Physics 72.1

Magnetic Field of a Solenoid Rose Anne E. Acedera, John Patrick de Guzman, Harley Lacbawan, and Joey Lutao College of Engineering, University of the Philippines, Diliman, Quezon City

Abstract The experiment primarily aims to study the magnetic field of a solenoid. To achieve this, the experiment was divided into four parts- (1) the identification of direction of the magnetic field, (2) the measurement of the magnetic field strength along different points in the solenoid, (3) the determination of the relationship of the magnetic field and the current, and (4) the determination of the relationship of the magnetic field and the number of turns per unit length of the solenoid. At the end of experiment, it was found out that the current, as well as the number of turns per unit length, varies linearly with the field intensity. The value of the permeability constant, µ0, was also calculate using the data obtained.

1. Introduction 2. Methodology The materials used in the experiment were PASCO Power Supply, magnetic field sensor, LabQuest, slinky, meter stick, connectors and tape. The experiment is basically divided into four parts, identification of the direction of the field, measurement of the magnetic field at different locations along the solenoid, identification of the relationship between magnetic field and current and lastly, identification of the relationship between magnetic field and spacing of turns. The first part was the identification of the direction of the magnetic field. For this part, a solenoid was stretched to 1m and was connected power supply with a current of 2.0 A. Then a Vernier Magnetic Field Sensor was placed in the center of the center of the solenoid facing one end of the solenoid. One the power supply was turned on, the sensor was turned to face the other side of the solenoid and back again to the origin. The magnitude of the magnetic field detected by the sensor was recorded and graphed using a Lab Quest. The orientation which gave the most positive magnetic field reading was then used all throughout the experiment. In the second part of the experiment, the sensor was placed in various locations along the solenoid. Then the strength of the magnetic observed for 15 seconds field was one again recorded and graphed using Lab Quest. For the third part, the strength of the magnetic field was measured at different currents by adjusting the power supply. While for the last part of the experiment, the current was again held constant, however, the degree by which the solenoid is being stretched was adjusted. All result were then recorded and graphed following the same procedure used in the second part of the experiment.

3. Results and Discussion In the first part of the experiment, the effect of the position of the sensor to the magnitude of the magnetic field was examined. This was made by measuring the magnetic field strength along different points along the solenoid. A plot of the data obtained is shown in Figure 1.

Magnetic Field Strength

Magnetic Field Strength vs. Position along the Solenoid

-0.20

0.2 0.15 0.1 0.05 0 0.00

0.20

0.40

0.60

0.80

1.00

1.20

Position along the solenoid, m

Figure 1 A plot of magnetic field strength against locations along the axis of the solenoid

Figure 1 shows that magnetic field as a function of location along the axis of the solenoid will yield a graph with fluctuating magnetic field values. Ideally, a long slinky can be treated as an infinite solenoid. The magnetic field inside the latter remains constant. The reason behind this is that the magnetic field being produced by the current is parallel to the axis of the solenoid. The magnetic field in other directions are cancelled by the opposing fields. On the other hand, as the sensor moves outside the solenoid, the magnetic field strength gradually decreases. The magnetic field lines spread out as it moves out of the solenoid. Keep in mind that the sensor can only pick up the magnetic field parallel to the axis of the solenoid. From the data above, although the magnetic field strengths for each location are not the same, it is apparent that they have close values, making it a plausible data. Additionally, it has been proven by the graph that the magnetic field strength decreases as the sensor moves outside the solenoid. In the second part of the experiment, the linear relationship between the magnitude of the magnetic field of a solenoid and the current passing through, as observed from the formula for the magnetic field of a solenoid, was verified. This was done by holding the effective length constant while varying the current. Doing so, gave the following data.

Magnetic Field Strength, mT

Table 1. Magnetic Field Strength and Current Data I (Amperes) B (mT) 0.5 0.048 1.0 0.081 1.5 0.115 2.0 0.141 2.5 0.164 Plotting those data gives a more picturesque view of the magnetic field strength and the current’s relationship.

0.2

Magnetic Field Strength vs. Current through the Solenoid

0.15 0.1 0.05

y = 0.0584x + 0.0222 R² = 0.993

0 0.00

1.00

2.00

Current, A

3.00

Figure 2. The values of current plotted against the measured magnetic field

Figure 2 shows that increasing the current increases the magnitude of the magnetic field. The curve has a regression value of 0.993, further confirming the direct proportional relationship of magnetic field and current. On the other hand, the last part of the experiment examined the direct proportionality of the magnetic field strength and the spacing of turns. Spacing of turns, in turn, affects the number of turns per unit length of the solenoid. Contrary to the preceding part, this objective was achieved by letting the current constant while varying the effective length of the solenoid. Again, doing so yield the following.

Length (m) 0.25 0.50 0.75 1.00 1.25

Table 2. Magnetic Field Strength as a Function of the Number of Turns per Unit Length Number of Turns B (mT) Turns per unit Length 77 0.453 308 78 0.182 156 78 0.131 104 79 0.101 79 79 0.091 63.2

A plot of these data is shown in Figure 3.

Magnetic Field Strength, mT

Magnetic Field Strength vs. Turns per unit Length 0.5 0.4 0.3 0.2 y = 0.0015x - 0.0219 R² = 0.9842

0.1 0 0.0

100.0

200.0

300.0

Turns per unit length, N/m

400.0

Figure 2. The number of turns per unit length plotted against the measured magnetic field

The plot has a correlation factor equal to 0.9842 which is approximately close to 1. This further affirms the direct proportionality of the number of turns per unit length of a solenoid and the magnetic field strength. Aside from verifying the direct proportionality of the magnetic field and the current or the number of turns, the regression curves in Figures 2 and 3 can be used to calculate the value of the permeability constant, µ0. The regression curve in Figure 2 has the equation, y = 0.0584x + 0.0222.When this equation is compared analogously to equation of the magnetic field of a solenoid, the slope is comparable to the product of µ0 and the number of turns, n. Thus, the slope should have a unit of mT/ A. The calculated value for the permeability constant is reflected in Table 3. On the other hand, the regression curve in Figure has the best fit line equation of y = 0.0015x - 0.0219. Comparing this again to the equation for the magnetic field, the slope signifies the product of µ 0 and the current. Therefore, it should have a unit of mT-m. Table 3. Calculated Permeability Constant Values

Regression Curve Equation y = 0.0584x + 0.0222 y = 0.0015x - 0.0219

Slope 0.0584 0.0015

µ0 7 x 10-7 1x 10-6

Percent Deviation 44.29% 20.43%

Though the objectives of the experiment were generally achieved, errors are still inevitable in this experiment, as for all analysis. These are the possible sources of error. First, the sensor might have been rotated as it was moved in the slinky. This would have made the runs unreasonable to compare because the sensor had a different normal level as it was rotated. Another possible source of error are the equipment discrepancies. Due to some electric effects in

the equipment, reading may be deviant from what is really measured. This could have skewed the data. Lastly, an error might have also been observed when the direction of the coil was changed during the experiment. Changing the direction of the coil influences the magnetic field sensor because the sensor has a different normal at each point on the compass.

4. Conclusion At the end of the experiment, the following conclusions were made. First, the magnetic field intensity varies at different locations along the solenoid, finding out that it was greatest at the center of the solenoid and least at the points outside. Second, the number of turns, as well as the current, varies linearly with the field strength. Lastly, the permeability constant, µ0, can be obtained from the slopes of the plots generated by the data obtained. For the second part it was found to be 7 x10-7, with deviation of 44.59%, and for the last part it was calculated to be 1x 10 -6, deviant of about 20.43% relative to the theoretical value.