# Magnetic Fields

Magnetic Fields Introduction: This lab was completed to investigate and map the magnetic field pattern of a single straight conductor by comparing it with the earth’s magnetic field - **Magnetic Fields** introduction. Quantitatively, the purpose of this lab was to determine the horizontal component of the earth’s magnetic field. Magnetic field and electric currents are naturally closely related because anytime a current runs through a wire, a magnetic field around the wire is created. These magnetic fields can be expressed in terms of both magnitude and direction and are, therefore, vector fields.

The magnitude of a magnetic field from a straight wire can be determine with the following equation: B = (? 0I)/(2? y), where B is the magnitude in units Tesla, ? 0 is the permeability of free space with a value of 4?? 10-7 Tm/A, I is the current running through the wire in amperes, and y is the perpendicular distance in meters away from the wire to the point where the magnetic field is being measured. Distance away from the wire and the strength of the magnetic field are inversely related. That’s is, as the distance increases, the magnetic field decreases.

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Additionally, although a Tesla is the standard unit of measurement for magnetic fields, a Tesla is a rather large unit of magnetic field. Therefore, magnetic fields are also commonly measured in “Gauss,” which are equal to 10-4 Tesla. In order to determine the direction of the magnetic field around a wire, an easy trick to use is the right hand rule. Simply place your right thumb along the wire to point in the same direction of the current. Curl your fingers around the wire. The resulting direction that your fingers circle around and point corresponds to the direction of the magnetic field.

A schematic of the right hand rule (left side) and current and magnetic field direction (right side) is seen below (Figure 1). Figure 1: right hand rule (left) and current/magnetic field direction (right) In this lab, a magnetic compass is used to measure the magnetic field produced by the electrical current flowing through the conductor (BI) by comparing the value to the horizontal component of the earth’s magnetic field (BHoriz Earth). At the initial position, the compass points directly at the conductor/wire with no current running through the wire.

Here, BI is perpendicular to BHoriz Earth because BI is always tangent to a circle centered on the conductor. When the current flows through the conductor, and I is a non-zero value, the compass needle moves to align its arrow with the total magnetic field that is produced: Btotal = BI + BHoriz Earth. Since the compass is in a position so that BI is perpendicular to BHoriz Earth, the needle points at an angle ? in relation to BHoriz Earth. This relationship can be expressed mathematically by: tan ? = BI / BHoriz Earth. Including the general equation for the strength of a magnetic field, the equation can be written as: tan ? (? 0I) / (2? yBHoriz Earth). This relationship and orientation of the compass, BI, BHoriz Earth, the angle ? , and current can be seen in Figure 2 below. Figure 2: Magnetic field setup Procedure: A couple of special precautions must be considered before beginning the lab. First, the power supply may not be set to run a current any higher than 8 Amps, although much lower (around 5) is probably recommended. Second, there may be significant variation in ambient magnetic field from table to table throughout the room, especially at tables made of steel and considering that research is conducted in the same building.

In the set up of this lab, a long straight conductor is connected to an ammeter and a power supply. A horizontal clear plastic plate is attached to allow compass measurements to be done near the center of the conductor. First, the compass was positioned so that the compass needle pointed directly towards the wire, that is, so that BI and BHoriz Earth were perpendicular to each other once current ran through the conductor. The compass was then rotates until the needle was on zero degrees to simplify the process of measuring change.

It is important to include that the red compass needle tip points opposite to the conventional magnetic field direction. Figure 3 to the right depicts the basic initial setup of the experiment. Figure 3: Experiment magnetic field setup To determine BHoriz Earth, the compass was positioned at a set distance, y (5 cm), from the center of the conductor. Then, ? was measure for five different values of I, while ensuring that ? did not exceed 60°. Result were recorded, organized into a table, and then graphed to visually display the linear relationship demonstrated between tan ? nd I. The slope of this graph is tan ? /I. Slope was then taken and substituted into the following equation to determine an experimentally measured value of BHoriz Earth: BHoriz Earth = (? 0I)/(2? ytan? ) = ? 0/(2? y ? slope) BHoriz Earth was calculated using SI units and where ? 0 = 4?? 10-7 Tm/A. A percentage difference was calculated for the experimental value of the horizontal component of the magnetic field strength of the earth compared to the theoretically established value for this geographical region (3? 10-5 T). In the next part of the lab, it was verified that tan ? s inversely proportional to y. This was experimentally done by setting the power supply to a fixed current (2. 54 amps) and moving the compass to various distances from the center of the conductor where y = 4, 5, 7, and 9 cm. Results were recorded, organized into a table and graphed so that y? tan? was on the y-axis and y was on the x-axis. Theoretically, the graph should demonstrate that tan? is proportional to 1/y. Data & Analysis: Table 1: Resultant Angles and Corresponding tan? from Varying the Current in the Wire I (amps)| ? (degrees)| Tan ? | 0| 0| 0| | 18| 0. 325| 1. 82| 29| 0. 554| 2. 05| 32| 0. 625| 2. 52| 38| 0. 781| 3. 02| 46| 1. 036| Graph 1: Tan ? vs. Current BHoriz Earth = (? 0I)/(2? ytan? ) = ? 0/(2? y ? slope) = (4?? 10-7 Tm/A) / (2? ?0. 05? 0. 329) = 1. 216? 10-5 T experimentally measured magnetic field % Difference when compared to established value for BHoriz Earth (3? 10-5 T) = 84. 63% _________________________________________________________________________________________________________ Table 2: Angles and Resultant Calculations at Various Distances (y) with Constant Current of 2. 54 amps

Distance (y) (m)| ? (degrees)| Tan ? | y? (Tan ? )| 0. 04| 45| 1| 0. 04| 0. 05| 42| 0. 9| 0. 045| 0. 07| 34| 0. 675| 0. 047| 0. 09| 26| 0. 488| 0. 0439| Average y? (tan ? ) = 0. 044 Average BHoriz Earth = (? 0I)/(2? ? Avg. (ytan? )) = (4?? 10-7 Tm/A ? 2. 54 A) / (2? ?0. 044) = 1. 154? 10-5 T % Difference when compared to other experimental value of BHoriz Earth from first part of lab = 5. 23% Conclusion: This lab demonstrated that the strength of magnetic field is proportional to the current through the conductor and inversely related to the distance away from the conductor.

In the first part of the experiment, the value of BHoriz Earth was calculated to be 1. 216? 10-5 T, yielding a percent difference of 84. 63% when compared to the established value for BHoriz Earth for this geographical region (3? 10-5 T). While this seems like a rather large percent difference, theoretically it is not a terrible result. The two values are within the same magnitude. The difference between the two results from the varying magnetic fields in the surroundings throughout the room. The results could have been entirely difference if the experiment was performed on the neighboring steel table.

Additionally, the large magnetic lab located nearby the room where this experiment was done could have had a significant effect on the results. More important to the success of the lab, was whether the two experimentally determined values of BHoriz Earth were equal. In the second part of the lab the current was held constant and distance was varied. Using an average value of y? (tan ? ), BHoriz Earth was calculated to be 1. 154? 10-5 T, yielding a percent difference of 5. 23% when compared to the experimental value when distance was the fixed variable and current was varied.

The small percent difference between the two experimental values confirms that the magnetic field of a wire is inversely proportional to the distance away from the conductor AND directly related to the current through the wire. That is, the experiment proved: Bwire = ? 0/2? = I/y. This lab could be improved upon by using a compass that is easier to discern small changes in degrees of an angle. Additionally, if the experiment were conducted in a room or building with less technological and magnetic interferences, more accurate results could be achieved.