Download 10 Magnetic Fields

Course 1 Laboratory
Second Semester
Experiment:
Magnetic Fields
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Magnetic Fields
1 Introduction
The purpose of these experiments is to determine the magnetic flux density, B, in the
neighbourhood of current carrying coils. These coils will be carrying an alternating
current and the induced voltage (commonly referred to (incorrectly) as the
electromotive force emf) in a search coil, of known area, placed at a point can be used
to calculate the magnetic field. The experimental measurements of B can be compared
with values calculated from the theoretical formulae for coils of known dimensions
carrying a known current.
2 Theoretical Background
Starting with the magnetic field of a short element (Biot-Savart law (Lectures 13&14
of EMI) it is possible to derive the equations describing the magnetic field
distributions for any geometry. One of the simplest geometries is the axial fields of
simple coil configurations, e.g. solenoids or flat coils.
Length
2.1 Solenoids
Figure 1 (a) schematically shows
the cross-section of a solenoid. At
any point, P, on the axis of a
solenoid, the magnetic field vector
B is parallel to the axis and its
magnitude is given by
B=
µ 0 nI
(cos α − cos β )
2
(1)
β
α
P
B-field
where the angles are defined in
Figure 1, n is the number of turns
per unit length of the solenoid, I is
the current, and µ0 is the magnetic
constant, which has a value of
4π×10-7 T m A-1. Figure 1 (b)
shows the variation of the B-field along the
axis of the solenoid.
2.2 Plane Circular Coils
Figure 2 (a) is an illustration of a coil of N
turns, radius a carrying a current I. At a point
P on the axis of this circular coil the magnetic
field vector is parallel to the axis and has a
magnitude given by
Figure 1. (a) The angles defining the magnetic
field for a solenoid and (b)the associated magnetic
field.
a
P
x
I
Figure 2. (a) A plane circular coil of N turns.
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Distance
B=
µ 0 NI
a2
2 ( a 2 + x 2 )3 / 2
(2)
B-field
where x is the distance of P from the plane of
the coil. Figure 2 (b) shows a plot of the field
profile for such a coil.
Equations (1) and (2) show that it is possible
to simply calculate the magnetic field
strength for either of these geometries if the
details of the coil or solenoid is known.
In the present experiment an alternating
current is used. The coils do not contain any
magnetic material, so B is a linear function of
I. Your measuring instrument displays the
root-mean-square value of a sinusoidal
input so the simplest thing is to calculate Brms,
the root-mean-square value of B
x
Figure 2. (b) Schematic of the magnetic field profile
on the axis of a plane coil as a function of distance.
2.3 Finding the B-field from the magnitude of the emf induced in the search coil
If a search-coil consisting of n turns, each of area S is placed with its plane
perpendicular to a uniform magnetic field of magnitude B, the magnetic flux linking
the search-coil is φ, where
φ = BnS
(3)
If B depends on time, an emf ε is induced in the coil. This is given by
ε=−
dφ
dB
= −nS
dt
dt
(4)
If the current responsible for the production of the magnetic field is a sinusoidal
function of time, with frequency ω/2π, and if the circuit does not contain any
magnetic material, B can be written as
B = B0 sin( ωt )
ε = − ωnSB0 sin( ωt )
εrms = − ωnSBrms
(5)
(6)
Thus if εrms, ω, n and S are measured Brms can be calculated.
3 Apparatus and Experimental Procedure
An oscillator producing a high quality sinusoidal output feeds a signal of
approximately 10 V (rms) to each of three experimental stations, where is passes
through a resistor of 28 Ω and appears at the blue and green terminals on a black box.
The resistor limits the current that you can draw, thus ensuring that one experimenter
does not affect another. The stable frequency of the signal is recorded on the black
box and can be checked with a frequency meter if required. You are also given a 0.01
Ω four terminal standard resistor labelled with its exact value. The electronic
voltmeter, Levell type TM3B, is battery operated so please remember to switch off at
the end of your measurements! Make sure that you understand the controls on the
TM3B voltmeter ask a demonstrator if you are unsure. Set the controls of the
voltmeter as follows:
•
B.W. (Bandwidth) 10 Hz – 10 kHz
•
5 mV (Full scale deflection)
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Now wire up the circuit shown in figure 3.
Assuming that you have done everything correctly you should see a reading of about
2 mV on the meter with the standard resistor in circuit; that is, with the switch (figure
3) in position “a”; what is the current in the solenoid?
Now switch the meter from the standard resistor to the search coil. On inserting the
search coil into the solenoid you should see a reading of a few mV. It may be
necessary to change ranges on the voltmeter from time to time and you should check
the current through the solenoid at intervals to see whether it is remaining constant.
4 terminal resistor
Bench
Power
Supply
433 Hz
C1
Green Terminal
C
P1
P
Solenoid
Switch box
b
a
Search Coil
Coaxial cable to
TM38 input
Figure 3. The experimental arrangement for the connections to the
solenoid and search coil.
3.1 Experiment 1
Use the search coil to investigate the variation of B, the magnetic flux density, along
the axis of the solenoid. Do a preliminary run to decide how many observations you
need; where the field is varying slowly observations can be fairly widely spaced but
you need more where it is changing rapidly with position. Take care over the
measurement of the distance of the search coil from the ends of the solenoid using the
callipers provided. It is a sensible precaution to remove all metal objects e.g. rulers
and callipers from the vicinity of the coils before taking the voltmeter readings. Why?
Information about all the coils used in this experiment can be found in section 4.
Please read and observe the precautions mentioned at the beginning of section 4
when using metal measuring instruments on any of the coils.
Measure the appropriate dimensions of the solenoid (see section 4) and use these,
together with the measured value of the current, so calculate values of Brms for each
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position of the search coil. Use the Mathcad programme solenoid to calculate the Bfields from your measured emfs. But before using the programme you must make two
“hand” calculations – one for a position near the centre of the solenoid and one for a
position just outside the windings. If these show reasonable agreement with the
search-coil results, proceed with the computer programme and make sure that you are
using it properly.
In an ideal situation, values of Brms determined experimentally and theoretically
should, of course, agree with each other to within the limits of experimental error; but
practical arrangements do not always conform to ideal specifications so you should be
on the look-out for, and try to explain, any significant discrepancies between the
results obtained by the two methods.
3.2 Experiment 2
Replace the solenoid by one pair of plane coils and investigate the magnetic field
along its axis. Modify the Mathcad programme solenoid to calculate the magnetic
field for this case.
3.3 Experiment 3 (only if time allows)
Connect both plane coils in series, with the current circulating in the same sense in
both. Make investigations of this well-known (Helmholtz) arrangement. What useful
properties does it have? What happens if you reverse the connections to one coil?
Again, modify the Mathcad programme solenoid to calculate the magnetic field for
the two coils.
4 Dimensions of Coils
These coils are easily damaged by metal measuring instruments. Please ensure that
your measurements do not remove the enamel or distort the wire!
Search coil data: The search coils were wound on a former of diameter 17.32 mm.
You may take this as the internal diameter of the coil. The coils have a single layer
winding, which has 10 turns (34 standard wire gauge (s.w.g.), diameter 0.2337 mm,
enamel covered).
Solenoid: You may determine all you need to know by direct observation. The
solenoid has a single layer of wire (30 s.w.g., diameter 0.3150 mm, enamel covered).
Plane coils: Each of the plane coils was wound on a former of diameter 20.0 cm.
Note the relationship between this and the separation of the two coils as mounted. The
number of turns on each coil is 72, wound in three layers (24 s.w.g., diameter 0.5588
mm, enamel covered.
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