UMEÅ UNIVERSITET
Institutionen för fysik
February, 2014
Avazeh Hashemloo & Dmitry Kobyakov
Oscillating magnetic dipole in an
inhomogeneous magnetic field
1
Introduction:
This lab is based on the lab instruction by M. Holmlund, R. Röding and M. Forsberg
(Umeå University, 2009-02-20) and on the following article:
[1] Oscillation of a dipole in a magnetic field: An experiment, Juan Bisqert et al.,
Am.J.Physics, 58(9), 1990. See also Erratum: “Oscillation of a dipole in a magnetic
field: An experiment” [Am.J.Physics, 58(9), 1990], Juan Bisqert et al., Am.J.Physics,
59(6), 1991.
The purpose of this lab is to compare the theoretical prediction for the frequency
of the magnetic dipole oscillation with the experimental value.
Experimental setup:
In this experiment you will need
1.
2.
3.
4.
5.
6.
7.
8.
9.
Air track
A glider
Air generator
A Helmholtz coil
Magnetic dipole
Current source
An ammeter
Magnetometer
Stopwatch
Figure 1: Experimental setup [1].
2
Questions:
1. Draw the electric current and the magnetic field lines produced by that
current in the Helmholtz coil.
2. What is the energy of the interaction of a point-like magnetic dipole
𝝁(𝐫 ′ ) = 𝝁0 δ(𝐫 ′ − 𝐫) with the magnetic field 𝑩(𝐫′)?
𝑈(𝒓) =
3. Draw the energy of the magnetic dipole as a function of its position on x-axis
𝜋
when the angle between 𝝁 and 𝑩 is 0, and 𝜋. When does the oscillation of
2
the dipole moment happen and why?
3
4. You will calculate the frequency of the oscillation theoretically and you will
also measure it experimentally. In one of the theoretical models, the
magnetic dipole is assumed to be a point-like magnetic dipole, and in the
other model, its dimensions are not neglected. By virtue of the
approximation made in the model, is the theoretical prediction of the
frequency expected to be less or larger than the experimental value? Why?
5. Using Biot-Savart law, derive 𝑩(𝑥′) of a coil.
6. Calculate the energy of the interaction between the density of the magnetic
dipole moment 𝝁(𝑥 ′ ), centered at the point 𝑥, and the magnetic field 𝑩(𝑥′ ),
for 𝑥 ≪ 𝑅, where 𝑅 is radius of the coil. Derive equation of motion of the
magnetic dipole. Using the measurements of mass and of the magnetic field
strength of magnetic dipole, calculate frequency of oscillations for point-like
and finite-size magnetic dipoles.
Hint: Only the dimension along 𝑥 axis of the magnet should be considered,
and other dimensions are neglected. The magnetic dipole is represented as
sum of thin slices
𝑙
+𝑥
𝝁0 = ∫2𝑙
−2+𝑥
𝑑𝝁0
where 𝑑𝝁0 = 𝝁0
energy is
𝑙
+𝑥
2
𝑙
−2+𝑥
𝑈(𝑥) = ∫
𝝁0
𝑈(𝑥) = ∑+∞
𝑛=0
𝑥𝑛
𝑑𝑥 ′
𝑙
𝑑𝑥 ′
𝑙
. Then the magnetic
𝑩(𝑥 ′ ).
For small displacement of the magnetic
dipole, from the center of the coil, the
energy can be approximated as the first
non-vanishing term in Taylor expansion of
the energy
𝜕𝑛
𝑈(𝑥 ′ )�
.
𝑛! (𝜕𝑥 ′ )𝑛
𝑥 ′ =0
4
Measurements:
1. Measure the frequency of the oscillation, 𝜔𝑒𝑥𝑝 , for 5 different values of
current 𝐼, in the Helmholtz coil.
2. Measure 𝜇⃗ by magnetometer then calculate the frequency of the oscillation
theoretically (a): for a point-like dipole moment (𝜔𝑝𝑜𝑖𝑛𝑡 ) and (b): for a finitesize dipole moment (𝜔𝑓𝑖𝑛𝑖𝑡𝑒 ).
Current (𝐼)
(A)
𝜔𝑒𝑥𝑝
(1/s)
𝜔𝑝𝑜𝑖𝑛𝑡
(1/s)
5
𝜔𝑓𝑖𝑛𝑖𝑡𝑒
(1/s)
UMEÅ UNIVERSITET
Institutionen för fysik
Madelen Holmlund / Robert Röding /Mats Forsberg
2009-02-20
Oscillating magnetic dipole
in an inhomogenous
magnetic field
Introduction
This laboration is based on the two following attached articles:
Oscillations of a dipole in a magnetic field: An experiment, Juan Bisqert et al., Am.J.Physics,
58(9), 1990,
and
Erratum: "Oscillations of a dipole in a magnetic field: An experiment" [Am.J.Physics, 58(9),
1990], Juan Bisqert et al., Am.J.Physics, 59(6), 1991.
The equations (7) and (9) in the first article should be replaced with the equations (1) and (2),
respectively, in the second article.
Study these articles carefully before you start the laboration. It is essential that you understand
the ideas and physics in the different steps in the derivations. The derived expressions are
quite long and complicated, so it is convenient to write some sort of program to make the
calculations easier.
The derivations should be made using two different approximatation schemes, as described in
the article.
Experimental Setup
In this experiment you will use
•
Air track with Helmholtz coil and glider
•
Permanent magnet
•
Current source and ammeter
•
Magnetometer and measurement table, for measurement of magnetic dipole moment
•
Computer
•
Stopwatch
Things to think about
Describe how the two models differs. In both models certain approximations are made. Which
ones?
Which approximation has been made in the derivation of the equations (4) and (5), and which
experimental consequence follows from it?
Describe, using words, how equation (2) in "Erratum:..." has been derived. Which important
assumption has been made?
Describe how you determine the magnetic dipole moment experimentally. What important
source of error should be particularly important, and how should it be dealt with?
Give equation (23) a physical interpretation.
Equation (13) is substituted in equation (23) and the result is approximated to 'lowest order in
xm' . What consequence follows from this approximation in the experiment?
Explain what is meant with the second sentence from the top on page 842: 'The coefficient of
xm in the force expression will be (-M) times the square of our final calculated value for the
frequency of the harmonic oscillation, ωII'.
These questions should be answered in the written report.
The Report
The report should contain answers to the questions above. The structure should be as follows:
Abstract – Describe, in a short and concise way, what you have done and what results you
have achieved.
Introduction – The background of the experiment. Describe the article and compare with
your experiment.
Theory – Describe all theoretical calculations used in the laboration. Below the 'Things to
think about' section there are hints on relations that describe the experiment. Include figures
for clarity.
Experimental setup – Show, using figures, which components you are using in your
experiment. Do not forget to specify magnitudes of distances, currents, voltages an so on.
Method – Describe how you got your result.
Calculations – Show which calculations you had to do in order to achieve yuor result. Do not
forget about performing an error estimation.
Discussion – Discuss your result and compare with the results gotten by the authors of the
article, and the theoretical value. What can be done to improve the laboration?
References – State all references used to write your report.