Download The Dependency of the Magnetic Field on the Diameter of a

The Dependency of the
Magnetic Field on the
Diameter of a Solenoid
ENS L202 – Scientific Communication
Almohaned Aljahdali
Eder Bastos
John Baycroft
Nat Steinsultz
Pavel Vasilev
Summary
An expression for the magnetic field produced by a solenoid can be obtained using
Ampere’s Law, which gives a proportional relationship between the current through a solenoid
and the magnetic field produced by it. Ampere’s Law also dictates an inversely proportional
relationship between the diameter of the solenoid and its magnetic field. By comparing three
different solenoids of diameter 2.2 cm, 4.2 cm and 8.9 cm, at 50mV, 100mV and 150 mV of
voltage, this experiment verified those relationships to be true with 2% to 36% error.
Introduction:
In this experiment, the magnetic field within three solenoids of differing diameters is
studied. The definition of a magnetic field is a field of force produced by a magnetic object or
particle. The field is detected by the force it exerts on other magnetic materials and moving
electric charges. At any given point it is specified by both a direction and a magnitude; as such it
is a vector field.
The term solenoid refers to a loop of wire, often wrapped around a core in the form of a
tightly packed helix, which produces a magnetic field when an electric current is passed through
it (Figure.1) Solenoids are important because they can create controlled magnetic fields and can
be used as electromagnets. The term also refers specifically to a magnet designed to produce a
uniform magnetic field in a volume of space.
Figure 1 Magnetic Field Lines in a Solenoid
In order to measure the magnetic field, we used a Vernier Magnetic Field Sensor (Figure
3) which uses a Hall Effect transducer. The Hall Effect is the production of a voltage difference
across an electrical conductor, transverse to an electric current in the conductor and a magnetic
field perpendicular to the current.
Figure 2 - Hall Effect Transducer
This sensor is sensitive enough to measure the Earth’s magnetic field and it can also be
used to study the field around permanent magnets, coils, and electrical devices.
The magnetic field within a solenoid can be predicted using Ampere’s law
 B  dl  
I
0 enc
(1)
Where B is the magnetic field, dl is the differential length along the path integral, μ0 is the
permeability of free space, and Ienc is the current enclosed by the path integral. Under the
idealized condition of an infinitely long solenoid, the magnetic field inside the solenoid is
B=μ0nI
(2)
where n is the number of turns per unit length, sometimes called the "turns density", μ0 is
the permeability of free space, and I is the current running through the solenoid. When the
length, L, is not significantly greater than the diameter, D, of the solenoid, the equation for the
magnetic field becomes
B= μ0nI*L/√(L2 + D2)
(3)
Experimental Setup
The experiment consisted of five major components: the multi-meter, power source, solenoids,
resistor and magnetometer (Figure 4). The multi-meter was used to measure the voltage across the
solenoid, which was generated by the power source. A 50Ω resistor was placed in series with the solenoid
and the power supply. This was used to limit the current through the circuit so that it did not overheat...
The magnetic field created within the solenoids was recorded through a USB connection to a PC. Using
solenoids with three different diameters (2.1cm, 4.2cm, and 8.8cm) we were able to see how the magnetic
field varied with respect to voltage as well as diameter.
Figure 3 - Vernier magnetic field sensor
Figure 4 - Photograph of the experimental set up
Procedure
First, we connected the power supply to the 50ohm resistor and the first solenoid in series. We
placed the multimeter across the terminals of the solenoid, and supplied voltage to the circuit. With the
solenoid powered, we placed the magnetometer in the center of the solenoid, and observed the results on
the computer attached to the magnetometer. These steps were repeated for each combination of the three
solenoids and three voltages: 50mV, 100mV, and 150mV.
Results
The measured characteristics of each solenoid are summarized in Table 1.
# of turns
L (cm)
D (cm)
Resistance (Ohm)
1
88
11.53
2.186
0.409
2
90
11.57
4.235
0.728
3
97
11.9
8.876
1.49
Table. 1
In order to calculate the hypothetical magnetic field, the current running through the solenoid must be
used. Since voltage and resistance across the solenoids were measured, current can be calculated using
Ohm’s Law. The measured voltage, the calculated current and the measured magnetic field inside the
solenoid are summarized in Table 2.
2.2cm
4.2cm
8.9cm
V (mV)
I (mA)
B (mT) measured
0.05088
0.124400978
0.160
0.10016
0.244889976
0.315
0.15056
0.368117359
0.475
0.04954
0.068049451
0.072
0.100838
0.138513736
0.134
0.15074
0.20706044
0.228
0.0503678
0.033803893
0.027
0.1009
0.067718121
0.060
0.15045
0.100973154
0.090
Table. 2
The theoretical field inside the solenoid can be calculated using the equation
 0 NI
L2 + D 2
(4)
. The ideal magnetic field was calculated and used to determine the percent error for each trial. The results
are summarized in Table 3.
B (mT) calculated
2.2cm
4.2cm
8.9cm
Error
%Error
0.117224562
-0.042775438
-36.4901667
0.230762817
-0.084237183
-36.50379392
0.346881487
-0.128118513
-36.9343762
0.062465602
-0.009534398
-15.26343676
0.127147889
-0.006852111
-5.389087788
0.190069941
-0.037930059
-19.95584283
0.027755499
0.000755499
2.721980267
0.055601592
-0.004398408
-7.910579062
-0.007093563
-8.556105955
0.082906437
Table. 3
As illustrated by Figures 5, 6, and 7, each solenoid demonstrates a linear relationship with
current, which is the expected behavior. The errors for the two largest solenoids were close to the
instrument error associated with the magnetometer. However, the measurements for the smallest solenoid
were consistently imprecise. The fact that the three errors for this solenoid were so close to each other
suggests that it is the result of a systematic error. It is possible that this error comes from an inaccurate
measurement of the solenoid characteristics, or a fault in the construction of the solenoid
Figure 5 – The calculated and measured magnetic field vs. current for a 2.2 cm diameter solenoid
Figure 6 – The calculated and measured magnetic field vs. current for a 4.2 cm diameter solenoid
Figure 7 – The calculated and measured magnetic field vs. current for a 2.2 cm diameter solenoid
Figure 8 - The measured magnetic field vs. the current for all three solenoids and a linear trend line for
each separate data set
By using Equation 4, the slope of the linear relationship between current and magnetic field for
each solenoid can be determined. The predicted slope is compared with the slope from a least squares fit
of the data is summarized in Table 4.
Predicted Slope
2.2cm
4.2cm
8.9cm
Measured Slope
0.940775948
0.915357709
0.817769286
Error in Measured Slope
1.292508
0.00346
1.121084
0.141833
0.938045
0.020471
Table. 4
The measured slope does not exactly agree with the calculated slope, but the inverse variation of
slope with diameter does hold consistent. The difference between the calculated and measured slopes is
probably the result of not having enough data points to get a truly accurate measurement of the slope.
Repeating this experiment with more data points should yield a more accurate slope.
Conclusion
The objective of our experiment was to attempt to observe the behavior predicted by Ampere's
law, and we did accomplish this. As expected, the smallest solenoid produced the most magnetic field,
and the largest produced the least. What did not match up as precisely were the exact results, which had
varying amounts of error. It is impossible to pinpoint what caused all the error, because the environment
in which the experiment was performed was riddled with potential disruptions. These included computers
with their spinning hard drives, the lack of a way to clamp the magnetometer in place, the lack of a core
in the solenoid, and many other omissions that reflect the lack of resources we had in performing the
experiment. An improved version of the experiment might use solenoids with metallic cores, which would
ideally be wound by machines for a greater number of turns; it would also likely take place with a
purpose-built apparatus and environment.
References
http://myweb.msoe.edu/~schenstr/Course_Web_Pages/2009-10-2-Winter_PH-2020/Lab%20Instructions/PH2020%20Exp%2007%20-%20The%20Magnetic%20Field%20Produced%20by%20a%20Solenoid.pdf
http://www.vernier.com/probes/mg-bta.html
http://en.wikipedia.org/wiki/Magnetic_field
http://www.netdenizen.com/emagnet/solenoids/thinsolenoid.htm