1. science
2. mathematics
3. geometry

The Curlmeter
[AUTHOR(S)]
[SUBHEAD]
BUILDING BLOCKS | Undergraduate Research Projects
Experiments with toroids
by Armian Hanelli, Cyrus
Northern1Virginia
Comm
111Equation Chapter 1 Section
111Equation
1 [SECTION]
Chapter 1 Section
[SECTION]
[AUTHOR(S)]
Building Blocks
Building Blocks
[BODY]
The
Magnetic
James Clerk Maxwell, th
by Armian
TK
[HEAD]Hanelli, Cyrus Hossainian,
[HEAD]
specifically, for donut-sha
Northern Virginia Community College, Annandale
The Curlmeter
The Curlmeter
In 1861 he wrote, “Let th
[BODY]
covered wire. It may be s
[SUBHEAD]
[SUBHEAD]
effect will
that—
ofm
a
James Clerk Maxwell, the father of electrodynamics, had athe
fondness
for be
donuts
Experiments
with toroids objects
Experiments
toroids
specifically,
for donut-shaped
wrappedwith
in wire.
Maxwell’s wire-covered r
properties
anduniformly
rich physw
[AUTHOR(S)]
In 1861
he wrote, “Let there be a [AUTHOR(S)]
circular ring of uniform section,
lapped
covered wire. It may be shown that if an electric current is passed
through
thiscalled
wire
multipole
moment
the by
effect
will be
that ofCyrus
a magnet
bent
round
till its two
poles
are in contact.”
simplest
of multipolar
cu
Armian
Hanelli,
Hossainian,
by
Armian
TKHanelli,
Cyrus
Hossainian,
TK
Northern Virginia Community Northern
College, Annandale
Virginia Community College, Annandale
Maxwell’s
wire-covered
ring, or
solenoid,
exhibits
aStudying
number Colof
interesting
MEMBERS
OF THE
SPS CHAPTER
attoroidal
the Northern
Virginia
Community
toroids
has yiel
properties
and rich physics.
The
current
flowingHossainian.
in its windings
is
characterized
by a
[BODY]
[BODY]
lege (NVCC)
in Annandale,
VA. Third
from
left: Cyrus
To the
rightlike miniatur
can
behave
multipole
moment
called
a
toroidal
dipole,
or
an
anapole,
moment.
Toroidal
current
by Armian Hanelli and Cyrus Hossainian
electromagnetic
of him: Walerian Majewski. Seventh from left: Armian Hanelli. Image
courtesy of media a
James
Maxwell,
the father
James
of electrodynamics,
Clerk
Maxwell,
the
hadfather
a fondness
of electrodynamic
for donuts
simplest
of Clerk
multipolar
currents
that
produce
only
finite-range
magnetic
fields.
Northern Virginia Community College, Annandale
NVCC.
specifically, for donut-shapedspecifically,
objects wrapped
for donut-shaped
in wire.
objects wrapped in w
We investigated the simp
Studying toroids has yielded new insights into fundamental“curlmeter”
physics. Atoms,
for
exam
using
a magn
In 1861 he wrote, “Let there be
In a1861
circular
he wrote,
ring of“Let
uniform
theresection,
be a circular
lapped
ring
uniform
of un
can behave like miniature toroids. Toroids are also finding applications in
covered wire. It may be shown
covered
that if an
wire.
electric
It may
current
be shown
is passed
that if through
an electric
thiscu
w
electromagnetic
hot
in
thermonuclear
reactors.
Toroidal
moment
the effect willmedia
be thatand
of aconfining
magnet
the effect
bentplasma
round
will betill
that
its of
two
a magnet
poles are
bent
in contact.”
round till its
James Clerk Maxwell, the father of electrodynamics, had a
We Maxwell’s
investigated
the simplestring,
toroidal
multipole:
the dipole.
Weaor
began
byofof
making
ae
Just
as
loops
wire are
wire-covered
or
Maxwell’s
toroidal
wire-covered
solenoid,
exhibits
ring,
number
toroidal
solenoid,
interestin
fondness for donuts—more specifically, for donut-shaped
“curlmeter”
using
magnetic
model,
a ring of
12rich
neodymium
bemagnets.
usedcurrent
curlmeters
to
properties
andarich
physics.
The
properties
current
and
flowing
in
physics.
its windings
The
isascharacterized
flowing in
Just as loops of wire are used as Gaussmeters to measure
objects wrapped in wire.
amoment.
vectordipole,
characterizing
multipole moment called a toroidal
multipole
dipole,
moment
or an called
anapole,
aistoroidal
Toroidal
or an
curr
an
magnetic
fields,
toroidscurrents
may be
used
as
to detect
theproduce
In 1861 he wrote, “Let there be a circular ring of uniform section, Toroidal
moment
calculated
as a fields.
vector
simplest
of multipolar
simplest
that
produce
of curlmeters
multipolar
only finite-range
currents
that
magnetic
only pr
fi
curl of magnetic fields. The curl of a magnetic field B isofa Bvector
lapped uniformly with covered wire. It may be shown that if an elecin the Cartesian coo
Just as loops of wire are used as Gaussmeters to measure magnetic fields, toroids
the has
amount
of circulation,
orinto
vorticity,
in thenew
Bphysics.
field.
tric current is passed through this wire. . . . the effect will be that of a characterizing
Studying toroids
yielded
new
Studying
insights
toroids
has
fundamental
yielded
insights
Atoms,
into fund
for
be used as curlmeters to detect the curl of magnetic fields. The curl of a magnetic
¶ inare also
¶ f
can
behave like
toroids.
can behave
Toroids
like
areminiature
also finding
toroids.
applications
Toroids
It is
calculated
as miniature
a vector product
of the differential
operator
magnet bent round till its two poles are in contact.”
ˆ
ˆ
Ñ
=
x
+
y
+
is a vector characterizing the amount of circulation, or vorticity, in the B field.
It is
¶
¶
x
y in
electromagnetic
media and
confining
electromagnetic
hot plasma
media
in thermonuclear
and confining
reactors.
hot plasma
acting
onas
the
components
in the
Cartesian
coordinate
system.
Maxwell's wire-covered ring or its equivalent, a magnet bent into calculated
a vector
productofofBthe
differential
operator
> acting
on the compon
Ñ
a loop, exhibits a number of interesting properties and rich physics. of B in the Cartesian coordinate system.
We investigated the simplest We
toroidal
investigated
multipole:
the
the
simplest
dipole. toroidal
We began
multipole:
by making
the
(1)
The current flowing in its windings is characterized by a multipole
“curlmeter”¶using ¶a magnetic
“curlmeter”
model, a ring
using
of 12
a neodymium
magnetic model,
magnets.
a ring of 12 n
¶
(1)
moment called a toroidal dipole, or an anapole, moment. Toroidal
Ñ = xˆ
+ yˆ
+ zˆ
¶x
¶y
¶z
Toroidal moment
Toroidal moment
current is the simplest of multipolar currents that produce only finiterange magnetic fields.
Just as loops of wire are usedJust
as Gaussmeters
as loops of wire
to measure
are used magnetic
as Gaussmeters
fields, toro
to
Current flowing in a coil or on the surface of a magnetic torus
Studying toroids has yielded new insights into fundamental
be used as curlmeters to detect
be the
used
curl
asof
curlmeters
magnetic to
fields.
detect
The
thecurl
curlofofa magnet
magne
directed
along the axis
of symmetry
generates
a characterizing
toroidal moment
physics. Atoms, for example, can behave like miniature toroids.
m
is a vector
thetis
amount
a vector
of circulation,
characterizing
or vorticity,
the
amount
in the
of B
circulation
field. It i
running
through
object’s
donut
hole.
This
moment
m is of
Toroids are also finding applications in electromagnetic media and
calculated
as athe
vector
product
calculated
of the
differential
as
a vector
operator
product
> acting
the differential
on the com
op
Ñ system.
analogous
the magnetic
dipole
moment
produced
by current
of B in thetoCartesian
coordinate
of Bsystem.
in the Cartesian
coordinate
confining hot plasma in thermonuclear reactors.
(1)
flowing through a loop of wire and interacting
with B via a torque (1)
We investigated the simplest toroidal multipole: the dipole. We
¶
¶
¶
¶
¶
¶
ˆ
ˆ
ˆ
ˆ
x B.
τ = tm × ( Ñ x= B)
began by making a “curlmeter” using a magnetic model, a ring of
x with
+ yenergy
+ z U = -tm · Ñ =
x By+ analogy
yˆ
+ zˆ with the
¶x
¶y
¶z
¶x
¶y
¶z
magnetic dipole moment, the toroidal moment of a thin toroid with
12 circumferentially magnetized neodymium magnets.
Curlmeter
EXPERIMENTS WITH TOROIDS
TOROIDAL MOMENT
LEFT: We passed a linear wire through the hole of our suspended magnetic model to measure its dipole moment.
ABOVE: A magnetization M creates a surface current density J and
magnetic field B in a torus, illustrated here. Images courtesy of NVCC.
10
Fall 2014 / The SPS Observer
xˆ = B+intaA,
yˆbe equal
Ñ
+ zˆ toA tand
be equal B
toint=tmshould
where
a int
are,
the torus’s areas of the torus’s
=B
aA,respectively,
where A andareas
a are,ofrespectively,
¶x
¶y
¶z m
and of the
crosshole
section
“donut”
and of its
the“limb.”
cross section of its “limb.”
toroidal moment tm directed along the axis of symmetry running through the object’s
donut hole. This moment is analogous to the magnetic dipole moment produced by
ic toroid,toroidal
withmagnetic
approximately
circumferential
magnetization
M, has
a toroidal
Our
toroid,
with approximately
circumferential
magnetization
a toroidal
moment
tm directed
along the
axis
of symmetry
running
throughM,
thehas
object’s
current flowing through a loop of wire. An external magnetic field B can exert a torque τ
ent tm that
can hole.
be
written
directly
inisbe
terms
of M:
dipole
moment
that can
written
directly
terms ofdipole
M: moment produced by
donut
Thistmmoment
analogous
to the in
magnetic
= tm × B on the toroid and give it the potential energy Um = -m × B. By analogy with the
(2)
(2)
current
flowing
through
a loop
of wire.be
Anequal
external
B can
a torque
τ
internal
magnetic
field
Bthe
should
to tmagnetic
= BintaA,field
where
A exert
of
its magnetic
field B on the magnetic dipole moment of our torus
int toroidal moment of amthin toroid
magnetic
dipole
moment,
with internal
magnetic
field
1
1 toroid
tand
×B
on
the
and
give
it
the
potential
energy
U
mdV
m = -m × B. By analogy with the
r ´ (=
M
t
=
r
´
M
dV
)
a are,
respectively,
the torus’s
hole and ofareas
the of became
much larger than the torque on the toroidal moment from
Bint should
equal( to)tm =areas
BintaA,ofwhere
A and“donut”
a are, respectively,
the torus’s
m be
2
2 moment, the toroidal moment of a thin toroid with internal magnetic field
magnetic dipole
“donut”
andofofits
the“limb.”
cross section of its “limb.”
cross hole
section
curl B.
Bint should be equal to tm = BintaA, where A and a are, respectively, areas of the torus’s
Our
magnetic
toroid,
approximately
In a future experiment we will try to detect fields that, accord“donut”
hole
and of the
crosswith
section
of its “limb.”circumferential magOur
magnetic
approximately
circumferential
magnetization
a toroidal
be written M, has
netization
M,toroid,
has awith
toroidal
dipole moment
tm that can
ing
to Nobel laureate Vitaly Ginzburg, appear outside of a toroidal
dipole moment tm that can be written directly in terms of M:
directly
in terms
ofwith
M and
the positioncircumferential
r inside the toroid:
immersed in an electromagnetic medium. (No such fields
Our
magnetic
toroid,
approximately
magnetization M, hasdipole
a toroidal
(2)
dipole moment
tm that can be written directly in terms of M:
appear in a vacuum.) We also plan to rotate our magnetic to1
ssumingEquation
then that
we
almost
and an
small
(as compared
with (as compared with
deal with
almost
ideal and small
dVthat
t m2Assuming
= deal
r ´with
( Mthen
)an
(2)we ideal
extent
roidfrom
around
f the external
field B)
toroid,
its interaction
with the
of B means,
from
theof B means,
the
of 12
the
external
field B) toroid,
itscurl
interaction
with the
curl
the its diameter and observe the toroid’s magnetic field
tm =
r ´with
dV
( M )an
.
(2)
(normally
axwell lLaw,
an interaction
current
having
current
density
Ampere-–Maxwell
lLaw,
anexternal
interaction
withi,an
external
current
i, having current
densitylocked inside the toroid) escaping outside, creating an
2
eraction J,
with
the rate
of change
therate
electric
field E.of the electric field E.
and/or
interaction
withofthe
of change
electric-dipole-type toroidal antenna. //
ò
ò
ò
ò
Assuming then(3)that we deal with an almost ideal and small
This research was funded by a 2013 Sigma Pi Sigma
(3)
(as compared
with then
the extent
the external
field B)
toroid,
Undergraduate
Research Award.
Equation
with an almost
ideal
andits
small (as compared
with
¶E 2Assuming
¶that
E weofdeal
B = m0 J +the
m0eextent
B the
= m0 J + m0e 0field B) toroid, its interaction with the curl of B means, from the
0 Ñ ´of
interaction
withexternal
thethen
curl that
of
means,
fromanthe
Ampere–Maxwell
Law,
¶t 2Assuming
¶t Bwe
Equation
deal with
almost
ideal and small
(as compared with
Ampere-–Maxwell
lLaw,
an interaction
with
an external
current
i, having
current density
anextent
interaction
an external
current
having
current
density
J, B means,
the
of the with
external
field B) toroid,
itsi, interaction
with the
curl of
from the
J, and/or 3interaction with the rate of change of the electric field E.
Equation
and/or interaction
withan
the
rate of change
the electric
field
E.
Ampere-–Maxwell
lLaw,
interaction
with anofexternal
current
i, having
current density
uation Eq.
3 isthat
a differential
form
of athe
integral
law
we
see
in
our textbooks.
Note
equation Eq.
3 is
differential
form
theelectric
integral
lawE.
we see in our textbooks.
J,
and/or
interaction
with
the
rate
of change
ofofthe
field
Get Money for
(3)
¶E
(3)
Ñ´B = m J + m e
(3)
0
0 0
Chapter Research!
¶¶Et
¶t
SPS chapters are eligible for up to $2,000 in funding
for research projects through the SPS Chapter ReEquation 3
Notethat
that
Eq. 3 isEq.
a differential
form ofform
the of
integral
law we
see
ourin oursearch
Award (formerly the Sigma Pi Sigma UnderNote
equation
3 is a differential
the integral
law
weinsee
textbooks.
Equation
3
textbooks.
graduate Research Award). Applications are due
Note that equation Eq. 3 is a differential form of the integral law we see in our textbooks.
November 15th. For details see www.spsnational.org/
(4)
(4)
programs/awards/research.htm.
1 ¶
1 ¶
ds = m0i +(4)
E
s=m i+
B × ddA
E × dA
Ñ ´ B = m0 J + m0e 0
º
c 2 ¶t ò
c 2 ¶t ò
0
CORRECTION
OUR EXPERIMENTS
“Get Inspired!” on page 17 of the Spring 2014 issue of The SPS
Observer referred incorrectly to physics demos put on by “Juanita
Community College.” The actual name of the school is Juniata
College, and it is a bachelor’s granting institution. We also incorrectly stated that their most popular demo includes smashing a
Equation
4Our experiments
We paramaterized
the torque resulting from the interaction of the
cement block on the chest of a math professor—we should have
current density in i with the toroid as τ = t x i. It rotated the toroid's
said a physics professor! We apologize for these errors. See beTo axis
create
a curlmeter
capable
interacting with the conduction current density in
to align
with the
wire’s of
current.
low for the demo in action as college president Jim Troha whacks
equation Eq. 3, we ran a linear wire with current i through the hole of our toroid. The
To find t, “the effective” toroidal moment, we used the method
physics
torque resulting from the interaction of the current and the toroid, τ = t x i, pushed
the professor and SPS advisor Jim Borgardt. //
that Gauss
ago to
make the first measurement of
toroid’s
axis toused
align180
withyears
the wire’s
current.
the Earth’s magnetic field: we measured the frequency f of oscillaTotions
find t,
toroidal moment,
we used
method
of“the
our effective”
freely suspended
toroid under
the the
influence
ofthat
curlGauss
B. used 180
years ago to make the first measurement of the Earth’s magnetic field: we measured the
From thef torsional
equation
motion
of our toroid
weFrom
foundthe
f (Eq.
5) equation of
frequency
of oscillations
of ouroffreely
suspended
toroid.
torsional
2
,
as a function
of i, we
the found
torus’ fmoment
inertia,
1.01 × 10
motion
of our toroid
(equationofEq.
5) asI a= function
of–3i, kg
themtorus’
moment of
inertia,
I = suspension
1.01 × 10–3 kg
m2, and
the suspension
and the
wire’s
torsional
constant k:wire’s torsional constant k.
(5)
(5)
f 2 = t / 4p 2 I i + k / 4p 2 I
(4)
1 ¶
= m0i + 2capable
E × dA
To create
of (4)
interacting with the conduction
ò B ×adscurlmeter
c1 ¶¶t ò
current B
density
in0i Eq.
3, weEran
a linear wire with current i
×
s
=
+
×
d
d
m
A
2
ò
c ¶toroid,
tò
through the hole of our
creating a non-zero curl B there.
(
)
(
)
/4π2I)i + k/(4π2I)We were ultimately able to extract the crucial measurement for our
device,
the ultimately
empirical toroidal
t =crucial
1.20 ×measurement
10–5 Nm/A. Wfor
ire currents that did not
We were
able to moment,
extract the
pass
through
the
donut
hole
in
the
torus
exerted
no
torque,
in
–5 agreement with the theory.
our device, the empirical toroidal moment, t = 1.20 × 10 Nm/A.
Wire currents that did not pass through the donut hole in the
We then tried to replicate our findings with electromagnetic toroids connected to dc or ac
torus exerted
no torque,
agreement
with the
theory;
from the
voltage.
This proved
difficultinbecause
the single
layer
of windings
around the toroid
perspective
ofaour
toroid,
their fields
werea curl-less.
created
not only
toroidal
moment
but also
net magnetic dipole moment. As the
current
the linear
wire
threaded
through
thewith
toroid’s
holetoroids
increased,
Weinthen
tried to
replicate
our
findings
electric
con-the effect of its
JUNIATA
magnetic
field
on
the
magnetic
dipole
of
our
torus
became
much
larger
torque COLLEGE PRESIDENT JIM TROHA puts the faculty
nected to dc or ac voltage. This proved difficult because the singlethan the
onlayer
the toroidal
moment:
oscillations
of
the
toroid
became
unobservable
as
the
torus’
in
line axis
by hitting physics professor James Borgardt with a sledge
of windings around the toroid created not only a toroidal moaligned with the magnetic field of the wire.
hammer
as SPS students "Danger" Dave Milligan (right) and
ment, but also a net magnetic dipole moment. As the current in the
Caitlin Everhart (left) look on. Photo by Rick Hamilton.
wire
threaded we
through
toroid’sfields
holethat,
increased,
thetoeffect
Inlinear
a future
experiment
will trythe
to detect
according
Nobel laureate Vitaly
Ginzburg, appear outside of a toroidal dipole immersed in an electromagnetic medium.
(No such fields appear in a vacuum.) We also plan to rotate our magnetic toroid around
its diameter and observe the toroid’s magnetic field (normally locked inside the toroid)
escaping outside, creating an electric-dipole-type toroidal antenna.
The SPS Observer / Fall 2014 11