The Magnetic Field in a Slinky by Ashley, Matt, Matt, and Sana Hypothesis: If we insert a Magnetic Field Sensor between the coils of a Slinky with an electric current running through it, then the magnetic field will increase linearly as the current increases because B = µ 0 nI , where B is the magnetic field, µ 0 is the permeability constant, n is the number of loops of wire, and I is current. The magnetic field will also increase as we compress the slinky because of B = µ 0 nI . We can also use this equation to find the permeability constant if we measure B, n, and I. If we we measure the magnetic field outside of the slinky, then it will be close to zero because the components of the magnetic field in other directions are cancelled by opposing fields fields from neighboring coils. If we measure the magnetic field at the ends of the slinky, then it will be half of the magnetic field at the center of the slinky. This This is because there is slinky on only one side of the end, as contrasted to the middle where there is slinky on both sides. Lab Questions: Determine the relationship between magnetic field and the current in a solenoid. • • Determine the relationship between magnetic field and the nu mber of turns per meter in a solenoid. • Study how the field varies inside and outside a solenoid. • Determine the value of µ0, the permeability constant. Preliminary Questions: 1. Hold the switch switch closed. closed. The current current should be 2.0 2.0 A. A. Place the the Magnetic Field Field Sensor between between the turns of the Slinky near its center. Rotate the sensor (graph sensor (graph 5) and determine which direction gives the largest magnetic field reading. What direction is the white dot on the sensor pointing? The sensor reads a maximum value when the white dot is pointing parallel to the slinky. 2. What happens happens if you you rotate rotate the white white dot to to point the opposi opposite te way? way? (graph 6) The readings become negative. What happens if you rotate the white dot so it points perpendicular to the axis of the solenoid? It decreases dramatically. 3. Stick Stick the Magnetic Magnetic Field Field Sensor through through differe different nt locations along along the Slinky Slinky to explore explore how the field varies along the length. Always orient the sensor to read the maximum magnetic field at that point along the Slinky. How does the magnetic field inside the solenoid seem to vary along its length? (graph 7) The field is at maximum value near the middle of the slinky, but at a minimum value near the ends of the slinky. 4. Check Check the magnetic magnetic field field intensi intensity ty just just outside outside the the solenoid. solenoid.(graph (graph 8) The magnetic field decreases in intensity as the sensor moves farther away from the center of the slinky. Data Tables: Part I Current in solenoid I Magnetic field B (A) (mT) 0.5 0.064
1.0
0.128
1.5
0.200
2.0
0.263
Length of solenoid (m) Number of turns
0.74
Physics with Computers
81
29 - 1
Experiment 29
109
Turns/m (m –1) Part II Length of of solenoid
Turns/meter n
Magnetic field B
(m) 0.5
–1 163
(mT) 0.28
1.0
81
0.13
1.5
54
0.10
2.0
40
0.08
Number of turns in Slinky
81
Analysis: 1. Plot Plot a graph graph of of magn magnet etic ic fie field ld B vs. the current I through the solenoid. Use either Graphical Analysis or graph paper. (See graph 12) 2. How is magnetic magnetic field field relate related d to the current current through through the the solenoid solenoid?? The magnetic field increases linearly as the current increases. 3. Determine Determine the the equation equation of the best-f best-fit it line, line, includi including ng the the y-intercept. -intercept. Note the constants and their units. Y=0. Y=0.133799x 133799x + 0.00 0.003319. 3319. 0.003319 0.003319 is the y-intercept, or the magnitude of the magnetic field in mT when the current is at 0 Amps. 0.133799 0.133799 is the increase in magnetic field mT per 1 Amp increase. 4. For each each of the measurement measurementss of Part II, calcula calculate te the number of turns turns per meter. meter. Enter Enter these these values in the data table. 5. Plot Plot a graph graph of of magn magnet etic ic fie field ld B vs. the length of the solenoid (m). Use Use either Graphical Analysis or graph paper. (See graph 13) 6. How is magnetic magnetic field field relate related d to the turns/me turns/meter ter of the the solenoid solenoid?? The magnetic field is directly proportional to the turns of the slinky. 7. Determine Determine the equation equation of the best-f best-fit it line to your your graph. graph. Note Note the constants constants and their units units . Y=0.001641x + 0.017132. 0.017132 is the y-intercept, or the magnitude of the magnetic field in mT when the current is at 0 Amps. 0.0016 0.001641 41 is the increase in magnetic field mT per 1 Amp increase. 8. From Ampere’s Ampere’s law, law, it can be shown shown that the magnetic field field B inside a long solenoid is B = µ 0 nI where µ 0 is the permeability the permeability constant. Do your results agree with this equation? Explain. The equation suggests that B is proportional to I. The graph in part I is consistent with this prediction. The equation also suggests that B is proportional to n. The graph in part II is consistent with this prediction. 9. Assuming Assuming the the equation equation in the previous previous question question applies applies for your your solenoid, solenoid, calculate calculate the value value of µ B vs. n. 1.3 x 10^-6 T*m/A µ 0 using your graph of B 10. Look up the value of µ0, the permeability the permeability constant. 4pi x 10^-7, which is about 1.26 x 10^-6 T*m/A. Compare it to your experimental value. Our experimental value is very close, with a 3% error. 11. Was Was your Slinky positioned along an east-west, east-west, north-south, or on some other axis? Along the north axis. Will this have any effect on your readings? No, readings? No, because we zero-ed the sensor before we started. Extensions: 29 - 2
Physics with Computers
The Magnetic Field in a Slinky
1. Carefully Carefully measure the magnetic field at the end of the solenoid. How does it compare to the value at the center of the solenoid? It’s about half of what it is in the middle. (graph # 7) Try to prove what the value at the end should be. The value at the end of the coil should be half the value at the center. 2. Study Study the magnetic magnetic field field strength strength inside inside and around a toroid, a circular-sha circular-shaped ped solenoid solenoid.. According to the internet, the magnetic field B in the center of the toroid is equal to the number of loops times the current in each loop. Which means that B2piR= µ0 NI, or B= µ0 NI/2piR. Since R=0.2m,, I=3A, and N=81, our hypothetical B is 0.1143mT. R=0.2m 0.1143mT. Experimenting, we get an average magnetic field of 0.1 0.1256mT 256mT,, which gives a percentage error of 4%. 3. If you you have studied studied calculus, calculus, refer refer to a calculus-based calculus-based physics physics text text to see how the the equation equation for
the field of a solenoid can be derived from Ampere’s law.
4. If you you look up the permeabi permeability lity constant constant in a reference, reference, you may may find it listed listed in units units of henry/meter. Show that these units are the same as tesla-meter/ampere. tesla-meter/ampere. Henry = 1 tesla meter^2/ampere,, so henry/meter*(tesla meter^2/ampere henry/meter*(tesla meter^2/ampere meter^2/ampere)/henry )/henry = tesla meter/ampere. 5. Take Take data on the the magnet magnetic ic field field intens intensity ity vs. position along the length of the solenoid. Check the field intensity at several distances along the axis of the Slinky past past the end. Note any patterns you see. Plot a graph of magnetic field ( B B) vs. distance from center. Use either Graphical Analysis or graph paper. How does the value at the end of the solenoid compare to that at the center? It’s about half of what it is in the middle (see data table). How does the value change as you move away from the end of the solenoid? It continues to decrease (see data table). Distance from the Magnetic field B center of slinky (m) (mT) 0.0 (the middle) 0.124 0.15 0.102 0.30
0.089
0.45 (the end)
0.061
0.60
0.034
0.75
0.022
0.90
0.010
1.05
0.003
1.20
0.001
(See graph 14) 6. Insert Insert a steel steel or iron rod inside the the solenoid solenoid and see what effect effect that that has on the field field intensity. intensity. It increases dramatically at all points (at the middle, the magnetic field is .305mT .305mT and at the ends, magnetic field is .159mT). .159mT). Be careful that the rod does not short out with the coils of the Slinky. You may need to change the range of the Magnetic Field Sensor.
Physics with Computers
29 - 3
Experiment 29 µ 0. It is the slope of the best-fit 7. Use the graph graph obtaine obtained d in Part Part I to to determine determine the the value value of µ regression regressi on line, 0.133799 T*m/A. Conclusion: 1. Our hypothesis correctly answers the lab questions. 2. Our hypothesis was correct! As the current increased in intervals of 0.5 Amps, the magnetic field did in fact increase, from 0.064mT to 0.128mT to 0.200mT to 0.263mT. As we stretched out the slinky in intervals of 0.5 meters, the magnetic field did in fact decrease, from 0.28mT to 0.13mT to 0.10mT to 0.08mT. We were also able to calculate the permeability constant by measuring magnetic field B, number of turns n, and current I. We estimated estimated 1.26 x 10^-6 T*m/A, T*m/A, which is very close to the actual value 4pi x 10^10^-7 7 T*m/A with a 3% error. Also, as we moved the magnetic field sensor towards the end of the slinky in intervals of 0.15m, the magnetic field decreased until it reached ½ the value of the middle at the ends. This is evidenced evidenced by the data 0.124mT 0.1 24mT at the middle, to 0.102mT, 0.102mT, to 0.089mT, to 0.061mT at the end. It continued to decrease past the ends (0.034mT to 0.022mT to 0.010 mT to 0.003mT to 0.001mT). 3. There are three possible sources of error for this lab. One is that the magnetic field sensor could have rotated as we moved its location in slinky. This This would have made our trials unreasonable to compare, as the sensor has a differe different nt normal level as it is rotated. This could be alleviated by moving the magnetic field sensor along a roller track over the slinky that is perfectly perfectly parallel with the slinky. Another Another possible source of error is that the current reading on the current sensor jumped around a lot, so we weren’t sure if the current going through the wire was really what we recorded it as. This could have skewed the data. This issue may be alleviated by using a battery/current sensor that doesn’t jump around a lot. A final source of error could be b e that the direction of the coil moved during the experiment. This would have influenced the magnetic field sensor because the sensor has a different normal at each point on the compass. As a result, our trials would be unreasonable to compare. This could be alleviated by fixing the coil to the table so that it doesn’t move. 4. We use the magnetic field generated from metal coils and electricity in everyday life. life. In a car or truck, a solenoid is part of the starting system. The solenoid receives a large electric current from the car battery and a small electric current from the ignition switch. switch. When the ignition switch switch is turned on by turning the key, the small electric current forces the solenoid to close a pair of heavy contacts, thus relaying the large electric current to the motor . http://en.wikiped http://e n.wikipedia.org/wiki ia.org/wiki/Solenoid#Applications /Solenoid#Applications
29 - 4
Physics with Computers
