Download Magnetic Field Map 1 Equipment 2 Theory

Physics 16
Magnetic Field Map
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Magnetic Field Map
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Equipment
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Theory
Faraday’s law states that if you have a coil of wire of n turns through which
there is a changing magnetic flux, then there will be an induced emf (measured in volts) in the coil of magnitude
= −n
dφ
dt
(1)
where φ is the magnetic flux (φ = B · dA) through the loop. In this
laboratory we will make use of Faraday’s law to map the magnitude of the
magnetic fields created in a single coil of conduction wire and in a matched
pair of coils (Helmholtz coils) when an alternating current is passed through
them.
R
2.1
The Single Coil
The formula for the magnetic field on the axis of a coil is;
B=
µo N IR2
2(R2 + Z 2 )3/2
(2)
where µo = 1.26 × 10−6 H/m is the magnetic permeability , N is the number
of turns of a coil, R is the radius, and z is the distance along the z-axis.
In this experiment, the current through the coils will be alternating and
can be expressed as I(t) = Io sin(ωt) where ω = 2π60. Since both the
magnitude and direction of the current will be varying sinusoidally at a rate
of 60 Hz, the magnitude and direction of the magnetic field at a point along
the axis of the coil will also be varying sinusoidally at 60 Hz. To reflect this
time dependence, equation (2) can be rewritten as B(t) = Bo sin(ωt) where
Bo =
µo N Io R2
2(r2 + z 2 )3/2
(3)
If Io is in amperes, mo in H/m and all length measurements are in meters,
Bo will be in Tesla.
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Figure 1: Single Coil
To check the validity of equation (3) and to map the magnitude of the
magnetic field along the axis, we need an independent way to measure Bo .
Consider then a small search coil consisting of n turns of conducting wire
and cross sectional area A placed along the axis of the large coil so that the
normal to A lies along the axis of the large coil as shown in figure 2. The
flux φ through this small search coil is
φ=
Z
~ · dA
~ = ABo sin(ωt).
B(t)
Figure 2: Single Coil with “search coil”
Since φ is varying sinusoidally there will be an induced emf in the search
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coil; and an application of Faraday’s law yields an expression for its magnitude
dφ
d(sin(ωt))
(t) = −n
= −nABo
dt
dt
or
(t) = −nωABo cos(ωt) .
(4)
If the ends of the search coil are connected to the input of an oscilloscope
and (t) displayed on the screen, the amplitude of (t),
o = −nωABo
(5)
can be measured. Since n, ω and A can also be easily measured, equation
(5) provides us with an independent way of measuring Bo . If o is in volts
and A is in m2 , then Bo will be in Tesla. We will use this method to measure
Bo in the laboratory.
Note that if the axis of the search coil is rotated so that it makes an angle
θ with the axis of the large coil, the amplitude of (t) becomes
o = −nωABo cos(θ) .
(6)
Note also that the current in the large coil and the induced emf are 90o out
of phase.
2.2
The Helmholtz Coils
Helmholtz coils are a matched pair (same number of coils, same radii, etc.)
spaced a distance apart equal to their radii. The field on the axis at the
midpoint between them (point P in figure 3) can be found from equation (2)
by multiplying by 2 (there are two coils each producing a field at P) and
setting Z = 1/2R (since the field point is at the same distance in front of
one and behind the other). Thus, B(t) = Bo sin(ωt) where
Bo =
4 3/2 µ
5
o N Io
R
≈
µo N Io
.
1.4 R
(7)
We will use the same method employed with the single coil to check equation
(7) and to map the magnetic field. Helmholtz coils provide a very uniform
field over a relatively large volume in the space between the coils and are,
therefore, a convenient and valuable apparatus for experiments which call for
a uniform magnetic field.
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Figure 3: Helmholtz Coils”
2.3
Earth’s Magnetic Field - A Challenge
Up to now we have been measuring a varying magnetic field. Clearly, if we
can produce an oscillating field with AC current, we can produce a steady
field with DC current. Ideally then we could produce any field we wanted,
and since we know the current and geometry of our coils, we can calculate
the direction and magnitude of out field.
A compass is a simple device which tells us only the direction of a magnetic field. It does not tell us the magnitude or the source of the field. Also
most compasses do not tell about the component of the field up or down,
they only tell us the direction in the plane perpendicular to gravity. So our
measurement will only be of the magnitude of the Earth’s field perpendicular to gravity. There is a special compass called a “dip compass”, which we
would need for the z-component.
To measure the Earth’s magnetic field we will add the field of the Earth
and the field of our coils. If these fields are not parallel, then the direction
of the sum will not be magnetic north, and a compass will show the vector
sum.
The challenge to you is to design a method for measuring the Earth’s
magnetic field. What are the parameters you can play with? You have
a compass, two coils, an current meter and a current supply which you can
vary. Arrange these as you think best, watch the compass, and vary geometry
and currents as YOU think best.
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It is a very good idea to discuss this with your lab partner before you
arrive at your lab class.
3
Experimental Purpose
The specific experimental purposes of this laboratory are the following:
1. to experimentally measure the magnitude of the magnetic field, Bo ,
at the center of a single coil and to compare that value to the value
predicted by equation (3);
2. to experimentally measure the magnitude of the magnetic field, Bo , at
a point on the axis midway between the two coils of a set of Helmholtz
coils and to compare that value to the value predicted by equation (7);
3. to map the magnetic field strengths of a single coil and of a set of
Helmholtz coils along both their transverse and longitudinal axes by
measuring Bo at various positions along those axes;
4. to confirm that the emf induced in the search coil is 90o out of phase
with the current in the coils creating the magnetic fields;
5. to experimentally confirm that the emf induced in a search coil which
has its longitudinal axis at an angle θ to the direction of a uniform
magnetic field is given by equation (6);
6. to make a rough estimate of the volume within a given set of Helmholtz
coils in which the magnetic field in uniform; and
7. to map the shape of the magnetic field of a single coil in a plane perpendicular to the plane of the coil and passing through the center of
the coil.
8. challenge to design a method using a controlled magnetic field and a
compass to measure the Earth’s magnetic field.
In all cases except the challenge, the measurement of Bo will be made
with the use of a search coil and equations (5) and (6).
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Procedure
The apparatus is illustrated in figure 4 on the next page. An oscilloscope
is used to measure the amplitude of the sinusoidal voltages. The current in
the large coils comes from a voltage reducing transformer which is plugged
into the line voltage. The voltage output from the transformer is approximately 0.5 volts at 60 Hz and is taken from the terminals marked A. The
terminals marked B give the voltage across a 0.1Ω resistor which is in series
with the coils. The current in the coils can be determined from the voltage
across this 0.1Ω resistor and Ohm’s law. The search coil has 20 turns and
a cross-sectional area of 1cm2 . The Helmholtz coils have 60 turns and are
approximately 13.5cm in diameter. The actual dimensions of the large coils
should be measured. A scale is mounted in the middle of the coils so as to
measure distances along the axes of the coils.
Figure 4: Experimental Apparatus
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1. Connect the positive terminal of the transformer output (the red jack
of terminals A) to the red jack on the Helmholtz coils. Connect the
negative terminal of the transformer output to the bare wire joining the
wire in the front coil to the rear coil. Connect the B terminals of the
transformer to channel 2(Y) of the oscilloscope. Connect the search coil
to channel 1(X) of the oscilloscope. In this configuration only one coil
will have a current passing through it. Channel 1(X) of the oscilloscope
can be used to measure the induced emf and channel 2(Y) can be used
to measure the current in the coil. Plug in the transformer and turn
on the oscilloscope. Place the scale along the axis of the coils making
it level with its top edge 5 mm below the mid-line (the approximate
radius of the search coil).
2. Keeping the plane of the search coil parallel with the plane of the coil,
hold it at the center of the coil carrying the current and measure the
amplitude of the induced emf. Then measure the current in the coils.
Use equation (3) to calculate the magnitude of Bo from Io and use
equation (5) to calculate the magnitude of Bo from o . Compare these
two values with each other and with the given calibrated value.
3. Adjust the vertical gain controls on channels 1(X) and 2(B) so that the
two are about equal. What is the phase difference? Set the scope to
the XY mode and observe the Lissajous figure. Note what you observe
and give a brief explanation.
4. Measure Bo at 5 mm intervals along the longitudinal axis starting 3 cm
on one side of the coil and ending 3 cm from the other side. Make a
plot of B as a function of position along the axis.
5. Place the scale along the transverse axis and measure Bo along this
line at 5 mm intervals starting at one edge of the coil and ending
at the other edge. Note that the curvature of the field along the two
directions (transverse and longitudinal) is opposite in the two directions
so that the center point is neither an absolute maximum nor an absolute
minimum. Make a plot of B as a function of position along this axis.
6. Connect the positive terminal of the transformer output to the positive jack of the Helmholtz coils. Connect the negative terminal of the
transformer output to the negative jack of the Helmholtz coils. Repeat
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the measurement of steps 2-5. Make a rough estimate of the volume
within the coils where B is constant to ±5%. Compare the homogeneity of the field of a single coil with that obtainable from the Helmholtz
arrangement.
7. Measure the change in the induced emf that occurs when the search
coil is held at an angle to the field so that the flux is reduced by a factor
cos(θ) in equation (6). To do this, place a piece of polar graph paper
under the coils with the origin at the center. Hold the search coil at
the center and looking down form the top bring the stick out at 45o to
the axis of the coils. Repeat for angles of 30o and 60o . Note that this
could be used to find the direction of the field at any point in space
around the coils. How? Set up the single coil again. Use this method
to find the shape of the magnetic field in a plane parallel to the table
top which intersects the center of the coils.
Challenge
You will want to control the magnitude of the field from you coil. You do
this by controlling the current from the DC supply. Wire the coil as shown
in Figure 5. If you use one or two coils is you choice. How you orient the
compass and coils depends on your design.
Note: the Earth’s field varies between 0.3 and 0.6 Gauss (1 Gauss = 1
milliTesla). You can produce a field of similar magnitude with this coil with
less then 100 mA. Do not exceed 100 mA.
current meter
DC supply
Coils
(1 or 2 coils - your choice)
Figure 5: Experimental Apparatus for Earth’s magnetic field measurement
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Lab Report
Follow the usual lab notebook format. Your lab report should include the
answers to all of the questions asked in the introduction or procedure, all
raw and derived data, and an estimate of the magnitude and sources of
error in any data recorded. When answering any question or when giving
any comparison or explanation, always refer to specific data to support your
statements. Be sure to include the following in your notebook:
1. all raw data;
2. samples of all calculations done;
3. the results of the comparisons in step (2);
4. the results of the observation in step (3);
5. plots of B vs position for the single coil and Helmholtz coils along their
transverse and longitudinal axes;
6. a comparison of the field homogeneity obtainable from a single coil with
that of a pair of Helmholtz coils;
7. an estimate of the volume within the Helmholtz coils where B is constant within ±5%;
8. the results of the calculations and mapping done in step 7;
9. a discussion of the sources of error in this experiment with an estimate
of their effect on the results.
For the challenge, you should include a very detailed description. Remember your solution may be very different then anyone else’s. This means
that you may need to come up with a unique analysis.
In this last section you will report the magnitude of the Earth’s magnetic
field AND the uncertainty in your measurement. The uncertainty analysis
should follow what was presented in class, and what is in a separate write-up.