NOTES AND DISCUSSIONS
The magnetic field produced at a focus of a current-carrying conductor
in the shape of a conic section
C. Christodoulidesa兲
Department of Physics, School of Mathematical and Physical Sciences, National Technical University
of Athens, Zografou, GR 157 80 Athens, Greece
共Received 21 June 2009; accepted 30 June 2009兲
We determine the magnetic field at the focus of a conic section due to a current along the conic
section. For a given current I, it is shown that this magnetic field is the same and equal to ␮0I / 2p
for all conics with the same semilatus rectum p. © 2009 American Association of Physics Teachers.
关DOI: 10.1119/1.3183888兴
I. INTRODUCTION
The calculation of magnetic fields in textbooks using the
Biot–Savart law is limited mainly to current-currying conductors consisting of linear or circular sections. In rare cases
the field at the focus of a current-carrying elliptical or parabolic conductor is calculated.1 Recently the field of a more
general plane current loop was considered.2 The purpose of
this paper is to demonstrate an interesting common property
of conic sections regarding the magnetic field produced at
one of the foci of a current-carrying conic section.
II. ANALYSIS
We consider a current-carrying conductor in the shape of a
conic section 共a circle, ellipse, parabola, or hyperbola兲. Polar
coordinates 共r , ␪兲 are used to describe the curves, with a
focus of the conic section taken to be the origin O, and the
axis of symmetry of the curve coinciding with the polar axis
共see Fig. 1兲. There is a constant current I in the conductor in
the direction of increasing ␪. A current element Idr with
position vector r relative to O produces a magnetic field dB
at O, which is given by the Biot–Savart law as
dB =
␮0I r ⫻ dr
.
4␲ r3
共1兲
dr̂
= e␪
d␪
共4兲
is the unit vector in the direction of increasing ␪. Therefore
r̂ ⫻
p
dr
=
r̂ ⫻ e␪
d␪ 1 + e cos ␪
共5兲
r̂ ⫻
p
dr
=
ẑ,
d␪ 1 + e cos ␪
共6兲
or
where ẑ = r̂ ⫻ e␪ is the unit vector normal to both r̂ and e␪,
and is out of the plane of the curve in Fig. 1.
Equations 共2兲 and 共6兲 and the definition r = rr̂ give
r⫻
冉
p
dr
=
d␪
1 + e cos ␪
冊
2
ẑ
共7兲
so that the ratio in Eq. 共1兲 becomes
冉
冊
1 e
r ⫻ dr
+ cos ␪ d␪ẑ.
=
r3
p p
共8兲
The total magnetic field at O for a circle, ellipse, or parabola is found by integrating Eq. 共1兲 from ␪ = −␲ to ␲
For the geometry and current in Fig. 1, the direction of dB is
normal to the plane of the figure and outward.
In polar coordinates the general equation of conic sections
共position vector of a point on the curve兲 is3
r=
p
r̂,
1 + e cos ␪
共2兲
where r̂ is a unit vector from O to the current element, the
length p is the semilatus rectum of the conic, and e is its
eccentricity 共e = 0 for a circle, 0 ⬍ e ⬍ 1 for an ellipse, e = 1
for a parabola, and e ⬎ 1 for the left branch of a hyperbola兲.
If we differentiate r with respect to ␪, we obtain
pe sin ␪
dr̂
p
dr
=
.
2 r̂ +
d␪ 共1 + e cos ␪兲
1 + e cos ␪ d␪
共3兲
Fig. 1. A current element Idr on a conic section and the magnetic field dB
it produces at O, the focus of the curve.
The quantity
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Am. J. Phys. 77 共12兲, December 2009
http://aapt.org/ajp
© 2009 American Association of Physics Teachers
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Fig. 2. The two branches of a hyperbola.
B=
␮ 0I
ẑ
4␲
冕冉
␲
−␲
冊
1 e
+ cos ␪ d␪ .
p p
Fig. 3. Conic sections with a common focus O and the same value of the
latus rectum 2p.
共9兲
If instead of the equation of a conic such as that in Eq. 共2兲,
we had used the general form
r = r共␪兲r̂
共10兲
for a curve in plane polar coordinates, Eq. 共9兲 would have the
more general form
B=
␮ 0I
ẑ
4␲
冖
d␪
,
r
r=
B− = − 2
derived in Ref. 2. We perform the integration in Eq. 共9兲 and
obtain
共12兲
for a circle, ellipse, or parabola.
The hyperbola has to be treated separately because it has
two branches between its two asymptotes, which form angles
of ␪0 and ␲ − ␪0 with its axis of symmetry 共see Fig. 2兲. The
value of ␪0 is given by cos ␪0 = −1 / e because this value of
cos ␪ makes r infinite. The field produced by the left branch
of the hyperbola is
B+ = 2
␮ 0I
ẑ
4␲
冕冉
␪0
0
冊
1 e
␮ 0I
+ cos ␪ d␪ =
共␪0 + e sin ␪0兲ẑ.
p p
2␲ p
共13兲
Equation 共2兲 produces negative values of r for ␪0 ⱕ ␪
⬍ 2␲ − ␪0. We can keep r positive if we change the sign of
the right-hand side of Eq. 共2兲 and can limit ␪ to the range
−共␲ − ␪0兲 ⱕ ␪ ⱕ ␲ − ␪0 for the right branch of the hyperbola if
we write cos共␪ − ␲兲 instead of cos ␪ in Eq. 共2兲. The right
branch of the hyperbola is then given by
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Am. J. Phys., Vol. 77, No. 12, December 2009
共14兲
for −共␲ − ␪0兲 ⱕ ␪ ⱕ ␲ − ␪0.
For the current direction shown in Fig. 2, the magnetic
field produced at O by the right branch of the hyperbola is
共11兲
␮ 0I
ẑ
B=
2p
p
r̂
e cos ␪ − 1
=
␮ 0I
ẑ
4␲
冕 冉
␲−␪0
0
−
冊
1 e
+ cos ␪ d␪
p p
␮ 0I
共␲ − ␪0 − e sin ␪0兲ẑ.
2␲ p
共15兲
The total magnetic field produced by a hyperbolic conductor at its focus is therefore
B = B+ + B− =
␮ 0I
ẑ,
2p
共16兲
the same as for the other conic sections. The field at the
right-hand focus of the hyperbola is also given by Eq. 共16兲.
We conclude that conic sections with the same semilatus
rectum p, carrying the same current I, produce the same
magnetic field ␮0I / 2p at their foci. A group of such conic
sections is shown in Fig. 3 with their common focus at O.
a兲
Electronic mail: cchrist@central.ntua.gr
J. D. Kraus, Electromagnetics, 4th ed. 共McGraw-Hill, New York, 1991兲,
Problems 6-3-10 and 6-3-12.
2
J. A. Miranda, “Magnetic field calculation for arbitrarily shaped planar
wires,” Am. J. Phys. 68共3兲, 254–258 共2000兲.
3
M. R. Spiegel, Mathematical Handbook of Formulas and Tables
共McGraw-Hill, New York, 1968兲, pp. 37–39.
1
Notes and Discussions
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