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Course 1 Laboratory
Second Semester
Module: Magnetic Fields
Units:2
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 emf induced 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) 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
β
α
2.1 Solenoids
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=
µ0nI
2
(cos α − cos β )
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 2 shows
the variation of the B-field along
the axis of the solenoid.
P
Figure 1. The angles defining the magnetic
field for a solenoid.
(1)
Length
β
α
P
B-field
Distance
Figure 2. The magnetic field for a solenoid.
2.2 Plane Circular Coils
At a point P on the axis of a circular coil
consisting of N turns, each of radius a the
magnetic field vector is parallel to the axis
and has a magnitude given by
B=
µ0 NI
2
a2
( a 2 + x 2 )3 / 2
a
P
x
( 2)
I
where x is the distance of P from the plane
of the coil and I is the current flowing
through each turn. Figure 4 shows a
plot of the field profile for such a
coil.
Figure 3. A plane circular coil
B-field
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.
x
Figure 4. The magnetic field profile on the axis of
a plane coil.
In the present experiment the current is an ac current, 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
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 E is induced in the coil. This is given by
dφ
dB
E=−
= − nS
( 4)
dt
dt
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, the instances value of B can be written as
B = B0 sin(ωt )
Hence
E = −ωnSB0 sin(ωt )
(5)
and
Erms = −ωnSBrms
( 6)
Thus if Erms, 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 ohms 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
given also a 0.01 ohm four terminal standard resistor labelled with its exact value
(why has it got four terminals?). The electronic voltmeter, Levell type TM3B, is
battery operated so please remember to switch off at the end of your
measurements!
4 terminal resistor
Bench
Power
Supply
433 Hz
C1
Green Terminal
C
Solenoid
Switch box
b
a
Search Coil
Coaxial cable to
TM38 input
Figure 5. The experimental arrangement for the connections to the
solenoid and search coil.
P1
P
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)
Now wire up the circuit shown in figure 5.
Assuming that you have done everything correctly you should see a reading of about
2 m V on the meter with the standard resistor in circuit; that is, with the switch (figure
5) 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 and can proceed
with the following investigations. 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.
3.1 Experiment 1
Use the search coil to investigate the variation of B, the magnetic flux density, along
the axis of the solenoid (see equation 6). 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
position of the search coil. Use equation (1). Use the mathcad programme solenoid to
calculate the B-fields 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 to 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
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 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.