Download Teaching Magnetism with Home

Teaching Magnetism with Home-Made
Experiments
SOMNATH DATTA
656, ”Snehalata”, 13th Main, 4th Stage, T K Layout, Mysore 570009, India
Email: [email protected]
Website “Physics for Pleasure” :http:// www.geocities.com/somdatta_2k
ABSTRACT
The article presents a methodology for teaching the basics of magnetism
in which concept-centred experiments (CCEs) made in a home workshop
form an integral part. The author has first presented a brief review of the
basic principles of magnetism followed by the CCEs that may facilitate
this learning process. He has mentioned four experiments, one of which
had been published earlier. After a cursory reference to that experiment
the author has presented details of the other three experiments, the
concepts they illuminate, and the data obtained from these experiments.
1 Introduction
Over the last two decades we have been
contributing low-cost concept-forming physics
experiments in the areas of mechanics and
electromagnetism, most of which had been
published in this journal. A round-up of these
experiments was also published in this journal
under the title “Concept-Centred Experiments
in Physics” (CCEs).1 The present article is a
continuation of that dialogue.
In this article we would like to present a
Physics Education • January − March 2007
combination of CCEs and theory that a teacher
at the +2 or college level may find useful in his
classroom teaching of the fundamentals of
magnetism. The experiments are “homemade”, a phrase we would like to stress to
highlight our belief that experiments that can
hold students in absorbing attention and at the
same time convey a lot of “physics with
conviction”, can be made in as little a space as
a garage, by a teacher himself, using some
basic tools and at a very small expenditure. The
reason we have chosen magnetism is that since
243
publication of the earlier article we have
designed a set of three new experiments
intended to facilitate understanding of this
particular topic.
There is this need for devising simple lowcost experiments which will fulfil a particular
objective of a teacher who is struggling to
convey the full meaning of a concept or an
equation to his young audience. The CCEs we
devised over a span of time were born out of
such needs felt in the classroom. Therefore
these experiments should not be seen in
isolation, but as part of a set of lessons. In this
article we shall discuss our experiments in
juxtaposition with the concepts and principles
(i.e., the theory) they will illuminate.
2 The Fundamentals
We shall review what we consider to be the
fundamentals in the theory of magnetism.
Figure 1. Explaining: Biot-Savart’s Law in (a) and (b); Ampere’s Law in (c).
Fundamental 1. Magnetism originates due to
the flow of electric currents, which can be
externally manifest (i.e., measurable with a
meter) as in conduction currents flowing in a
metallic wire, or can be internal currents,
effectively due to the orientation of the spin
and orbital angular momenta of the atomic
electrons, as in the case of magnetized
materials.
Fundamental 2. The magnetic force is a
velocity-dependent transverse force. By this we
mean that an electrically charged particle
carrying charge q and placed in a magnetic
field B will experience a magnetic force Fm
only when it is in motion, and that this force
will be perpendicular to the direction of its
velocity v. This force is given by the Lorentz
force equation:
244
Fm = qv × B.
(1)
Corollary 2A: If we place the straight segment
of a conductor carrying current I in a
“uniform” magnetic field B produced in the
region between the pole pieces of two magnets
(such that the N-pole of one faces the S-pole of
the other with a gap in between), then the
conductor experiences the force
Fm = I(L × B),
(2)
where L is the “length vector” of this segment
lying within the pole pieces of the magnets, its
magnitude being L and its direction being the
direction of the current.
From this point onward we shall restrict
ourselves to static magnetism. A static
magnetic field can be produced by a constant
electric current flowing through a closed coil.
Physics Education • January − March 2007
Fundamental
3.
The
time-independent
magnetic field B(r) produced at a point r is
given by Biot-Savart’s law which can be
expressed in either of the following two forms
μ I
B (r ) = 0
4π
μ0
B(r ) =
4π
ds ′ × ( r − r ′ )
.
(a )
C
| r − r ′|3
J (r ′) × (r − r ′) 3
d r ′. (b)
V
| r − r ′|3
∫
∫∫∫
(3)
In the above I represents a steady current
flowing through a closed conducting wire C,
ds′ is a directed line segment of C at r′ (Figure
1a), J(r′) is steady current density at r′ (Figure
1b), d3r′ is a volume element at r′. The
constant μ0 is called permeability of free space,
whose value can be expressed in the
convenient form:
μ0
= 10–7 T.m/A
4π
(4)
At this point, and for subsequent discourse
on electromagnetism it is essential that the
teacher introduces the concept of an oriented
surface and its directed boundary, without
which it is impossible to teach the basics of
electrodynamics, in particular, Ampere’s law
and Faraday’s law of electromagnetic
induction.
Figure 2. Application of Ampere’s Law for computing B (a) near a long straight
current carrying conductor, (b) inside a long solenoid of arbitrary cross section.
Definition of the Directed Boundary of an
Oriented Surface
S is an oriented (in general, curved) surface as
shown in Figure 1c. By this we mean that it has
a definite positive side and a definite negative
side. (The teacher should recognize that there
are non-orientable surfaces.3) We have
indicated the positive side of S with the + sign.
This surface S is not closed, so that it has a
Physics Education • January − March 2007
“boundary” Γ which is a closed curve. We
assign Γ a positive direction (indicated with an
arrow) which is taken to be anti-clockwise
when looked at from the positive side of the
surface.
This strange “asymmetric” direction
relationship can be traced to the right-handrule (RHR) associated with vector “cross
product”. The statement in the above
paragraph, translated into RHR would be read
245
as follows. If the thumb of the right hand points
in the direction of the positive direction of the
surface, then the other fingers would be seen
curling in the positive direction of the
boundary curve.
We now come to Ampere’s Law which can
be derived from Eq. (3b).
Corollary 3A: Ampere’s Law. Let there be
steady currents I1, I2, such that the net current
crossing S from the negative side to the
positive side is ΣI. Then this distribution of
currents creates a B field whose line integral
around the closed path Γ, taken in the positive
direction, is given by Ampere’s Circuital
formula:
∫
Γ
B.dr = μ 0 ∑ I .
(5)
For the example depicted in Figure 1b, ΣI =
I2 – I3 + I4.
Evaluation of B by performing the line
integral (5) is often difficult. In certain
instances of direct concern to us, however, we
come across distributions of electric current
which present a kind of “symmetry”.
Simultaneous application of Ampere’s law and
the symmetry consideration can lead to a
simplified, even if approximate, computation
of the B field. Two instances of such
computation have a bearing on two of the
experiments that will follow.
The first example is the magnetic field in
the vicinity of a long straight segment AB of a
closed coil ABCD (Figure 2a) carrying electric
current I. It is assumed that the adjacent sides
BC and DA are perpendicular to AB and long,
so that the segments BC, CD and DA make
little contribution to the field at a point P near
the centre of AB. Let r be the distance of P
from AB and L be the length of AB. It is
assumed that r
L. Then “by symmetry” the
magnetic field lines make coaxial circles
around the straight wire. A path C that passes
through P and also coincides with a field line
246
has a path length L = 2πr, and the current
crossing the surface is I. Hence from Eq. (5)
B≈
μ0I
.
2 πr
(6)
The direction of this B field is given by the
curling fingers of the right hand with the thumb
pointing in the direction of the current I.
The second example is of a long solenoid
(of arbitrary cross-section) whose lateral
dimension b is very small compared with its
length L (Figure 2b). This solenoid is carrying
a current I which flows through N turns. The
field inside the solenoid is strong and nearly
parallel to its axis, having an average value Bin.
The contribution from that part of the path C
which lies just outside the solenoid is rather
small. The left side of (5) is approximately BinL
whereas the right side is μ0IN. Hence,
Bin ≈
μ 0 IN
L
(7)
The direction of the B field is such that if
the right hand thumb aligns along it, the other
fingers would be curling in the direction of the
current I.
We shall now note here one important
aspect of current-current interaction that
follows by combining Eq. (2) with (6).
Corollary 3B. Let there be two parallel
segments AB and ab which are parts of two
closed circuits ABCD and abcd carrying
currents I and i respectively. Let the lengths of
these segments be L and l , and let them be
l L
separated by a distance r such that r
(Figure 2a). Then between these two segments
there will be (i) a force of attraction if their
currents are in the same direction, (ii) a force
of repulsion if their currents are in opposite
directions. The magnitude of this force, in
either case, is approximately given as
Physics Education • January − March 2007
Fm ≈
μ 0 Iil
.
2 πr
(8)
Note that the expression above ignores the
length L of the larger conductor.
3 Experiment No. 1: The Fundamental
Experiment in Magnetism − Equivalence
between a Bar Magnet and a Current
Carrying Solenoid
To a school child magnetism means
phenomena associated with “magnets”. He has
seen magnets in different shapes, e.g., bar
magnets, horse-shoe magnets, fridge magnets
(used to stick paper the fridge door), has played
with them for fun and is familiar with some of
their properties, e.g., that they attract iron. Yet
when we teach magnetism to a beginner, our
teaching strategy demands a paradigm shift
from magnets to electric currents. The first
lesson on magnetism should start with the field
produced by an electric current and the force
such current exerts on another current. This
means that it is the Fundamental 1 that should
form the basis for all subsequent discourses on
magnetism.
Figure 3: Solenoidal compass and its accessories.
The first lesson in magnetism should
therefore start by first establishing the
Equivalence between a Bar magnet and a
Current Carrying Solenoid by demonstrating
that a current carrying solenoid replicates all
the properties of a bar magnet, viz., (1) it has a
north-seeking pole and a south-seeking pole,
(2) it attracts iron and (3) like poles repel and
unlike poles attract. We have designed the
Solenoidal Compass and its Accessory to meet
Physics Education • January − March 2007
precisely this teaching objective.
We have designed the Solenoidal Compass
and its Accessory to demonstrate what we shall
call the Fundamental Experiment in
Magnetism, aimed at establishing the above
equivalence. We had presented a detailed
account of the device, and its pedagogical uses
in an earlier article.4 Therefore we shall avoid
further discussion of this experiment. We have,
however, presented a schematic diagram of the
247
revised design of this device in Figure 3 and a
still from a UGC TV program in Figure 4.
Figure 4. Still from a UGC TV program.
Figure 5. Schematic design of the apparatus for demonstration of the transverse
nature of the magnetic force.
248
Physics Education • January − March 2007
4 Experiment No. 2: The Transverse
Nature of Magnetic Force
One can give a convincing demonstration of
the transverse nature of the magnetic force
using one of those olden day cathode ray tubes.
In this apparatus the electrons leave a trace of
their path by ionizing the residual air inside the
evacuated glass tube. When a bar magnet is
brought close to this tube, its axis being held
perpendicular to the path of the electrons, the
ray is deflected giving a clear indication that
the magnetic force is perpendicular to the
velocity of the electrons.
However, such an experiment will not
come under the “home-made” category. The
home-experiment that we have fabricated is
described schematically in Figure 5. It uses a
rectangular coil W made of 70 turns of winding
wire, and provided with heat sinks S1 and S2 in
the form of thin aluminium foils stitched and
glued to the coil. The electric current enters
through the terminal P1 and leaves through the
terminal P2 (Figure 5a).
The horizontal segment HH at the bottom
of this coil is placed between the N-pole of the
top magnet M1 and the S-pole of bottom
magnet M2 of the magnet assembly shown in
Figure 5b. This assembly is made of a pair of
commonly
available
ceramic
magnets
(available in kids’ toy shops), each about 4 cm
long. To hold them together we have used a
single lamina collected from the choke of a
fluorescent tube (whose multifarious uses had
been highlighted in one of our previous
articles2). With some sawing, trimming and
gluing we have given it the shape shown in the
figure.
The force experienced by the segment HH
placed in the B field between M1 and M2 when
a current of 0.5A flows through the coil is
sufficiently large for the purpose of our
experiment since the effective current is Ieff =
0.5A × 70 turns = 35A. We have two modes of
placing the segment HH between the pole
Physics Education • January − March 2007
pieces as shown in Figsure 5c and d. In the first
mode the B field is vertical, and the segment
HH suffers horizontal deflection, i.e., left or
right. In the second mode the B field is
horizontal and the segment HH suffers vertical
deflection, i.e., up or down.
In order to hold the rectangular coil in place
we have a platform (Figure 5e) and a springlever assembly (Figure 5f). Two vertically
fixed aluminium strips AC and BD are
electrically connected through the ports A and
B to the DC power supply PS#1 (Figure 5g).
The horizontal pins C and D (Figure 5e)
transmit the voltage to the coil through the
spring lever assembly. The lever consists of
two parallel aluminium strips each having four
holes and separated from each other by
insulating plastic stubs inserted through the
holes E and F and glued to them. The lever is
supported on the pins C and D at the holes J1
and J2. The other two holes K1 and K2 have
been used for supporting the rectangular coil
by its pins P1 and P2, thereby transmitting the
voltage from PS#1 to the rectangular coil.
The “pins” P1, P2, C and D are made of
silver plated copper wire of diameter 1.5 mm.
The rectangular coil of Figure 5a has been
fabricated using 70 turns of 35 gauge winding
wire around a wooden block cut to the required
size, and then gluing them together with a thin
thread and an adhesive (Araldite).
The complete assembly is shown in
Figure5g. An actual photograph of the device
is shown in Figure 6.
The experimental set-up proposed in this
section gives a qualitative demonstration of the
Corollary 2B written in the form of Eq.(2). In
both modes of operation (see previous page)
the vector L is horizontal and perpendicular to
the vector B so that the magnitude of the
magnetic force is Fm = IeffLB, where Ieff ≈ 35A
and L ≈ 6.5 cm.
5 Experiment No. 3: Demonstration of
Attraction/Repulsion Between Parallel/
249
Anti-parallel Currents; Measurement of
the Force of Attraction, and Approximate
Evaluation of μ0.
The device here is a Swinging Coil Cs
suspended vertically and deflecting in the
magnetic field B of a Horizontal Coil Ch. The
objective of this device is two-fold. (1) It
demonstrates attraction/repulsion between two
parallel wires when the currents flowing in
them are in the same/opposite directions
(Corollary 3B). (2) It gives an approximate
evaluation of the coupling constant μ0/4π from
observation of the angle of deflection θ.
A schematic design of the device is shown
in Figure 7a. The multi-line rectangle ABCD
represents the coil Cs. It has Ns = 60 turns of
35 gauge winding wire. Its terminals are
soldered to two horizontal supporting pins S
and S made of 1.5 mm brass rods. They rest on
two holes drilled into a pair of inclined
aluminium bars K and K which are firmly fixed
to a flat wooden base. Leads from the DC
power supply PS#1 are connected to the bars K
and K thereby providing electric current Is (of
about 1 A) flowing through this coil. In order
to dissipate the heat generated by this current a
heat sink HS1 (not shown in the drawing) made
of thin aluminium foil is stitched to Cs.
Figure 6: A photograph of the apparatus for demonstration of the transverse nature of
the magnetic force. C = Rectangular Current Carrying Coil, M1 and M2 = top and bottom
magnets, F = Lever frame, L= lamina supporting the magnets, P = Platform.
The other rectangle EFGH represents Ch. It
has Nh = 300 turns of wire of the same gauge.
The DC Power supply PS#2 provides current Ih
of the order of 0.75A. It is loosely clamped to
the heat sink HS2 made of aluminium plate
(Figure 7b). The plate is bent at the edges to
250
make walls on three sides EF, EH and FG (we
have shown only one of these walls, running
parallel to EF, and labelled W). With this
design HS2 serves three objectives. (1) The
primary objective of dissipating the heat
generated by Ih. (2) The walls on the sides EH
Physics Education • January − March 2007
and FG provide “handle” for sliding Ch
forward and backward (we have provided two
parallel wooden guide walls so as to prevent
sidewise movement of the plate) in order to
bring the segment EF of Ch close to the
segment AB of Cs, either during repulsion or
during attraction. (3) The wall W makes a
barrier so that the closest approach between the
arms AB of Cs and EF of Ch is a “fixed
distance” r.
At the top of the vertical arm BC of the coil
Cs a pointer is provided. In the “attraction”
mode, this pointer moves against an angular
scale giving a visual measure of the deflection
θ in degrees.
Figure 8 shows an actual photograph of the
device with the coil Cs deflected by an angle of
approximately 30 degrees.
Figure 7. Schematic design of the apparatus for measurement of μ0 using ”swinging coil”.
In the discussion to follow Cs and Ch will
mean the parallel arms AB and EF,
respectively, of the two coils. When the
currents Is and Ih are in opposite directions the
force Fm between Cs and Ch is repulsive, so
that Cs swings away from Ch, tilting towards
the inclined bars KK.
When the currents are in the same direction
this force is attractive. Consequently, Cs
swings towards Ch and presses against W. By
sliding Ch the angle of deflection of Cs with the
vertical is allowed to increase until Cs separates
from W at some angle θ. This θ is the required
angle of deflection for currents Is and Ih. The
three values θ, Is and Ih are required for
evaluation of μ0.
Physics Education • January − March 2007
Figure 9 explains the geometrical and other
parameters used in deriving the formula for μ0.
The coil Cs has mass M, G is its centre of
gravity, w is its width. It swings about the line
SS which is at a distance h above G. Taking the
distance between the centres of the coil
segments AB and EF to be r, and assuming that
w is less than the length d of Ch, the magnetic
field produced by Ch along the axis of AB is
obtained from Eq. (6) to be
B ≈ μ0
Nh Ih
.
2 πr
(9)
For the force between the coils we adopt Eq.
(8).
251
Fm = N s I s wB ≈
μ0 ⎛ 2 N s Nh Is Ih w⎞
⎜
⎟.
⎠
r
4π ⎝
(10)
The force Fm makes the coil Cs tilt by an
angle θ at which the torque due to the force of
gravity Fg = Mg balances the torque due to the
magnetic force Fm. Hence at equilibrium
Mghsinθ = Fmucosθ.
(11)
Combining (11) with (10) we get an
approximate formula for determination of μ0.
μ 0 ⎡ Mghr ⎤ ⎛ tan θ ⎞
≈⎢
⎟ = ζχ ,
⎥⎜
4 π ⎣ 2 N h N s wu ⎦ ⎝ I h I s ⎠
(12)
Figure 8. A photograph of the ”swinging coil” apparatus.
⎡ Mghr ⎤
⎛ tan θ ⎞
where ζ = ⎢
⎟.
⎥; χ = ⎜
2
N
N
wu
⎝ Ih Is ⎠
⎣ h s ⎦
It should be remembered that the
assumption behind the “approximate formula”
(12) which is based on (8) is not valid (since
the two coils are nearly equal.) Applying a
more “adequate formula”5 which takes care of
the magnetic force between all the four arms of
the swinging coil Cs and all the four arms of
252
the horizontal coil Ch one gets a better estimate
of μ0/4π.
We have tabulated (Table 1) below our
experimental data and “estimates” of the value
of of μ0/4π using both the “approximate”
formula and as well as the “adequate formula”
in Table 1, for the sake of comparison with the
correct value given in Eq. (4). (We have taken
g = 9.8m/s2.)
Physics Education • January − March 2007
Figure 9. Explaining the measurement of μ0 using a “Swinging Coil”.
Table 1. Parameters and variables for the “Swing Coil” experiment.
M
(gm)
8.9
Ih
(A)
0.34
0.35
0.38
h
(cm)
0.8
Is
(A)
1.1
1.13
1.13
r
(cm)
1.0
w
(cm)
8.0
θ
(deg)
26
26
27
u
(cm)
4.5
tanθ
.4877
.4877
.5095
6 Experiment No. 4: Demonstration and
Measurement of the Magnetic Field
Inside a Solenoid, Leading to Evaluation
of μ0
The experiment we are going to describe is a
modification of the “current balance” described
in PSSC Physics.6 The current balance has a Ushaped conducting path etched on one half of a
rectangular flat plastic plate. A small mass m is
placed at the end of the other half of the plate.
The etched end of the plate is inserted inside a
large solenoid. The plate is supported along its
transverse axis by two horizontal brass rods.
Currents Is and Iu flowing through the solenoid
and the plate respectively are adjusted for
Physics Education • January − March 2007
Nh
300
χ
(A−2)
1.30
1.23
1.19
ζ
(×10−7Ν)
60
0.55
μo/4π(approx)
(×10−7Ν/Α2)
0.702
0.663
0.638
Ns
μo/4π(adequate)
(×10−7Ν/Α2)
1.026
0.970
0.934
balancing the mass m, leading to evaluation of
μ0. This experiment forms part of a PSSC kit
that had been distributed across our country
about 50 years ago.
Our experiment replaces the U-shaped
conducting path (allowing a single layer of
current) with a rectangular coil having 160
turns (thereby allowing 160 layers of current),
so that we can get better effect with much less
current drawn from our DC power supply. Also
we have replaced the cylindrical solenoid with
a solenoid with rectangular cross section for
greater freedom of movement of the coil inside
it. We shall describe our experiment.
The mention of “solenoid” normally brings
us the picture of a cylinder with circular cross
section and windings around it, thanks to
253
innumerable diagrams we have seen in text
books. However, under the usual assumption
that the B field is concentrated mostly inside
the solenoid and is approximately uniform, the
strength of Bin averaged across the cross
section should be nearly independent of the
shape of the cross section (unless the cross
section has some “odd shape”). Under the
above assumption the magnetic field Bin inside
the solenoid is approximately given as
Bin ≈
μ 0 IN
,
L
(13)
where I, N, L represent the current, number of
turns and the length of the solenoid.
Figure 10. Schematic design of the apparatus for measurement of μ0 using ”rectangular
coil inside rectangular solenoid”.
We have devised an experimental set up in
which a solenoid Σ with rectangular cross
section has been used (Figure 10a). One reason
for this choice is that rectangular aluminium
pipes are abundantly available in local
aluminium shops and such pipes come handy
for acting as the frame on which the solenoid is
to be wound. Also this aluminium base makes
an excellent heat sink to dissipate the heat
generated by the current Is which will flow
through the coil. The strongest reason however
is that when the rectangular coil Ρ (marked
abcd in the diagram) is inserted inside the
solenoid for detection of the Bin field, its right
hand end, marked cd, will find greater freedom
to move up and down under the magnetic force
Fm as the current Ir will flow through it. A
254
circular cross section would have obstructed
this movement due to its curved wall.
Our solenoid Σ is hand-wound with Ns =
1160 turns of 29 gauge winding wire on a 10
cm long rectangular frame cut out from a 6.3
cm×3.7 cm aluminium pipe (Figure 10b). The
field Bin is created by the current Is that flows
through Σ when the terminals are connected to
the DC Power supply PS#1.
The average of the magnetic field Bin
generated inside the solenoid by the current Is
will be directed parallel to the axis. We have
designed a rectangular coil Ρ having Nr = 160
turns of 35 gauge winding wire to sense this
field (Figure 10c). The wire is first handwound on a 12.3 cm × 4.8 cm and about 0.5 cm
thick rectangular wooden plate (so that the
Physics Education • January − March 2007
dimensions of Ρ is slightly larger, i.e., about
12.6 cm×5.1 cm.) The coil is then removed
from the frame and sewed onto a thick
aluminium foil with thread after curling and
wrapping up the edges of the foil around the
coil (Figure 10d). The foil serves several
purposes. (1) It acts as a heat sink. (2) It
provides some rigidity to the coil Ρ by acting as
a frame. (3) It provides a platform on which a
counterbalancing mass m is to be placed. (4) It
provides a means for fixing two insulated
vertical pins P and Q (we have used the
ubiquitous pins used by clerks for fastening
papers) for acting as the terminals to which the
voltage from the DC Power supply PS#2 is
transmitted. This is achieved by first soldering
the ends of the coil Ρ to P and Q, and then
resting these pins on two “bridge shaped”
aluminium bases Ω - Ω (with grooves indented
on it) which are electrically connected to the
DC Power Supply PS#2.
Figure 11. Photograph of the apparatus for measurement of μ0 using ”rectangular
coil inside rectangular solenoid”.
The polarities of the wires drawn from the
power supplies PS#1 and PS#2 are chosen in
such a way that Bin is directed from left to right
and the current Ir is directed away from us, i.e.,
in the direction of cd (Figure 10 a). As a
consequence the magnetic force Fm on the right
cross-arm cd will be downward. To
counterbalance this force a small mass m is
placed on the aluminium foil just inside the left
arm 4-1. (We have incorrectly shown m outside
in Figure 10a for convenience of drawing, but
Physics Education • January − March 2007
shown the correct position in Figure 10c).
Initially the left end cross-arm ab is resting
on a wooden block ‘W’ and the arm bc is
horizontal. The mass m is placed close to the
cross-arm ab just inside the coil on the
aluminium foil. PS#1 is switched on so that a
steady current Is flows through Σ. Next, PS#2
is switched on and the current flowing through
Ρ is gradually increased until, at some current
Ir the right hand end just dips downward. These
255
two currents Ir and Ir are noted for
determination of μ0/4π.
We have presented a photograph of the
device in Figure 11.
Let us take l for the length of the crossarm cd of Ρ which experiences the magnetic
force Fm and L for the length of the solenoid Σ.
The magnetic force Fmis obtained by
combining Eqs. (2) and (9). Its magnitude is
then
Fm = Bin lN r I r ≈ μ 0
N s I r lN r I r
L
(14)
Assuming that the pins P and Q are located
midway between the ends ab and cd, the
gravitational force mg that balances Fm has the
same magnitude. Hence, Fm = mg.
We shall use “small weights” for m in the
range of 1-3 gm. Therefore we define the
parameter η as follows.
η=
gL
× 10 −3 ,
4 πN s N r l
(15)
with the explicit understanding that the masses
of the counterweights will be in grams. Then
μ0
m
=η
.
4π
Is Ir
(16)
The data obtained from our experiment
have been tabulated in Table 2.
Figure 12. IsIr versus m.
Table 2. Values of parameters and variables for “Coil-inside-Solenoid” experiment.
l
(cm)
5.0
L
(cm)
Nr
Ns
8.5
160
1160
η
(×10−7)
0.0714
For evaluation of μ0/4π we plotted IsIr
versus m as shown in Figure 12. The data
shown in Table 2 on a straight line having a
small positive intercept on the IsIr axis. The
slope that the straight line makes with the
vertical axis is the ratio x/y equal to 12.8. From
this we find the experimental value μ0/4π =
12:8×.0714×10–7 = 0:914×10–7 N/A2.
256
m
(gm)
Is
(A)
Ir
(A)
IsIr
2
(A )
1
0.38
0.25
.095
2
3
0.39
0..39
0.45
0.65
.1755
.2535
Acknowledgements
The author is indebted to Mr. Vinay Babu and
Prof. S.R. Madhu Rao for their constant
assistance in overcoming all problems with his
computer (used for preparing the manuscript).
He is grateful to Prof. R. Srinivasan for his
encouragement and for being instrumental in
Physics Education • January − March 2007
providing the author with a small grant from
the Meera Memorial Trust, Palghat, Kerala.
Unfortunately the author could utilize only a
small part of that grant within the stipulated
time period. The cooperation received from Mr
B. N. Nanjundaswamy, even though it was for
a brief period, is also thankfully acknowledged.
It was at his suggestion that the author used
rectangular solenoid for Expt. No.4. The author
is also grateful to Prof. S.R. Madhu Rao for
going through the first draft of the manuscript
and pointing out some errors. The author has
used the digital camera presented by his son-inlaw Michael Murphy for preparing Figures 6, 8
and 11.
The author has used LATEX2 “for
composing this manuscript, has used freesoftware like Xfig 3.2.5, Gimp 2.2 (one of the
packages in Debian Linux) for preparing the
drawings, for editing the digital photographs
and for incorporating these into the text of the
manuscript. The anonymous authors of such
software have been an inspiration for him to
serve the physics community with his own
kind of service without expecting a reward.
Physics Education • January − March 2007
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257