Helmholtz Coils I Introduction ................................................. 2 Review Theory ............................................ 3 The Experiment ........................................... 4 Getting Started ........................................ 4 Prepare for Data Collection ..................... 5 Calibrate the Pendulum ........................... 6 Measure the Pendulum Period ................ 7 Find the Magnetic Field Strength ............. 7 Compare Theory with Experiment ........... 8 Examine the Axial Field Strength ............. 8 Team Discussion ......................................... 9 Finish Up ................................................... 10 Supplies and Materials ...............................11

VISUAL E&M

Copyright ©2000-2004 J. A. Panitz

eLAB

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A FOCUSSED CONCEPT

Introduction Figure 1 shows two coils of wire, fixed in position, with their center-to-center separation equal to their radius. When current is passed through each coil (in the same direction) a magnetic field is created. This configuration is called a pair of “Helmholtz coils”. The magnetic field is very uniform over a fairly large volume at the center of the Helmholtz coils. The goal, today, is to use a torsional pendulum to measure the magnetic field strength at the center of the coils as a function of the current in the coils.

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Figure 1. Helmholtz coilapparatus.

Review Theory

If the current in each coil is identical, and flows in the same direction, the axial component of the magnetic field strength from each coil will add together. A particularly useful configuration occurs when the two coils are separated by their radius (d = Rc). This configuration is called a pair of “Helmholtz coils”. For Helmholtz coils:

Suppose a current (I) is passed through a coil of radius (Rc) that contains (N) turns of wire. Center the origin of a coordinate system on the axis of the coil at a distance (d/2) from its center. Then the axial component of the magnetic field strength (B) at a distance (x) from the origin becomes: È 2 Ê d ˆ2˘ 2 ÍRC + + x ˙ Ë2 ¯ ˚ Î

3/2

Consider an identical coil, parallel to the first coil and separated from it by a distance (d). For this coil:

m 0 N I RC

3/2

Notice that the field increases linearly with current. At the center of the coils (x = 0), the magnetic field strength becomes:

(32p x 10 ) N I B= -7

2

È 2 Êd ˆ ˘ 2 ÍRC + - x ˙ Ë2 ¯ ˚ Î 2

5 3/2 RC

3/2

If the current is reversed, the direction of the magnetic field will be reversed. 3

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Copyright ©2000-2004 J. A. Panitz

2 È 2 Ê RC ˆ ˘ -x ˙ ÍRC + Ë ¯ ˚ 2 Î

+

(2p x10-7 ) N I RC2

where m0 is called the “permeability of free space”. In the SI system of units the permeability is defined as m0 = 4p x 10-7.

B=

È 2 Ê RC ˆ ˘ + + R x Í C Ë ˙ ¯ 2 Î ˚

3/2

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B=

B=

m 0 N I RC 2

(2p x10-7 ) N I RC2

The Experiment

HELMHOLTZ COIL APPARATUS

Getting Started COIL 1

A torsional pendulum will be used to find the magnetic field strength at the center of the coils as a function of the current in the coils. The results will be compared to Helmholtz theory.

2.

3.

COIL 2

FIELD

POWER SUPPLY

+

Discuss the theory and the experiment. Plan a way to share the workload. Managers: select a discussion question.

MULTIMETER I

Figure 2. Schematic diagram of the Helmholtz apparatus.

Connect the components as shown in Figure 2. The power supply could generate a magnetic field that may interfere with your measurements. Locate the power supply two or three feet from the Helmholtz coil apparatus. Record Figure 2 in your notebook. Use a compass to align the axis of the Helmholtz coil apparatus (the axis of coil 1 and coil 2) with magnetic north. 4

3.

Turn on the power supply. Adjust the current to 0.400 ± 0.001 A. Confirm the reading with a multimeter.

4.

Use a compass to find the position of several magnetic field lines inside and outside the coils. Sketch the field lines and the position of the Helmholtz coils in your notebook. Indicate the direction of magnetic north.

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Copyright ©2000-2004 J. A. Panitz

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1.

I

Prepare for Data Collection The Helmholtz coil apparatus must be aligned so that its magnetic field will point in the direction of magnetic north. Devise a way to perform the alignment. Before you continue verify with your instructor that the alignment is correct.

MAGNETIC NORTH

2.

Set the current to zero (± 0.001 A).

3.

Move the pendulum support to align the plumb bob with the center of the socket in the Helmholtz coil apparatus. See Figure 3. Do not move the apparatus!

4.

COLOR PLUMB BOB

Identify your magnet by the color of its plastic sleeve. SOCKET

5.

Decide on a format to record data and observations in your lab notebook. Include the SI units for all parameters.

MAGNET

HELMHOLTZ COIL ASSEMBLY

Figure 3. Helmholtz apparatus.

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Copyright ©2000-2004 J. A. Panitz

PENDULUM SUPPORT

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1.

NYLON THREAD

6.

Calibrate the Pendulum

Move the nylon thread until the loop in the thread lies in the groove that held the plumb bob. The magnet should be centered in the Helmholtz coils (x = 0).

1.

Review the experimental method.

2.

Initiate a rotational displacement of the pendulum magnet that is less than 5∞. Use a stopwatch to measure the time for ten oscillations of the magnet about its equilibrium position. Divide by ten to find the average time for one oscillation. Record the data in your lab notebook.

3.

Repeat this procedure five times.

Caution: Use care when you move the loop. The nylon thread is fragile and can be easily broken.

4.

Find the mean period (T) from your measurements.

5.

Calculate the experimental value of the pendulum constant.

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In Albuquerque, NM: Be= 2.36 x 10-5 Tesla.

Measure the Pendulum Period Set the current to I = 0.100 ± 0.001 A.

2.

Initiate a rotational displacement of the pendulum magnet that is less than 5∞. Use a stopwatch to measure the time for ten oscillations of the magnet about its equilibrium position. Divide by ten to find the average time for one oscillation. Record the data in your lab notebook.

3.

The experimental value of the magnetic field strength (Bm) is the difference between the magnetic field strength found from the pendulum constant (C) and the horizontal component of the earth’s magnetic field: C Bm = ÈÍ 2 ˘˙ - Be ÎT ˚ 1.

Repeat this procedure five times.

4.

Find the mean period (T) from your measurements.

5.

Repeat steps 2–4 for I = 0.200, 0.300, and 0.400 ± 0.001 A in the coils.

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Copyright ©2000-2004 J. A. Panitz

Find the experimental value of the magnetic field strength (Bm) for each value of the current (I = 0.000 to I = 0.400 A).

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1.

Find the Magnetic Field Strength

Compare Theory with Experiment

Examine the Axial Field Strength

Compare the experimental results with Helmholtz theory. The theoretical values of the magnetic field strength and the % errors will be calculated for you.

The magnetic field is very uniform over a fairly large volume at the center of the Helmholtz coils. Figure 4 shows a calculation of the field strength along the x-axis.

1.

Compare the theoretical values of the magnetic field with the experimental values.

Show B-field B (Tesla)

Helmholtz Theory

4.300E-4 N

119

4.250E-4 I (A) 4.200E-4

0.400 Rc (m)

4.150E-4

0.102 4.100E-4

4.050E-4

4.000E-4

3.950E-4

3.900E-4 -0.050

-0.040

-0.030

-0.020

-0.010

0.000 x (m)

0.010

0.020

0.030

0.040

0.050

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Figure 4. Field strength calculated from Helmholtz theory.

Team Discussion Summarize the experiment and the conclusions that were reached. Go to the blackboard, sketch the essential features of the apparatus and compare the results from all the teams.

4.

Suppose the magnetic axis of the Helmholtz coils made an angle (q) with the horizontal component of the earth’s magnetic field. Discuss the change this would make in the calculation of (B).

2.

Consider how a power supply could generate an external magnetic field when it is turned off. Discuss the changes that could occur when current is drawn from the power supply.

5.

Estimate the per cent change in the magnetic field strength over the length of the pendulum magnet (ª 5 cm). Discuss its effect on the experimental value for the field strength at x = 0.

3.

Discuss the reason for aligning the pendulum magnet and the axis of the Helmholtz coils with the earth’s magnetic field. Consider the effect of reversing the direction of the current in the coils.

6.

A medical facility would like to measure the weak magnetic field generated by the human brain. Discuss a method, using Helmholtz coils, that would eliminate the effect of the earth’s magnetic field. Estimate the radius of the coils, their spacing, and determine their orientation in space. Enter a brief summary of the experiment and the team discussion in your lab notebook.

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Copyright ©2000-2004 J. A. Panitz

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1.

Finish Up 1.

2.

3.

Enter a brief summary of the experiment and the team discussion in your lab notebook. Quit the software menu.

Manager: Ensure that all components and hardware used by your team are returned to their original state and placed in their original location.

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Supplies and Materials Helmholtz apparatus. Figure 5. 1 each Plumb Bob. CENCO WLS-70-200. 1 each Helmholtz coil assembly. CENCO CP721267-01. 1 each Torsional pendulum. UNM161L-017

TORSIONAL PENDULUM

1 each Meter Stick. CENCO CP72717-80. 1 each Stopwatch. CENCO WLA5615

PLUMB BOB

1 each Compass. CENCO WLS-22400-19. HELMHOLTZ COIL ASSEMBLY

3 each Patch Cord. 36” Banana (Black). Pomona 1440-36-0. 1 each Power Supply (30 V @ 6 A). Good Will Instrument Co., Ltd. Model GPR-3060.

Figure 5. Helmholtz apparatus.

1 each Multimeter (Zmin =10 MW). 3 1/2 Digit. Wavetek 15XL. Copyright ©2000-2004 J. A. Panitz

11

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Resistance (0 - 2000 MW)

1. V

OFF

1.

Connect one test lead to the common input (COM) and another test lead to the Volt or Ohm input (V W).

2.

Set the function/range switch to (W), and to the highest resistance range: 2000M (2000 million ohms full scale).

3.

Place the ends of the test leads across the component.

4.

Read the resistance. Select a lower range (if feasible) to give a higher resolution. The display in the figure at the left indicates a resistance greater than the full scale value.

5.

Measure the resistance of the test leads by touching (shorting) the ends of the leads together. Deduct the test lead resistance from the resistance of the component that was measured.

6.

Set the function/range switch OFF.

V~

A ~ W

VW

A

COM

200 mA

10 A

Wavetek 15XL Digital Multimeter (M = 106, k = 103, m = 10-3, m = 10-6 ) Set the function/range switch to the OFF position when the multimeter is not in use. CAUTION: Be sure that no current will flow in the component(s) to be measured. Copyright ©2000-2003 J. A. Panitz

RETURN

DC Voltage (0 - 1000 V)

00.0 V

OFF

V~

A ~ W

1.

Connect one test lead to the common input (COM) and another test lead to the Volt or Ohm input (V W). Read the note at the meter input.

2.

Set the function/range switch to (V ), and to the highest DC voltage range: 1000 V (1000 volts full scale).

3.

Place the ends of the test leads across the circuit element.

4.

Read the voltage and the polarity. Select a lower range (if feasible) to give a higher resolution.

A

Range overload is indicated by a “1” or a “-1” in the display with all the other digits blanked. VW

COM

200 mA

10 A

Wavetek 15XL Digital Multimeter (M = 106, k = 103, m = 10-3, m = 10-6 )

5.

Select a higher range. If the highest voltage range is in use, interrupt the measurement immediately.

6.

Set the function/range switch OFF.

Set the function/range switch to the OFF position when the multimeter is not in use.

Copyright ©2000-2003 J. A. Panitz

RETURN

DC Current (0 - 200 mA) 1.

00.0 V

OFF

Connect one test lead to the common input (COM) and another test lead to the yellow input labeled 200 mA. The input has a 200 milliampere fuse installed. Check the fuse if current is not detected.

V~

2.

Set the function/range switch to (V ), and to the highest DC current range for this input: 200 mA (200 milliamperes full scale).

A ~ W

VW

Remove power from the circuit to be tested (no current should flow).

A

COM

200 mA

10 A

Wavetek 15XL Digital Multimeter (M = 106, k = 103, m = 10-3, m = 10-6 )

3.

Connect the test leads securely in series with a circuit element.

4.

Apply power to the circuit being tested.

5.

Read the current. Select a lower range (if feasible) to give a higher resolution.

6.

Set the function/range switch OFF.

Set the function/range switch to the OFF position when the multimeter is not in use.

Copyright ©2000-2003 J. A. Panitz

RETURN

DC Current (0 - 10 A) 1.

00.0 V

OFF

The input has a 10 ampere fuse installed. Check the fuse if current is not detected.

V~

2.

VW

Set the function/range switch to (V ), and to the only DC current range for this input: 10 A (10 amps full scale). Remove power from the circuit to be tested (no current should flow).

A ~ W

Connect one test lead to the common input (COM) and another test lead to the yellow input labeled 10 A.

A

COM

200 mA

10 A

Wavetek 15XL Digital Multimeter (M = 106, k = 103, m = 10-3, m = 10-6 ) Set the function/range switch to the OFF position when the multimeter is not in use.

Copyright ©2000-2003 J. A. Panitz

3.

Connect the test leads securely in series with a circuit element.

4.

Apply power to the circuit being tested.

5.

Read the current.

6.

Set the function/range switch OFF.

RETURN

AC Voltage (0 - 750 V)

00.0 V

OFF

V~

A ~ W

1.

Connect one test lead to the common input (COM) and another test lead to the Volt or Ohm input (V W). Read the note at the meter input.

2.

Set the function/range switch to (V~), and to the highest AC voltage range: 750 V (750 volts full scale).

3.

Place the ends of the test leads across the circuit element.

4.

Read the voltage and the polarity. Select a lower range (if feasible) to give a higher resolution.

A

Range overload is indicated by a “1” or a “-1” in the display with all the other digits blanked. VW

COM

200 mA

10 A

Wavetek 15XL Digital Multimeter (M = 106, k = 103, m = 10-3, m = 10-6 )

5.

Select a higher range. If the highest voltage range is in use, interrupt the measurement immediately.

6.

Set the function/range switch OFF.

Set the function/range switch to the OFF position when the multimeter is not in use.

Copyright ©2000-2004 J. A. Panitz

RETURN

A torsional pendulum can be made by suspending a bar magnet from a rigid support. If the pendulum is placed in a magnetic field it will act as a compass and the magnet will align itself with the field. When the magnet is rotated slightly it will oscillate about its equilibrium position. See Figure 1. The product of the horizontal component of the magnetic field strength (B) and the square of the period of oscillation (T) define the pendulum constant (C), where:

The equation for the pendulum constant is only valid for small rotational displacements from the equilibrium position of the magnet, typically less than 5∞.

C = B T2

Figure 1. Rotational displacement of the pendulum magnet.

An experimental value for the pendulum constant can be found by measuring the period of oscillation of the magnet in the horizontal component of the earth’s magnetic field (Be). That is:

The pendulum magnet must be level. If necessary, the magnet can be moved in its plastic sleeve until the magnet is leveled. At equilibrium, the magnet should be aligned with the horizontal component of the earth’s magnetic field. A compass can be used to determine the proper alignment.

C = Be T 2

The horizontal component of the earth’s magnetic field can be obtained from the United States Geological Survey website at: http://geomag.usgs.gov/ Copyright ©2000-2003 J. A. Panitz

5° Be

MAGNET

The pendulum may try to swing from side-to-side (or up and down) when the magnet rotates. These movements must be eliminated. RETURN

Monday, March 13, 2000

re: information request

Subject: re: information request Date: Mon, 13 Mar 2000 08:49:17 -0700 From: Jill Caldwell To: [email protected] Jill: We are preparing a laboratory exercise for physics undergraduate students. The lab requires the strength of the horizontal component of the earth's magnetic field in Albuquerque, NM. The value that has been suggested is: B(horiz) = 2.45 E-5 Tesla. Can you provide a more accurate value, or suggest a method to obtain one? Sincerely, John ************************************************** Hello JohnThe current value of the horizontal component of the geomagnetic field in the vicinity of Albuquerque, NM is 23627 nanoTeslas. This value is considered accurate to within +/- 200 nT. It is changing approximately -27 nT per year right now. In case it comes up, the remainder of the geomagnetic field components is given below: Declination= 10.5 degrees East Inclination= 62.5 degrees North Total Field Strength= 51120 nT Vertical Field Strength= 45332 nT This information is available from our Internet site by downloading our geomagnetic field value calculation program. Our Web site address is: http://geomag.usgs.gov click on "freeware" and it will walk you through the download proceedure. You do need an IBM compatible PC (no MacIntosh's), but the programs are free of charge and will give geomagnetic field values for anywhere in the world and quite a date range. You will need to know the latitude and longitde of your area of interest. We do hope to have an interactive program on our Web site within about a month, but for right now you do need to download the program to run it. I hope this information is helpful, but please do not hesitate to contact me again if you have any further questions. Sincerely, Jill Jill Caldwell National Geomagnetic Information Center U.S. Geological Survey Box 25046, MS 966, Federal Center Denver, CO 80225-0046 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ /\ Phone: 303.273.8486 /\ /\/^^\/\ Fax: 303.273.8506 /^^\/ /^^^/^^\/\ email: [email protected] / / /^^^/^^^/ \ ***************************************************

mailbox:/iMac/e-mail/Mail/161L.sbd/Lab06?id= 3.0.1.32.20000313084917.00919d60%40ghtmail.cr.usgs.gov&number=2

Page: 1

The Magnetic Field in the United States, 1995 250˚

˚ 240

˚

290

280˚

270˚

260˚

Total Intensity (F) -70

0

00

56

580

00

40˚

40˚

54 -60

00

0

00

-70

0

-80

56000

52

0

-6

50

54000

00

0

48

00

0

52000

-50

30˚

50000

48000 460

00

280˚

270˚

260˚

Total Intensity in thousands of nT Annual Change in nT

250˚

30˚

The Magnetic Field in the United States, 1995 250˚

˚ 240

˚

290

280˚

270˚

260˚

Inclination (I)

75

65

40˚

40˚

70 70

-4

-2

60

65 0

30˚

65

30˚

60

60

55

0

280˚

260˚

55

270˚

250˚

Inclination in degrees Annual change in minutes

The Magnetic Field in the United States, 1995 250˚

˚ 240

˚

290

280˚

270˚

260˚

Declination (D) 5 -2

-1

-4

-4

40˚

-2

40˚ -6 -10

5

15

0

10

-5

-6

-4

-2

0

30˚

30˚ -6

-8

260˚

280˚

270˚

250˚

Declination in degrees Annual change in minutes

1.

Locate the power switch. Turn the power supply off.

VO LTAG E CO URSE

Rotate both voltage controls to their minimum position (fully counter clockwise). See Figure 1.

3

Rotate both current controls to their maximum position (fully clockwise).

4.

Connect the load between the positive (+) and the negative (-) terminal.

5.

Connect the ground terminal (GND) as needed.

6.

Plug the line cord into a wall receptacle.

7.

Turn on the power supply.

8.

Rotate the voltage controls to obtain the desired current.

CURRENT

FINE

CO URSE

FINE

POWER

-

2.

G ND

+

Figure 1. Good Will Instrument Company, Ltd power supply (model GPR-3060).

Copyright ©2000-2003 J. A. Panitz

RETURN

NOTES: 1. MATERIAL: ALUMINUM (6061-T6). 2. FINISH: #32 OR BETTER, ALL OVER (NO SCRATCHES). 3. BLACK ANODIZE. LIGHT OIL (NO RESIDUE). 4. BREAK EDGES AND CORNERS .005 MIN. R (SMOOTH). 5. ALL DIMENSIONS ±0.005 UNLESS NOTED. 6. ALNICO MAGNET. MAGNETIC MOMENT = 0.1 A-m2. 7. NYLON MONOFILAMENT FISHING LINE (10 # TEST). LOOP AROUND MAGNET UNDER SHRINK TUBING. EXIT THROUGH PIN HOLE. HEAT TUBING TO SECURE.

ITEM 2

2 cm (REF)

SEE NOTE 7 PIN HOLE SHRINK TUBING

ITEM 1

1.00

30.00 cm

1.000

TYP.

Ø.188 REF.

2.00 REF. MAGNET DETAIL SCALE 1/1 SEE NOTE 6

CENTER LINE HELMHOLTZ COILS (Ø20.4 cm)

MAGNET

4.875 UNDERGRADUATE LABORATORIES Phone: (505) 277-5805 FAX: (505) 277-1520

ITEM 3 TITLE:

ALIGNMENT SLOT (ORIENT AS SHOWN)

PENDULUM SUPPORT

Prepared By: JAP

Dwg #:

Approved By:

Size: ANSI B

Scale: 1/2

Date: 07.11.00

161L-017

Rev: A Sheet: 1 OF 2

UNM PROPIETARY AND CONFIDENTIAL INFORMATION

Ø.516 REAM THRU

Ø.193 REAM THRU

NOTES: 1. MATERIAL: ALUMINUM (6061-T6). 2. FINISH: #32 OR BETTER, ALL OVER (NO SCRATCHES). 3. BLACK ANODIZE. LIGHT OIL (NO RESIDUE). 4. BREAK EDGES AND CORNERS .005 MIN. R (SMOOTH). 5. DECIMAL DIMENSIONS ±0.005. METRIC: ±0.01 cm. 6. TUBING: Ø0.500 x .063 WALL. PAPER O.D. (SMOOTH). 7. V-GROOVE. 0.020 WIDE x 0.020 DP. 8. TAP (Ø.138) 6-32 UNC-2B x 0.500 ± 0.125 DP. EACH END. 9. 0.020-0.030 WIDE x .031 DP. FILL WITH WHITE PAINT.

SEE NOTE 8

ITEM 1 (ROD) Ø1.000 4.375

11.811

CHAMFER 45° x .031

16.625

SEE NOTE 7

SEE NOTE 7

SEE NOTE 6

Ø.500 REF

1.929

7.874 10.000 ITEM 2 (TUBE) SCALE 1:1

CHAMFER 45° x .031 DP. EACH END

CHAMFER 0.063 x 45° A Ø.144 THRU. C'BORE Ø.375 x .250 DP.

ALIGNMENT SLOT SEE NOTE 9 Ø5.500

UNDERGRADUATE LABORATORIES Phone: (505) 277-5805 FAX: (505) 277-1520

TITLE: .500

A ITEM 3 (BASE)

PENDULUM SUPPORT

Prepared By: JAP

Dwg #:

Approved By:

Size: ANSI B

Scale: 1/2

Date: 07.12.00

161L-017

Rev: A Sheet: 2 OF 2

UNM PROPIETARY AND CONFIDENTIAL INFORMATION