Download Oscillating Magnetic Dipole in an Inhomogeneous Magnetic Field

Umeå University
Department of Physics
Petter Lundberg
2015-02-11
Oscillating Magnetic Dipole in an
Inhomogeneous Magnetic Field
- an instruction for the experimental lab
1
Introduction
Scientific work presented in journals are a common source of knowledge and inspiration. This
experimental lab is inspired by the article ”Oscillation of a dipole in a magnetic field: An experiment”, published in Am.J.Physics 58(9), 1990, and corresponding erratum (correction)1 : both of
which are based on the work carried out by Bisquert, J, et al.,[1, 2].
As the title of the article suggest, it considers an experiment of small ”frictionless” oscillations,
so that the concept of a simple harmonic oscillator can be applied, along the axis of a circular
coil carrying an electric current. This oscillator consists of a parallelepipedal permanent magnet
attached to a glider, which is allowed to oscillate close to friction free, on a linear air track
surrounded by a circular coil of rectangular cross section. The described system is analyzed
both theoretically and experimentally in the presented article, where the authors aim to provide
a clear example of how a ”judicious use of approximations are essential in order to apply the
general theory to a real world problem”. This is motivated by the use of different theoretical
models for the angular oscillation frequency: one deduced under the assumption of an infinitely
thin coil and neglecting the finite size of the magnet (ωI ), whereas the second model (ωII ) takes
these assumption, somewhat, into consideration. Finally both these models are compared to an
experimentally deduced value of the oscillation frequency (ωexp ).
As you will notice when reading the article, even though the presented experiment is suggested
to be suitable as a laboratory project at an intermediate/advanced undergraduate level, it contains
some troublesome calculations and derivations2 . Therefore you will be given additional theory,
see section 2, as a complement to the theory given in the article, as well as a more clearly stated
experimental procedure, see section 3.
However, before coming to the laboratory, you are strongly urged to read both the
article, its erratum,3 and the upcoming theory section (section 2) thoroughly. You are
required to complete the preparatory exercises given in section 2.3. It is of course also, as
always, important to read through the entire lab instruction, before the lab, so that you have a
clear picture in mind of the work at hand. The examination criteria (report structure, deadline
etc.) for this experimental lab are given at the end of this instruction, see section 4.
1.1
Aim
The magnetic dipole and harmonic oscillator are two very important, and commonly used, physical models. The aim of this experimental lab is therefore to enhance your understanding of these
models, their underlying approximations, and how these approximations limits their practical
applications.
Since the exercises in this lab are inspired by the the work carried out by Bisquert, J, et al.
[1, 2], an additional aim is to introduce you to scientific reading.
1
See attached article and erratum at the end of this lab instruction.
Some which even the authors got wrong initially and hence the erratum.
3
Note that eq.(7) and eq.(9) in [1] should be replaced by eq.(1) and eq.(2) in [2], respectively.
2
Umeå University
Department of Physics
2
Oscillating Magnetic Dipole in an Inhomogeneous Magnetic Field
- an instruction for the experimental lab
Theory
The majority of underlying theory for this lab is given in [1]. However, as mentioned above,
this section aims to facilitate your reading and understanding of the work described therein, and
thereby also your own experimental work.
2.1
Magnetic dipole moment
When examining the the magnetic field around a small permanent magnet, it shares many attributes with those of a magnetic dipole. A magnetic dipole can be viewed upon as a very small
loop, carrying a current I . The loop radius is very small, in relation to all other distances. The
dipole can be described by its magnetic moment m, according to
m = nI An̂,
(1)
where n is the number of loops around the edge of an certain cross sectional area A, with n̂
being the normal direction. The magnetic field of a dipole Bdipole may, after quite troublesome
calculations, using the relation between magnetic field and vector potential, be expressed as
Bdipole (r) =
µ0
(3
(m
·
r̂)
r̂
−
m
,
4πr3
(2)
where µ0 is the permeability of free space and r = rr̂ is the radial distance to the dipole.
The magnetic field lines for such a current loop is illustrated in figure 1, where the current
goes in to (×), and out from (·), the page. If one introduces polar coordinates, with ẑ along n̂,
and only considers points along the axis r = zẑ, the magnetic field strength, Bdipolez (z), is given
by
µ0 m
,
(3)
Bdipolez (z) =
2πz 3
where m is the magnetic moment in ±ẑ.
Due to the resemblance between a dipole and a permanent
magnet, a magnet can be seen as a huge number of microscopic
dipoles, which are more or less parallel. However, since it is
impossible to measure I , in the loop around A = π2 , on an
atomic level, one has to determine the total magnetic moment
mp for the entire permanent magnet. Using superposition, mp
can be expressed as the vector sum of all the dipoles inside the
magnet.
There are a number of alternative procedures of measuring
mp . One method, which will be used in this lab, is to measure
the magnetic field created by the magnet along its symmetry
axis Bexpz and compare it with a theoretical deduced expression,
Bcalz . This theoretical expression will be expressed as a function
Figure 1: A sketch over the
of mp and the dimensions of the magnet. The way of deriving
magnetic dipole field of a small
this theoretical expression may differ, dependent on which ascurrent loop, figure from [3].
sumption one takes. The simplest approach is to assume that
one measures the field from a far enough distance so that the
entire magnet may be seen as a single dipole. This enables one to neglect its finite size and hence
express Bcalz = Bdipolez . Note however that this model only is valid for distances much larger
than the magnet size, i.e. at distances where the magnetic field is very weak and hence hard to
measure. Therefore, to get a reasonable accurate expression for Bcalz , close to the magnet, one
should take the magnets dimensions into consideration.
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Umeå University
Department of Physics
Oscillating Magnetic Dipole in an Inhomogeneous Magnetic Field
- an instruction for the experimental lab
In [1] they made use of the magnet-solenoid equivalence under the assumption of uniformed
magnetization of the magnet, when taking the magnets dimensions into consideration. As described, they carried out the derivation for a parallelepipedal-shaped magnet. However if changing
the geometry of the magnet, one will have to consider another expression.
If for example the magnet instead had a cylindrical shape, with length L and radius a, as
illustrated in figure 2, one can express the strength of the magnetic field at the axis of symmetry,
assuming it is homogenous magnetized, by integrating the magnetostatic potential Ap over the
entire magnet. This integration can be simplified by using cylindrical coordinates ({ρ, φ, z}) and
the symmetry of the system. When integrating for the potential along ẑ, the integration can be
reduced to an integration over two magnetic monopolar disks, separated by the distance L. Then
since Bpz = −∂Ap/∂z the final expression


L
L
z+ 2
z− 2
µ0 m p 
,
q
q
(4)
−
Bpz =
2πa2 L
L 2
L 2
2
2
z+ 2 +a
z− 2 +a
is derived, where z originates from the center of the magnet.
y
a
mp ρ
(0, 0, zM P )
φ
L
z
x
Figure 2: A permanent cylindrical magnet of length L, radius a and mp = mp ẑ, built up of
numerous of atomic current loops, has an certain magnetic field strength at the position (0, 0, zM P )
along its symmetry axis.
Within this lab you will also be using a larger current loop, a so called solenoid. If viewed
from a far away distance it could also be approximated as a magnetic dipole. However, since
you are going to investigate properties of an oscillating movement along its central axis, such an
approximation is invalid and hence is the coil dimensions non-neglectable. Two ways of expressing
the field strength of the coil along its central axis Bcoilz were considered in [1]. The first method
uses Biot-Savart law, assuming an infinitely thin coil of radius R and N number of turns carrying
the current I, which after some calculations (see exercise 1b, in section 2.3), gives the following
expression
µ0 N IR2
,
(5)
BcoilIz (z) =
2(R2 + z 2 )3/2
for the magnetic field strength as a function of distance [5]. Note that if R → 0 eq.(5)→eq.(3),
I
and also that Bcoil
(z) has its maximum at z = 0.
z
The second approach described in [1] considers a coil of finite thickness by approximating the
cross-section of the coil to be rectangular with the dimensions 2l1 × 2l2 . By parametrization of
the whole coil section, introducing the two coordinates y1 and y2 with their origin at the center
of the coil, located R from the coil center and limited by ±{l1 , l2 }, it is possible to compute
the current contribution dI from every infinitesimal area dA = dy1 dy2 . As it turns out, this
approach is equivalent to integrating eq.(5) over the entire cross section while letting {R, z, I} →
{R + y2 , z − y1 , dI}. The final expression, for the magnetic field strength of a coil of finite size
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Umeå University
Department of Physics
Oscillating Magnetic Dipole in an Inhomogeneous Magnetic Field
- an instruction for the experimental lab
and rectangular cross-section BcoilII
(z), after integration is
z
µ0 N I
(z − l1 ) ln c1 − (z + l1 ) ln c2 ,
8l1 l2
p
R + l2 + (R + l2 )2 + (z − l1 )2
p
c1 =
,
R − l2 + (R − l2 )2 + (z − l1 )2
p
R + l2 + (R + l2 )2 + (z + l1 )2
p
c2 =
,
R − l2 + (R − l2 )2 + (z + l1 )2
BcoilII
(z) =
z
(6)
where {c1 , c2 } are constants[1].
2.2
Harmonic oscillator
Before going into describing the two models used to express the frequency of a harmonic oscillating
dipole in an inhomogeneous magnetic field in [1], it is wise to take a look at the force acting on
a dipole in an external field. We known that the force dF on a small current element I dl is
orthogonal to the external magnetic field Bext , according to the Lorentz force, and given by
dF = I dl × Bext ,
(7)
where in our case dl = dφφ̂ and Bext = Br ρ̂ + Bz ẑ, using cylindrical coordinates. By completing
the cross-product in eq.(7), where I is constant, one can integrate over the entire atomic current
loop and thereby get the total force acting on the loop, by the external field. To carry out
the integration one may express ρ̂ in terms of φ and change to Cartesian coordinates, i.e. ρ̂ =
cos φx̂ + sin φŷ. The complete integration follows below
Z 2π
Z 2π
Br ẑdφ
F = I
Bz ρ̂dφ −
0
0


Z 2π
Z 2π



ẑdφ
= I Bz
(cos φx̂ + sin φŷ)dφ −Br

0
{z
}
|0
=0
= −2πI Br ẑ,
and as shown, we are left with
F = Fz ẑ = −2πI Br ẑ,
(8)
from which one can conclude that the magnetic field from the solenoid, i.e. eq.(5) must be studied
more closely, in order to express Br in terms of Bcoilz [4, 5], expressed by either eq.(5) or eq.(6).
One method of performing this study, is to consider the magnetic flux Φr through the shaded
areas4 , shown in figure 3, which can be expressed as Φr = πr2 (z)Bz . It can be shown that Φr (z)
is independent of ∆z given that r(z) is the distance to a certain magnetic field line. By a first
order Taylor expansion of Φr (z + ∆z), i.e.
dΦr (z)
d
= Φr (z) + ∆z
πr2 (z)Bcoilz (z)
dz dz
dr
dB
coilz
= Φr (z) + ∆zπ 2r Bcoilz + r2
,
dz
dz
Φr (z + ∆z) = Φr (z) + ∆z
(9)
one can conclude that in order for Φr to be independent of ∆z, the expression within brackets
must equal to zero. In other words
Bcoilz
4
dr r dBcoilz
+
= 0,
dz 2 dz
i.e. along the axis
4
(10)
Umeå University
Department of Physics
Oscillating Magnetic Dipole in an Inhomogeneous Magnetic Field
- an instruction for the experimental lab
−1
which in combination with the fact that all field lines are parallel with B, i.e dr dz −1 = Bcoilr Bcoil
z
as seen in figure 3, gives
r dBcoilz
,
(11)
Bcoilr (r) = −
2 dz
i.e. an expression of the relation between the different components of the magnetic field. Since
we only are interested in the field lines in the interval r ∈ [0, ], we now have the possibility to
express the force along the axis, acting on a current loop, as a function of Bz , by substituting
eq.(11) into eq.(8), that is
Fz = −2πI Br () = Iπ2
dBz
dBz
= [I → nI ] = m
,
dz
dz
(12)
and since a permanent magnet consists of a superposition of many small current loop, the same
equation holds for the entire magnet, as m → mp .
Now when we know the force acting on our magnet it is possible, by completing the derivative
I
in eq.(12), using the expression for Bcoil
stated in eq.(5) and Newton’s 2nd law of motion (see
z
exercise 2b, in section 2.3), to derive the following expression for the frequency of a harmonic
oscillator inside an inhomogeneous magnetic field as
r
3µ0 IN
mp ,
(13)
ωI =
2Mtot R3
given that R z and Mtot is the total mass of the oscillating object.
It is also possible to arrive at the same expression for ωI , shown in eq.(13), by comparing the
potential energy of a harmonic oscillation, i.e. 0.5Mtot ωI2 z 2 , with that of the potential energy of
a dipole in an external field, i.e. mp · B, under similar assumptions [1].
Br
I
B
Bz
r(z)
z
R
Figure 3: The magnetic field of a circular coil, of radius R carrying the current I, is illustrated.
Also indicated are the circular areas of radius r(z), where r(z) is the distance to a certain field
line.
Since there are a lot of underlying assumptions for the theoretical model for ωI , some of
which are more or less questionable, Bisquert, J, et al. also considers an additional model for the
oscillation frequency, aiming to correct for some of the previously taken assumptions. The second
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Umeå University
Department of Physics
Oscillating Magnetic Dipole in an Inhomogeneous Magnetic Field
- an instruction for the experimental lab
expression they end up with is far from pretty;
s
µ0 Ncoil Icoil mp
K−
ωII =
ln
+ A− − A+ ,
8L/2l1 l2 Mtot
K+
p
2
R± = R ± l2 ,
R+ + R+
+ L2±
p
,
K
=
±
2
L± = L/2 ± l1 ,
+ L2±
R− + R−
p
p
2
2
R
−
R
+
+
L
R
R2 + L2± − 4Rl2
−
+
±
−
p+
,
A± = L2± p
2
2
2
2
+ L2± R− R−
+ L2±
R+ R+
+ L2± + R+
+ L2± + R−
(14)
but after direct, but long, calculations it is possible to show that eq.(14)→eq.(13) in the limit
{L, l1 , l2 } → {0, 0, 0} [1].
2.3
Preparatory exercises
During this lab you will study two different theoretical models for the frequency of a harmonic
oscillating dipole in an external inhomogeneous magnetic field. To prepare you for this task it
is important to grasp how these two models differ, both in their underlying assumptions and
derivation. Hence you should complete the following exercises:
Exercise 1: Magnetic moment
a) Would you say that eq.(3) when m → mp overestimates or underestimates the magnetic field
strength, if you compare with eq.(4), assuming the later is more accurate? Make sure to
motivate your answer.
Hint: Given {mp , L, a}, how will the field strength according to eq.(3) differ from that of
eq.(4) along the z axis?
b) Derive eq.(5), i.e the field strength of a solenoid along its central axis, using Biot-Savart law,
assuming a infinitely thin coil of radius R and N number of turns carrying the current I.
c) In [1] they describe a method to check the correctness of eq.(3) (i.e. eq.(13) in [1]). Describe
how they perform this check.
Note: No calculation needed, only describe how they perform this check.
Exercise 2: Axial force and harmonic oscillations
a) With eq.(12) as a starting point, describe, both in words and figures, how the direction of the
force depends on the how m̂p = ±mp ẑ and Bcoilz are directed in respect to each other.
b) Complete the derivation of ωI using eq.(12) and eq.(5) as well as Fz = Mtot d2 z/dt2 , given that
R z and Mtot is the total mass of the oscillating object.
Hint: You will have to solve a differential equation on the form d2 z/dt2 + Cz = 0
c) What assumptions are made when deriving ωI and in what way do these assumption differ
from those used when deriving ωII ?
d) In the experiment described in [1] they use a parallelepipedal permanent magnet, whilst you
will be using a cylindrical. Would you, based on the underlying assumption and derivation
described in [1], say that the same expression for ωII , i.e. eq.(14), holds for a cylinder?
Make sure to motivate your answer.
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Oscillating Magnetic Dipole in an Inhomogeneous Magnetic Field
- an instruction for the experimental lab
Experimental procedure
3.1
Experimental setup
The experimental setup basically consists of:
- An air track, with glider, to which a solenoid (R = 0.2 m and N = 154), is attached.
- A current source and and ampere-meter, used to drive and measure I.
- A cylindrical permanent magnet, with a dipole moment mp that is to be determined.
- A Hall probe, used to measure magnetic fields.
- A stopwatch, used to time the oscillation frequencies.
In addition to the above stated equipment you will also have the possibility to use rulers, calipers
and balances, in order to carry out the experiment. A simplified schematic sketch of the fundamental parts of the experimental setup is illustrated in figure 4, indicating both the air track,
glider, magnetic dipole and solenoid.
y
z
x
Figure 4: A simple schematic sketch over the experimental setup, illustrating the air track (triangular) on which the glider (gray) may move close to frictionless. On top of the glider is a
cylindrical permanent magnet (red) centered and attached. This ”dipole” is able to oscillate in
the inhomogeneous magnetic field caused by the solenoid (circular), powered by a current source
(not illustrated in the figure), surrounding the center of the air track.
3.2
Experimental exercises
Exercise 1: Determine the magnetic moment of your permanent magnet by measuring
the magnetic field strength along its symmetry axis.
In order to compare the accuracy of the two theoretical models for the angular frequency ω it
is necessary to determine the magnetic moment of the permanent magnet used. One method to
get an experimental estimation of mp is to use a Hall probe to measure the magnetic field caused
by an assumed homogeneously magnetized dipole of finite size, at a given distance zM P from the
probe, see figure 2. By alternating this distance along a given axis (in our case ẑ), experimental
values for for the magnetic field at a given distance, i.e. Bexp (zM P ), can be expressed. The
magnetic moment may then be estimated as the slope of a linear fit between the experimentally
measured Bexp (zM P ) and the theoretically deduced m−1
p Bcal (zM P ). The procedure for carrying
out this experimental exercise follows below.
Step 1: Place the magnetic dipole within the measurement stand and measure the field strength
along its axis of symmetry. This measurement should be carried out for at least 10 distances,
e.g. zM P = {2, 4, 6, . . . , 20} cm or zM P = {8.0, 8.5, 9.0, . . . , 15.0} cm.
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Umeå University
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Oscillating Magnetic Dipole in an Inhomogeneous Magnetic Field
- an instruction for the experimental lab
Step 2: Measure the Earth’s magnetic field strength Bearth and make sure to compensate for it,
i.e. remove it from your Bexp (zM P ) data. One way of measuring an estimate of Bearth is to
measure Bexp (zM P ) from both end-sides of your magnet, for a couple of ZM P , from which
it is possible to compute an estimate for Bearth .
Step 3: Use eq.(4) to express Bcal (zM P ) and thereafter estimate mp , as the slope of a linear fit
between Bexp (zM P ) and m−1
p Bcal (zM P ). You will need to measure the magnets dimensions,
i.e. L and a.
Exercise 2: Determine the limiting oscillation amplitude, such that the approximation
R z does not affect the oscillation frequency significantly.
In order to determine the oscillation frequency within the limitations of the underlying assumptions of the theoretical frequency models one must investigate at which oscillation amplitude zamp ,
i.e. distance from equilibrium, the approximation R z breaks. The reason for this exercise is
determine which amplitude, that is start position, to use in the upcoming exercise. The procedure
for this experimental exercise is stated below.
Step 1: Mount and center your permanent magnet on the glider.
Step 2: Make sure that the air track is level.
Step 3: Turn on the air, so that the glider may move freely along the air track.
Step 4: Set Icoil = 5.00 A and place the center the glider 1 cm from its equilibrium point.
Step 5: When released, time a suitable number of oscillations, e.g. 10 oscillations, and determine
the oscillation frequency ωexp (zini ).
Hint: Try to decide if it is easier to time the oscillations at a end-side or at the midpoint
of the oscillating movement.
Step 6: Repeat Step 5 for increasing amplitudes, e.g. increase zinin+1 = zinin + 2 cm, until a clear
pattern is visible, tentatively until 15 cm.
Step 6: Use your measurement data to determine a suitable zamp to use in exercise 3. Make sure
to motivate your decision.
Exercise 3: Determine the magnetic moment of the dipole, by measuring the oscillation frequency in an inhomogeneous magnetic field.
Assuming that eq.(13) and eq.(14) holds, one should be able to determine mp , by experimentally
measure the oscillation frequency ωexp , over a range of currents. The procedure for carrying out
this experimental exercise follows below.
Step 1: Make sure that your magnet still is mounted and centered on the glider.
Step 2: Set the current Icoil = 1.00 A and let the glider oscillate from the experimentally determined suitable starting position from Exercise 2. Time a suitable number of oscillations,
e.g. 10 oscillations, and determine ωexp (Icoil )|zini =zamp , for Icoil = 1.00 : 0.50 : 5.00 A.
Step 3: Measure Mtot and the rectangular dimensions of the solenoid, i.e {l1 , l2 }.
Step 4: Use the same method as in Exercise 1 to determine mpexp using both eq.(13) and eq.(14)
and compare with the measured value in exercise 1. In other words plot ωexp towards both
2
−1 2
m−1
p ωI and mp ωII and determine mp as the linear slope.
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Oscillating Magnetic Dipole in an Inhomogeneous Magnetic Field
- an instruction for the experimental lab
Examination
Since you have been given the opportunity to practice your skills in scientific reading, it would
be wrong not to give you the chance to practice your scientific writing as well. therefore the
examination for this experimental lab will be a complete report, written in English and
following the structure suggested in section 4.1. To facilitate your writing you may consider both
the preparatory and experimental exercises as a guide, as well as the questions and comments
given in section 4.2. The report should be well written, with your own words, using proper
language and notations, so make sure to proofread your writing.
The deadline for handing in the report to your opponents is no later than 2015-03-22
at 11.59 p.m.When you have corrected your report, with respect to the comments
from your opponent, the final report, together with the additional document containing comments and corrections, should be sent in to either: [email protected]
or [email protected], no later than 2015-03-27 at 11.59 p.m.
4.1
Report structure
Below follows a suggested report structure, which you should look upon as a recipe, i.e. you are
free to adapt, add, or change to your liking, but the outcome should be at least as well-structured
as the suggestion.
Cover page - As always when handing in work, or publishing, it is important to know by whom
the work has been carried out, what they have done, when and why, as well as contact information to the authors. Therefore it is essential that your cover page contains a suitable
title, your names and contact information, which course you are taking and its credits, as well as the date of submission. It is also customary to state at which department
and university the course is given.
Abstract - Describe, in a short and concise way, what you have done, why, what results you
have achieved and any conclusion drawn. The abstract may (preferably) be included on the
cover page, or alternately at the beginning of your report.
Introduction - The background to the experiment and the work carried out. Introduce your
readers to what you have done.
Theory - Describe the theoretical models, calculations and assumption used to carry out the
experiment and computations. It is strongly recommended to include figures for clarity.
Experimental procedure - Explain, using your own words and clear figures, the experimental
setup (which components have you been using), method, and the measurement techniques,
used throughout your experiment. You should also include the computational steps carried
out in order to achieve the results presented later on. Do not forget to specify magnitudes of
distances, currents, voltages etc. Based on this section, one should be able to repeat
your experimental work and, ideally, end up with the same result.
Results - What results have you achieved from the experimental work carried out. Make sure
to always include error estimation when presenting measurement data and/or
results.
Discussion - Discuss your results, their accuracy, experimental errors and possible improvements
in your work. The discussion may preferably end with a few sentences of final conclusion.
References - Make sure to state all references when writing your report.
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Umeå University
Department of Physics
Oscillating Magnetic Dipole in an Inhomogeneous Magnetic Field
- an instruction for the experimental lab
Appendix - If you have large data sets, of e.g. experimental data, to present or harsh computations, e.g. Matlab-scrips, you may attach them in an appendix at the very end. This is
to facilitate reading your report, but do not forget that each included table, graph, script
must be referred to in the ”actual” report.
4.2
Things to think about
Throughout this instruction you have been given a lot of information considering various approaches for calculating and measuring magnetic dipole moments and frequencies of an oscillating
dipole in an inhomogeneous magnetic field. It may therefore be hard to know what to put into
your report, which is the reason for this section. Below follows a list of obvious and non-obvious
things that might be worth to consider while writing your report.
- The preparatory exercises do not have to be included as such, e.g. full derivations might be
unnecessary. However they might be a good indicator for what to introduce, explain and
discuss in your report.
- Are every figure and table referred to in the main text? Do they also have a captions, which
described the contents of the table or figure?
- Are the results presented in a nice and clear way?
- Make sure to clearly state what underlying assumptions there are for each and every model.
How do they differ?
- Do the more advanced models bring anything new to the table or do they only make computations unnecessarily troublesome?
- Can you give physical interpretation to any of the more advanced models, and if so, do they
make sense?
- Would you say that measuring oscillations of a permanent magnet is a good way of estimating
its magnetic moment?
- How accurate are your results? Is any model more accurate than the other?
- Which are the error sources of the experimental work carried out and how could one minimize
these?
References
[1] Bisquer, J, Hurtado, E, Mafé, S, Pina, J (1990). Oscillations of a dipole in a magnetic field:
An experiment. American association of Physics teachers, Vol. 58 No. 9.
[2] Bisquer, J Hurtado, E, Mafé, S, Pina, J (1991). Erratum: ”Oscillations of a dipole in a
magnetic field: An experiment”. American association of Physics teachers, Vol. 59 No.6.
[3] ”Wikipedia, Magnetic dipole. 2015 01 10. http://en.wikipedia.org/wiki/Magnetic dipole
[4] Nordling, C , Österman, J. (2006). Physics Handbook 8th . New Jersey, USA, Studentlitteratur.
[5] Griffiths, D, Soroka, M, Throop, W. (2003). Introduction to electrodynamics 3th . Lund, Sweden, Pearson Education.
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