on thin straight wire, revisited
,”
Am. J. Phys.
68
(
9
),
789
799
(
2000
).
9.
C.
Maxwell
, “
On the electrical capacity of a long narrow cylinder, and of a disk of sensible thickness
,”
Proc. London Math. Soc.
1
(
1
),
94
102
(
1877
).
10.
L.
Vainshtein
, “
Static boundary problems for a hollow cylinder of finite length. 3. Approximate formulas
,”
Sov. Phys.-Tech. Phys.
7
(
10
),
855
–866 (
1963
).
11.
J.
Lekner
,
Electrostatics of Conducting Cylinders and Spheres
(
AIP Publishing LLC
,
New York
,
2021
).
12.
O.
Ciftja
and
J.
Batle
, “
Coulomb potential and energy of a uniformly charged cylindrical shell
,”
J. Electrostat.
96
,
45
48
(
2018
).
13.
V.
Belevitch
and
J.
Boersma
, “
Some electrical problems for a torus
,”
Philips J. Res.
38
(
3
),
79
137
(
1983
).
14.
D. H.
Dubin
, “
Minimum energy state of the one-dimensional coulomb chain
,”
Phys. Rev. E
55
(
4
),
4017
(
1997
).
15.
P.
Amore
and
M.
Jacobo
, “
Thomson problem in one dimension: Minimal energy configurations of n charges on a curve
,”
Physica A
519
,
256
266
(
2019
).
16.
G.
Watson
, “
XVI. The sum of a series of cosecants
,”
London, Edinburgh, Dublin Philos. Mag. J. Sci.
31
(
182
),
111
118
(
1916
).
17.
S.
Fukuhara
, “
Energy of a knot
,” in
A Fête of Topology
(
Elsevier
,
New York
,
1988
), pp.
443
451
.
18.
J.
O'Hara
, “
Family of energy functionals of knots
,”
Topol. Appl.
48
(
2
),
147
161
(
1992
).
19.
J.
O'Hara
, “
Regularization of neumann and weber formulae for inductance
,”
J. Geom. Phys.
149
,
103567
(
2020
).
20.
O.
Ciftja
, “
Coulomb self-energy of a uniformly charged three-dimensional cylinder
,”
Physica B
407
(
14
),
2803
2807
(
2012
).
21.
C.
Wong
, “
Toroidal and spherical bubble nuclei
,”
Ann. Phys.
77
(
1–2
),
279
353
(
1973
).
22.
L. D.
Landau
,
The Classical Theory of Fields
, 4th ed. (
Elsevier
,
New York
,
1975
).
23.
O.
Ciftja
, “
Stored coulomb self-energy of a uniformly charged rectangular plate
,”
Adv. Math. Phys.
2016
,
7207536
.
24.
H. S.
Cohl
and
J. E.
Tohline
, “
A compact cylindrical Green's function expansion for the solution of potential problems
,”
Astrophys. J.
527
(
1
),
86
101
(
1999
).
25.
NIST Digital Library of Mathematical Functions
, edited by
F. W. J.
Olver
,
A. B.
Olde Daalhuis
,
D. W.
Lozier
,
B. I.
Schneider
,
R. F.
Boisvert
,
C. W.
Clark
,
B. R.
Miller
, and
B. V.
Saunders
(
NIST
,
Gaithersburg
,
2017
).
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