Exploring The Electric Field
Exploring The Electric Field
Exploring The Electric Field
Editors’ Suggestion
Exploring the electric field around a loop of static charge: Rectangles, stadiums, ellipses, and knots
We study the electric field around a continuous one-dimensional loop of static charge, under the assumption
that the charge is distributed uniformly along the loop. For rectangular or stadium-shaped loops in the plane, we
find that the electric field can undergo a symmetry-breaking pitchfork bifurcation as the loop is elongated; the
field can have either one or three zeros, depending on the loop’s aspect ratio. For knotted charge distributions
in three-dimensional space, we compute the electric field numerically and compare our results to previously
published theoretical bounds on the number of equilibrium points around charged knots. Our computations
reveal that the previous bounds are far from sharp. The numerics also suggest conjectures for the actual minimum
number of equilibrium points for all charged knots with five or fewer crossings. In addition, we provide the first
images of the equipotential surfaces around charged knots and visualize their topological transitions as the level
of the potential is varied.
DOI: 10.1103/PhysRevResearch.4.033249
I. INTRODUCTION Next we allow our charged loops to move out of the plane
and into three dimensions. In particular, we consider charged
In a first course on electricity and magnetism, students
trefoil knots, figure-eight knots, and other charged loops that
are often asked to solve problems in electrostatics [1]. For
have knots tied in them. Charged knots may sound at first like
example, calculating the electric field around an infinite line
a contrived thing to study, but they actually arise quite com-
of uniformly distributed static charge provides good practice
monly in nature; they are found in long closed polymers and
in working with Coulomb’s law or Gauss’s law. Another clas-
specifically in long loops of DNA [2–5]. Many new questions
sic exercise is to calculate the electric field at all points on
about equilibrium points and equipotential surfaces arise in
the symmetry axis above the center of a uniformly charged
the context of charged knots that we hope will appeal to the
circular ring.
general physics community.
Here we explore the electrostatics of one-dimensional
Before we proceed with the analysis, let us clarify how
charge distributions that are less standard than lines and cir-
our work differs from that recently presented elsewhere by
cles. First, working with charged loops confined to a plane,
the first author [6,7]. The articles [6,7] are aimed at spe-
we show that charged rectangles and stadiums can give rise to
cialists in topology, geometry, and knot theory, whereas the
symmetry-breaking bifurcations in their surrounding electric
present treatment is intended for physicists. We are also more
fields. Specifically, imagine stretching a rectangular charge
concerned here with giving physical, visual, and numerical
distribution along its major axis. When the charged rectangle
results and examples, whereas the earlier articles concentrated
becomes sufficiently elongated, the electric field it generates
on the precise statements and proofs of certain theorems we
can have three equilibria: one at the center of the rectangle,
cite below. For example, it was proven in Ref. [7] that the
and another two located on the major axis, symmetrically
electric field around any uniformly charged knot must have
placed on opposite sides of the center. These two additional
at least 2t + 1 equilibrium points, where t is a topological
equilibria emerge from a supercritical pitchfork bifurcation
invariant known as the knot’s tunnel number. We now show
at a critical aspect ratio that can be calculated explicitly.
numerically that this lower bound is not sharp, and we offer
Surprisingly, nothing like this happens for a charged ellipse;
conjectures for the actual minimum number of equilibrium
at all aspect ratios its surrounding electric field has only one
points for each type of knot with five or fewer crossings.
equilibrium point at its center.
Using computer graphics, we also visualize the equipotential
surfaces of charged knots, and we now discuss our results
regarding charged rectangles, stadiums, and ellipses.
*
ml2437@cornell.edu
†
townsend@cornell.edu
‡
strogatz@cornell.edu II. CHARGED LOOPS IN THE PLANE: RECTANGLES,
STADIUMS, AND ELLIPSES
Published by the American Physical Society under the terms of the
A. Rectangle
Creative Commons Attribution 4.0 International license. Further
distribution of this work must maintain attribution to the author(s) As a first example, consider a one-dimensional loop of
and the published article’s title, journal citation, and DOI. static charge in the xy plane, uniformly and continuously
(u, v) 1
(x, y)
−a a
−1
FIG. 2. Contour map of the electrostatic potential inside a uni-
form rectangular charge distribution. (a) Aspect ratio a = 1.1. The
FIG. 1. A uniform charge distribution in the shape of a rectangle. only critical point (also known as a zero or an equilibrium point of
Static charge is confined to this one-dimensional loop, and we seek the electric field) occurs at the center of the rectangle. (b) Aspect
to understand the electric field that surrounds it in three dimensions ratio a = 2.5. There are now three zeros.
distributed in the shape of a rectangle with aspect ratio a 1. obtain the potential everywhere inside the rectangle in the xy
As shown in Fig. 1, the rectangle consists of two horizontal plane. (We do not show the final result because it is unpleasant
line segments extending from x = −a to a, lying one unit to look at, but it is a sum of four terms like that above.)
above and below the x axis, capped off by vertical lines at Figure 2 shows a contour map of the potential φ(x, y, z)
both ends. restricted to the plane z = 0 for two values of the aspect ratio
One of the main topics of interest to us is the location of a. When a is close to 1 and the rectangle is almost square, the
the zeros of the electric field in the three-dimensional space only zero is at the origin [Fig. 2(a)]. However, if we increase a
around the rectangle. These zeros correspond to equilibrium to 2.5, the rectangle becomes more elongated and we now find
points where a test charge could remain at rest (though not three zeros: one at the origin, and a symmetric pair on either
stably, as a well-known theorem forbids the existence of stable side of the origin [Fig. 2(b)].
equilibria in an electrostatic field [1]). By symmetry, there To find the threshold value of a at which the bifurcation
must be a zero at the center of the rectangle. But can there occurs, we examine how the potential varies along the x axis.
be any other zeros? Fig. 3 shows that the potential on the x axis changes from
Clearly there cannot be any zeros above or below the xy having a minimum to a maximum at x = 0 as a increases.
plane, because all the charges in the rectangle would attract At the bifurcation value of a, the second derivative of the
or repel a test charge in the same direction relative to the z potential at the origin vanishes as the graph of the potential
axis. So it suffices to look for zeros in the xy plane. These cor- changes from concave up to concave down. By calculating
respond to critical points of the potential where the gradient this second derivative analytically, we find that the threshold
vanishes. To find such zeros, if they exist, let us calculate the value of a satisfies
potential on the xy plane generated by the rectangular charge 4 + 8a2 − 4a3 = 0,
distribution.
Consider the contribution to the electric potential from the whose unique positive root is a ≈ 2.205 57.
top side of the rectangle. Assume a linear charge density of
unit strength, for simplicity. As shown in Fig. 1, the distance B. Stadium
between a typical line element dx at (u, v) on the top and Our second example takes the form of a planar curve
a point (x, y) inside the rectangle is (u − x)2 + (1 − y)2 , shaped like a stadium (Fig. 4). This curve consists of two
since v = 1 on the top of the rectangle. For the inverse- equal parallel line segments that run from x = −a to x = a,
square electrical force corresponding to Coulomb’s law in with y values given by y = ±1. These line segments are
three dimensions, the contribution to the potential is inversely capped off by semicircles of unit radius at either end. The
proportional to this distance. Hence, by integrating over all resulting stadium curve is well known in studies of chaotic
the line elements on the top of the rectangle, we obtain a
contribution to the potential of
a
dx
.
−a (u − x) + (1 − y)2
2
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−(a + 1) −a a a+1
−1
FIG. 6. (a) Potential along x axis for stadium with a = 1. (b) Po-
FIG. 4. A stadium-shaped charge distribution. The stadium’s as- tential along x axis for stadium with a = 2. The three critical points
pect ratio is defined as a + 1, where a denotes the half-length of the where the slope vanishes correspond to electrostatic equilibrium
straight portion of the stadium. points, i.e., zeros of the electric field.
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FIG. 9. A selection of the topologically different equipotential surfaces for a charged figure-eight knot.
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unsolved and is just one of the many open problems about We use the same trapezoidal approximation for the electric
charged knots. field E (x). (There are more efficient approaches for evaluating
In what follows we propose conjectures for the minimum φ(x) and E (x) when N is very large; these are based on
number of zeros for all prime knots with up to five crossings, multipole expansions such as those used in the fast multipole
based on our numerical experiments. But proving (or disprov- method [47].)
ing) these conjectures—and extending them to a much wider As mentioned above, we are interested in finding the zeros
range of knots—remains a challenge. of the electric field. These are defined as points x ∗ ∈ R3 such
One may reasonably ask why we are so concerned with that E (x ∗ ) = 0. To find them, we start with initial guesses and
these zeros, given that they represent unstable equilibria and then refine the guesses iteratively using a multivariable New-
are therefore of questionable physical significance. Although ton method. To obtain reasonable initial guesses, we use an
it remains to be seen whether they are physically important, algorithm known as “3D marching cubes,” a computer graph-
from a mathematical standpoint they are definitely interest- ics algorithm for finding level sets of a scalar function [48].
ing for the following reason. The minimum number of zeros Here, we use marching cubes on each of the three components
for the electric field produced by a charged knot of a given of E (x) to find their zero level sets. The algorithm partitions
knot type provides a new knot invariant, and one based in a large cuboid containing the knot into small cubes; then, on
elementary physics. It may be a difficult invariant to calculate each cube, it uses a bilinear approximation of that component.
at this early stage. But we are intrigued by the possibility If the bilinear approximations of all three components of E
that subsequent research will find connections between it and pass through zero in the same cube, then we take the center
other, better established knot invariants. of that cube as an initial guess for our multivariable Newton
Our simulations also allow us to investigate how the method. Some initial guesses to Newton end up diverging or
equipotential surfaces change their topology as we vary the converging to a far away critical point, and we throw these
level of the potential. This sequence of topological transitions away. In contrast, the successful initial guesses quickly con-
can be characterized with a set of integers we call the Morse verge to the approximate locations of the zeros of E (x).
code for the knot [6]. We compute these transitions numeri- Along with the zeros, we are also interested in the equipo-
cally for some convenient embeddings of the simplest knots tential surfaces. These are given by φ −1 (v), where 0 < v < ∞
and illustrate them with computer graphics (Figs. 8 and 9). is some given voltage level. The relevant values of v range
from small positive values far from the knot, to large positive
values close to the knot. Let v ∗ = φ(x ∗ ) denote the potential
C. Numerical methods
at an equilibrium point. Recall that equipotential surfaces
To describe our computations, we introduce some notation undergo self-collisions at x ∗ and lose smoothness there. So
and terminology. Let the knot K be parametrized by a vector- to get a smooth surface, we perturb v ∗ to a nearby regular
valued function r(t ), where 0 t 2π . Because the knot value v at which the Hessian matrix of second derivatives
forms a closed loop, we also require that r(0) = r(2π ). Then, of φ (equivalent to the Jacobian of E ) has full rank. By the
from Coulomb’s law, the electric potential φ at a point x ∈ R3 implicit function theorem, this full rank condition ensures
away from the knot is given in dimensionless form by that φ −1 (v) is a smooth, orientable, compact surface without
2π boundary. We then use the marching cubes algorithm to render
|dr| |r (t )|dt the surface. By repeating this process for a range of v values,
φ(x) = = , (2)
r∈K |x − r| 0 |x − r(t )| we can explore how the equipotential surfaces change as we
vary the level of the potential.
where | · | denotes the magnitude of a vector quantity. (We
have written the potential in dimensionless form for conve-
nience; one could include physical parameters like the vacuum D. Results and conjectures for charged knots
permittivity 0 or the uniform charge density ρ along the knot, To illustrate the results obtained with this approach, con-
but we have chosen not to do so, as they play no role in sider the following parametrization of a trefoil knot:
our analysis.) The electric field associated with the potential
is given by E (x) = −∇φ(x). The zeros (i.e., the equilibrium r(t ) = (sin t + 2 sin 2t, cos t − 2 cos 2t, − sin 3t ). (4)
points) of the electric field are equivalent to the critical points
In our numerical simulations, we sampled a cubic domain
of the electric potential φ; as such, we will continue to use
of 30 × 30 × 30 initial guesses in a mesh surrounding the
the terms zeros, equilibrium points, and critical points inter-
knot and ran the multivariable Newton method to test for
changeably, as we did earlier in Sec. II.
convergence to a zero. We rejected iterations that grew too
For most knots, the potential φ(x) and its critical points
large, or were within a small distance threshold from another
cannot be calculated analytically. We must rely on numeri-
computed zero, indicating a duplicate.
cal integration and root-finding techniques. To perform these
Figure 10 shows that the electric field has seven zeros
computations, we replace the continuous knot by N + 1
for this particular embedding of a trefoil. By calculating the
unit point charges located at r(t0 ), . . . , r(tN ), where tk =
eigenvalues at these zeros, we can confirm that they are all
2π k/(N + 1), and use the following trapezoidal approxima-
saddle points and classify them by their indices (the dimen-
tion of (2):
sions of their stable manifolds).
Then, to obtain representatives of the equipotential sur-
2π |r (t j )|
N
φ(x) ≈ . (3) faces around the trefoil, we compute the critical values v ∗
N + 1 j=0 |x − r(t j )| at the zeros, perturb them to nearby values v, and take their
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First, we flatten K into a plane curve. The resulting curve are performed, altogether we get c additional zeros of index 2.
crosses itself and bounds a number of planar regions that Counting the c + 1 (or more) zeros of index 1 and the c zeros
we call holes (more properly, “holes” are bounded connected of index 2, we conclude the electric field around a squashed
components in the planar complement of the flattened knot). but not strictly planar version of K has at least 2c + 1 zeros.
The electric field in R3 produced by a charged planar curve The weakness of the “at least” part of the conclusion can
has a special property: at all points in the plane away from be traced back to the Poincaré–Hopf index theorem; that is
the curve, the electric field vector E lies within the same where the first “at least” qualifier popped up. If we could
plane. Moreover, at points very close to the charged curve, ensure exactly one zero in each hole, we would then be able to
the direction of E is nearly perpendicular to the curve (be- claim what we want: Z 2c + 1. We suspect the uniqueness
cause it receives its dominant contribution from the portion of the zero in each hole would follow if the holes were round
of the charged curve nearby). Thus the winding number of enough (neither too elongated nor too nonconvex), but this is
the electric field around the boundary of each hole is 1. The what remains to be properly formulated and proven in future
Poincaré–Hopf index theorem then implies that each hole con- work.
tains at least one source or sink zero of the planar vector field.
These planar zeros extend to zeros of the full electric field in
R3 , because the out-of-plane component of E is also zero, as IV. DISCUSSION
discussed above. In Ref. [6] it is proven that these zeros are
As we have tried to show in this article, the study of
all saddle points of index 1 in R3 . A counting argument then
knotted charge distributions opens up many new directions
shows that a planar curve with c crossings has c + 1 holes, and
for exploration. The questions are mainly motivated by their
since we just showed that each hole must contain at least one
conceptual simplicity and theoretical appeal, but they could
zero, we arrive at our first conclusion: the electric field around
have real-world implications. For instance, given that the ze-
a flattened knot has at least c + 1 zeros.
ros of electric and magnetic fields are relevant to problems of
But we are not done yet. A flattened knot is not an admis-
plasma confinement in nuclear fusion and to trapping of cold
sible knot because it has self-crossings. So the second step
atoms, related questions may be of experimental interest in
is to perturb the flattened knot by lifting one of its strands
those settings [53–55]. Charged knots also arise in molecular
up out of the plane, ever so slightly, at each crossing to
biology and polymer physics, where researchers study the
restore the topology of the original knot. By performing these
knottedness of charged DNA molecules and their interactions
lifting operations in tiny neighborhoods of the crossings that
with electric fields [2–5].
are sufficiently far away from the aforementioned zeros, we
are sure to preserve the existence and topological types of
the c + 1 (or more) zeros deduced in the first step, thanks
ACKNOWLEDGMENTS
to the structural stability of gradient vector fields [51].
Now comes the third and final step. By applying the Morse We thank Greg Buck for helpful discussions. M.L. was
inequalities [52], one can show [6] that each lifting performed supported in part by NSF RTG Grant No. DMS-1645643. A.T.
in the second step gives rise to a new zero at the associated was supported by National Science Foundation Grants No.
crossing, and this zero is of index 2. Since a total of c liftings DMS-1818757, No. DMS-1952757, and No. DMS-2045646.
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