MAGNETIC FIELDS












FORMAL REPORT
PCS110 - 007











SIMON WITHERS
973779499

In this experiment the magnetic fields of the earth and the fields produced by electric currents in a long strait wire as well as in three common coil configurations, Single Coil, Solenoid Coil and Heimholtz Coil, was investigated and compared to theory, in particular results derived from Ampère's Law for Steady Current. The potential advantages and disadvantages of the three coils are also examined. For the most part, once experimental error has been accounted for, the results measured agree with the accepted theoretical results.

Magnetic field is a vector quantity and so the earth's magnetic field strength and direction can be determined by taking measurements of the vertical, longitudinal and latitudinal components and then summing these results. The vertical component of the earth's magnetic field can be measured directly with a gaussmeter (a device for measuring the magnitude of the magnetic field in a chosen direction), or can also be calculated from B axis intercept of a B vs 1/R graph for the magnetic field strength (B) generated by a long strait wire at a horizontal radial distance (R). The measurements taken of the wire's magnetic field can also be used to determine the permeability of air which should not differ meaningfully from the permeability of a vacuum, known to be exactly 4  ×10^-7 Wb/Am.^[1]. Theory predicts that the magnetic field at a distance R from a wire of non-finite length, carrying a current I is given by the formula B=^µ0^I/_2  R, which can be compared with experimental evidence for a long but not infinate lenth wire. The results should be indistinguishably different, for a wire of sufficient length.

Three common magnetic coil configurations are the compact Single Coil, the Solenoid Coil and the Helmholtz Coil, a configuration which consists of two identical single coils set side by side in parallel, with the distance between them equal to their radius (fig. 2). A Solenoid Coil is very similar to a Single Coil, but is is extended, until it's length is much greater then it's diameter (fig. 3). Standard magnetic theory allows us to derive equations using Ampère's Law for Steady Current^[1,2], for the magnetic fields generated by these coils. Ampère's Law states that the value of the line integral:     B^.dl about any closed loop is equal to the product of µ₀ and the encircled current. For the Single Coil, Ampère's Law gives a magnetic field of B=^µ0^NI/_2R. For the Helmholtz Coils the magnetic field is given by B=^4µ0^NI/₍₁₂₅₎½_R. For the Solenoid Coil the equation is B=^µ0^IN/_L. For all equations, B is the magnetic field, I is the current running through the coil, µ₀, the permeability of vacuum, R is radius of the coil, N is the number of loops or turns the in coil and L is the length of the coil.

For measuring magnetic field strength and direction a hand held gaussmeter with a sensitivity of 10^-2mT was used. A long strait wire with a radius of 4.85mm connected to a power source was provided, as well as a 110 loop Solenoid Coil 250mm in length, a 100 loop Helmholtz Coil and a 100 loop Single Coil. All three coils had an radius of 55mm.

Figure 1: Single Coil

Figure 2: Heimholtz Coil

Figure 3: Solenoid Coil

Measurements of the vertical and horizontal components of the earth's magnetic field were taken by holding the gaussmeter as close to perpendicular to the component being measured as possible, with the gaussmeter stablised against a solid surface in order to prevent unexpected motion while making the reading.

The magnetic field surrounding the wire was again measured with the gaussmetter, by placing a sheet of paper against the wire, marking off distances and taking field strength readings at those distances. For the three coils the gaussmeter was placed at the center of the of coil under study, recording the field strength and then moving the gaussmeter away from the center of the coil first radially and then longitudinally to determine the distance from the center of the coil at which the field strength dropped by 5%

Table 1 details the measured directional components for the Earth's magnetic field. When summed as vectors these results give earth's magnetic field as being 38° below the horizontal and 8.9° East of North with a magnitude of 0.0648 mT

The current through the long strait wire in the second phase of the experiment was set to 19A. Results are tabulated in table 2 (with the radius of the wire included in the listed distances).

The magnetic field strength was plotted against the inverse of the distance as shown in fig. 4. It can be seen from the graph that the vertical component of the earth's magnetic field should be about 0.03 mT or 0.04 mT. Also from fig. 4 the values can be fit to a function of the form B=m¹/_R+b where b will be 0.04 mT and m will be 2.71×10^-9Tm. Comparing this value with the formula for the long strait wire presented earlier we see that 2.71×10^-9Tm=^µ0^I/₂   with the current being constant at 19A we get µ₀=2.85  ×10^-7Wb/Am which is about 71% the accepted value of µ₀=4  ×10^-7Wb/Am

Distance (inc. R) Magnetic Field Calculated Field 10.85mm 0.27mT 0.35mT 13.85mm 0.23mT 0.27mT 16.85mm 0.20mT 0.23mT 19.85mm 0.19mT 0.19mT 21.85mm 0.18mT 0.17mT 23.85mm 0.16mT 0.16mT 26.85mm 0.15mT 0.14mT 30.85mm 0.13mT 0.12mT 33.85mm 0.12mT 0.11mT 40.85mm 0.10mT 0.09mT Table 2: magnetic field strength and distance from a strait wire carrying 19A

Figure 4: magnetic field vs inverse of distance (B vs 1/R) for a strait wire carrying 19A

For the experiments with the Solenoid, Helmholtz and Single Coils, the current was maintained at 1.9A. Results are summarized in table 3.

From theory, using the equations presented above, the magnetic field at the center of the Solenoid Coil should be 1.05mT. Both radial and longitudinal measurements of the Solinoid Coil determined a magnetic field strength at the center of the coil of 1.05mT - the amount predicted by theory. The field decreased to 1.00mT, a decrease of just under 5%, 40mm from the center in both the radial and the longitudinal directions. This produces a sphere 2.68×10^-4m³ in volume inside of which the magnetic field is at least 95% the strength at the center.

Similar measurements and calculations were performed for the Helmholtz Coils. The magnetic field strength at the center, as calculated, should be 1.55mT, about 7% higher the value measured at the center of the two paired coils at 1.45mT. Radially, the field dropped 5% to 1.38mT, 35mm from the center and likewise longitudinally. This produces a sphere 1.80×10^-4m³ inside of which the field is at least 95% the center of the field's strength.

For the Single Coils, theory indicates that the magnetic field strength should be measured at 2.17mT, a difference of about 20% from the measured value of 1.79mT. The field strength dropped 5% at a distance of 25mm from the center of the coil, generating a volume of 0.65×10^-4m³

In considering any of these results it must be noted that the experiments were not performed under ideal conditions. At any given point in time, there were potentially as many as 8 or 9 other groups using the same laboratory space generating magnetic fields. While it is not likely that the field strengths were strong enough to cause any observanle deviation, the possibility remains.

The measurements for the magnetic field of the earth are relatively suspect. The horizontal components were very difficult to measure for a number of reasons. Primarily due to being in an enclosed room when taking the measurements, the actual East-West and North-South axis' had to be estimated, and thus the measurements reflect this estimation. The vertical component however does not suffer from this point of error and as is shown from the results from the long wire, which agree with the value of 0.04mT, this value should be close if not accurate.

The measurements for the magnetic field produced by the long strait wire are very close to the calculated values for the measurements taken from 19.85mm and farther, but the closer measurements deviate somewhat from what is expected. One possible reason for this deviation would be inaccuracy in measuring the radius of the wire. This would explain the decreasing deviation from the expected with distance, as the distance increases an error in the wire radius would effect the resulting calculation less. This however seems unlikely as the calculation from these figures for µ₀ was too low by 29%₀, and from the graph of B vs 1/R, fig. 4, the three measurements that most disagree with the results of theory play an important role in reducing the slope of the curve that was used to calculate µ₀. This would indicate that there are either unaccounted for forces acting in the region close to the wire, or more likely there the measurements taken closer to the wire were not as accurate as they should have been.

The magnetic field measurements for the three coils vary from matching exactly what theory predicts for the solenoid coil, to being off by 7% for the Helmholtz coil, to being as much as 20% for the single coil. The reason for the greater error on the single coil is probably due to the difficulty in accurately positioning the gaussmeter at the center of the coil, which in turn makes the reading somewhat uncertain. The observed volumes at which the field is at 95% strength or greater and theoretical magnetic field strengths are repeated in table 4, which serves to illustrate the differences in utility of the three coils.

What is clear that the strength of the magnetic field generated by the coil is inversely proportional to the volume at which the field is nearly at full strength. This phenomenon can be taken advantage of by considering the use to which a magnetic field is being applied. When a task requires a larger volume to be affected by the field the Solenoid Coil may be very appropriate, whereas when a smaller volume affected by a stronger field suggests the use of a Single Coil. The Helmholtz Coil on the other hand provides a good trade off between them. With the Solenoid Coil one must also consider the volume vs radius trade off. In these experements the sphere examined was for 95% magnetic field strength, and the radius of that field was very close to the radius of the coil. If a wider tollerance such as 80% or 70%magnetic field strength was being looked for, the volume would cease to be a sphere as it must remain contained within the coil, and in this way the volume becomes much more limited.

The author would like to thank the following two people for providing advice and information: Mr T. Deen and Mr A. Liang. Of most importance would be the assistance in using the Ryerson Polytechnic University laboratory equipment provided by Mr M. Asyani. Without their help this report would not be possible.